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    "signature": "(self) -> Poset",
    "signatures": [
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    "summary": "Return the poset recorded by this `CategoricalHomologyComponentSlice`.",
    "summary_source": "derived projection vocabulary"
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    "canonical_name": "category.CategoricalHomologyPresentation",
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    "doc": "Exact categorical-homology presentation over the coefficient monoid N.",
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    "file": "noether/category/__init__.pyi",
    "input_schema": {},
    "kind": "class",
    "line": 129,
    "module": "noether.category",
    "name": "CategoricalHomologyPresentation",
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    "qualified_name": "CategoricalHomologyPresentation",
    "releases_gil": false,
    "result_schema": {},
    "signature": "",
    "signatures": [],
    "summary": "Exact categorical-homology presentation over the coefficient monoid N.",
    "summary_source": "public docstring"
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    "canonical_name": "category.CategoricalHomologyPresentation.basepoint",
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    "doc": "",
    "domain": "category",
    "file": "noether/category/__init__.pyi",
    "input_schema": {
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      "properties": {},
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    "kind": "property",
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    "module": "noether.category",
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    "qualified_name": "CategoricalHomologyPresentation.basepoint",
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    "signature": "(self) -> SteinerCell | None",
    "signatures": [
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    "summary": "Return the basepoint recorded by this `CategoricalHomologyPresentation`.",
    "summary_source": "derived projection vocabulary"
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    "canonical_name": "category.CategoricalHomologyPresentation.coefficient_monoid",
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    "domain": "category",
    "file": "noether/category/__init__.pyi",
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      "properties": {},
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    "kind": "property",
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    "qualified_name": "CategoricalHomologyPresentation.coefficient_monoid",
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    "signature": "(self) -> str",
    "signatures": [
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    "summary": "Return the coefficient monoid recorded by this `CategoricalHomologyPresentation`.",
    "summary_source": "derived projection vocabulary"
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    "canonical_name": "category.CategoricalHomologyPresentation.component_poset_slice",
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    "doc": "",
    "domain": "category",
    "file": "noether/category/__init__.pyi",
    "input_schema": {
      "additionalProperties": false,
      "properties": {
        "particle_number": {
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      },
      "required": [
        "particle_number"
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      "type": "object"
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    "module": "noether.category",
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    "overloads": [
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          "type": "object"
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        ],
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    "result_schema": {
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    "signature": "(self, particle_number: int) -> CategoricalHomologyComponentSlice",
    "signatures": [
      "(self, particle_number: int) -> CategoricalHomologyComponentSlice"
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    "summary": "Compute component poset slice for this `CategoricalHomologyPresentation`.",
    "summary_source": "derived projection vocabulary"
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  {
    "canonical_name": "category.CategoricalHomologyPresentation.fundamental_poset",
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    "doc": "",
    "domain": "category",
    "file": "noether/category/__init__.pyi",
    "input_schema": {
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      "properties": {
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        "target_word"
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            "target_word": {
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    ],
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    "result_schema": {
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    "signatures": [
      "(self, source_word: Sequence[SteinerCell], target_word: Sequence[SteinerCell]) -> Poset"
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    "summary": "Compute fundamental poset for this `CategoricalHomologyPresentation`.",
    "summary_source": "derived projection vocabulary"
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    "canonical_name": "category.CategoricalHomologyPresentation.infinite_loop_model",
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    "doc": "",
    "domain": "category",
    "file": "noether/category/__init__.pyi",
    "input_schema": {
      "additionalProperties": false,
      "properties": {},
      "type": "object"
    },
    "kind": "property",
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    "module": "noether.category",
    "name": "infinite_loop_model",
    "overloads": [
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        "signature": "(self) -> str"
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    "qualified_name": "CategoricalHomologyPresentation.infinite_loop_model",
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    "result_schema": {
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    "signature": "(self) -> str",
    "signatures": [
      "(self) -> str"
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    "summary": "Return the infinite loop model recorded by this `CategoricalHomologyPresentation`.",
    "summary_source": "derived projection vocabulary"
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  {
    "canonical_name": "category.CategoricalHomologyPresentation.is_reduced",
    "determinism": "deterministic",
    "doc": "",
    "domain": "category",
    "file": "noether/category/__init__.pyi",
    "input_schema": {
      "additionalProperties": false,
      "properties": {},
      "type": "object"
    },
    "kind": "property",
    "line": 143,
    "module": "noether.category",
    "name": "is_reduced",
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          "properties": {},
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        "signature": "(self) -> bool"
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    ],
    "qualified_name": "CategoricalHomologyPresentation.is_reduced",
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    "result_schema": {
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    "signature": "(self) -> bool",
    "signatures": [
      "(self) -> bool"
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    "summary": "Test whether this `CategoricalHomologyPresentation` is reduced.",
    "summary_source": "derived projection vocabulary"
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  {
    "canonical_name": "category.CategoricalHomologyPresentation.model",
    "determinism": "deterministic",
    "doc": "",
    "domain": "category",
    "file": "noether/category/__init__.pyi",
    "input_schema": {
      "additionalProperties": false,
      "properties": {},
      "type": "object"
    },
    "kind": "property",
    "line": 133,
    "module": "noether.category",
    "name": "model",
    "overloads": [
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          "properties": {},
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        "parameters": [],
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        "signature": "(self) -> str"
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    ],
    "qualified_name": "CategoricalHomologyPresentation.model",
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    "result_schema": {
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    "signature": "(self) -> str",
    "signatures": [
      "(self) -> str"
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    "summary": "Return the model recorded by this `CategoricalHomologyPresentation`.",
    "summary_source": "derived projection vocabulary"
  },
  {
    "canonical_name": "category.CategoricalHomologyPresentation.particle_homotopy_poset",
    "determinism": "deterministic",
    "doc": "",
    "domain": "category",
    "file": "noether/category/__init__.pyi",
    "input_schema": {
      "additionalProperties": false,
      "properties": {
        "factors": {
          "items": {
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          "type": "array",
          "x-python-type": "Sequence[OrientedBasepoint]"
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      },
      "required": [
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      "type": "object"
    },
    "kind": "method",
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    "module": "noether.category",
    "name": "particle_homotopy_poset",
    "overloads": [
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        "input_schema": {
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          "properties": {
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              "x-python-type": "Sequence[OrientedBasepoint]"
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          },
          "required": [
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          ],
          "type": "object"
        },
        "parameters": [
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            "required": true,
            "schema": {
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              "type": "array",
              "x-python-type": "Sequence[OrientedBasepoint]"
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          }
        ],
        "result_schema": {
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        "signature": "(self, factors: Sequence[OrientedBasepoint]) -> Poset"
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    ],
    "qualified_name": "CategoricalHomologyPresentation.particle_homotopy_poset",
    "releases_gil": true,
    "result_schema": {
      "x-python-type": "Poset"
    },
    "signature": "(self, factors: Sequence[OrientedBasepoint]) -> Poset",
    "signatures": [
      "(self, factors: Sequence[OrientedBasepoint]) -> Poset"
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    "summary": "Compute particle homotopy poset for this `CategoricalHomologyPresentation`.",
    "summary_source": "derived projection vocabulary"
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  {
    "canonical_name": "category.CategoricalHomologyPresentation.skeleton",
    "determinism": "deterministic",
    "doc": "",
    "domain": "category",
    "file": "noether/category/__init__.pyi",
    "input_schema": {
      "additionalProperties": false,
      "properties": {
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      },
      "required": [
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      "type": "object"
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    "kind": "method",
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    "module": "noether.category",
    "name": "skeleton",
    "overloads": [
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          "properties": {
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        "parameters": [
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        ],
        "result_schema": {
          "x-python-type": "StreetNerveSkeleton"
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        "signature": "(self, max_degree: int) -> StreetNerveSkeleton"
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    ],
    "qualified_name": "CategoricalHomologyPresentation.skeleton",
    "releases_gil": true,
    "result_schema": {
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    "signature": "(self, max_degree: int) -> StreetNerveSkeleton",
    "signatures": [
      "(self, max_degree: int) -> StreetNerveSkeleton"
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    "summary": "Compute skeleton for this `CategoricalHomologyPresentation`.",
    "summary_source": "derived projection vocabulary"
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  {
    "canonical_name": "category.CategoricalHomologyPresentation.source",
    "determinism": "deterministic",
    "doc": "",
    "domain": "category",
    "file": "noether/category/__init__.pyi",
    "input_schema": {
      "additionalProperties": false,
      "properties": {},
      "type": "object"
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    "kind": "property",
    "line": 139,
    "module": "noether.category",
    "name": "source",
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        "signature": "(self) -> SteinerCategory"
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    "qualified_name": "CategoricalHomologyPresentation.source",
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    "result_schema": {
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    "signature": "(self) -> SteinerCategory",
    "signatures": [
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    "summary": "Return the source recorded by this `CategoricalHomologyPresentation`.",
    "summary_source": "derived projection vocabulary"
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  {
    "canonical_name": "category.CategoricalHomologyPresentation.spectrum_model",
    "determinism": "deterministic",
    "doc": "",
    "domain": "category",
    "file": "noether/category/__init__.pyi",
    "input_schema": {
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      "properties": {},
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    "kind": "property",
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    "module": "noether.category",
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    ],
    "qualified_name": "CategoricalHomologyPresentation.spectrum_model",
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    "result_schema": {
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    "signature": "(self) -> str",
    "signatures": [
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    "summary": "Return the spectrum model recorded by this `CategoricalHomologyPresentation`.",
    "summary_source": "derived projection vocabulary"
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    "canonical_name": "category.FiniteOmegaCategory",
    "determinism": "deterministic",
    "doc": "Explicit finite iterated mapping category.",
    "domain": "category",
    "file": "noether/category/__init__.pyi",
    "input_schema": {},
    "kind": "class",
    "line": 69,
    "module": "noether.category",
    "name": "FiniteOmegaCategory",
    "overloads": [],
    "qualified_name": "FiniteOmegaCategory",
    "releases_gil": false,
    "result_schema": {},
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    "signatures": [],
    "summary": "Explicit finite iterated mapping category.",
    "summary_source": "public docstring"
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    "canonical_name": "category.FiniteOmegaCategory.ambient",
    "determinism": "deterministic",
    "doc": "",
    "domain": "category",
    "file": "noether/category/__init__.pyi",
    "input_schema": {
      "additionalProperties": false,
      "properties": {},
      "type": "object"
    },
    "kind": "property",
    "line": 71,
    "module": "noether.category",
    "name": "ambient",
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        "result_schema": {
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        "signature": "(self) -> SteinerCategory"
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    ],
    "qualified_name": "FiniteOmegaCategory.ambient",
    "releases_gil": false,
    "result_schema": {
      "x-python-type": "SteinerCategory"
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    "signature": "(self) -> SteinerCategory",
    "signatures": [
      "(self) -> SteinerCategory"
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    "summary": "Return the ambient recorded by this `FiniteOmegaCategory`.",
    "summary_source": "derived projection vocabulary"
  },
  {
    "canonical_name": "category.FiniteOmegaCategory.component_poset",
    "determinism": "deterministic",
    "doc": "",
    "domain": "category",
    "file": "noether/category/__init__.pyi",
    "input_schema": {
      "additionalProperties": false,
      "properties": {},
      "type": "object"
    },
    "kind": "method",
    "line": 85,
    "module": "noether.category",
    "name": "component_poset",
    "overloads": [
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        "input_schema": {
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          "properties": {},
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        "parameters": [],
        "result_schema": {
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        "signature": "(self) -> Poset"
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    ],
    "qualified_name": "FiniteOmegaCategory.component_poset",
    "releases_gil": true,
    "result_schema": {
      "x-python-type": "Poset"
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    "signature": "(self) -> Poset",
    "signatures": [
      "(self) -> Poset"
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    "summary": "Compute component poset for this `FiniteOmegaCategory`.",
    "summary_source": "derived projection vocabulary"
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  {
    "canonical_name": "category.FiniteOmegaCategory.depth",
    "determinism": "deterministic",
    "doc": "",
    "domain": "category",
    "file": "noether/category/__init__.pyi",
    "input_schema": {
      "additionalProperties": false,
      "properties": {},
      "type": "object"
    },
    "kind": "property",
    "line": 73,
    "module": "noether.category",
    "name": "depth",
    "overloads": [
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          "properties": {},
          "type": "object"
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        "parameters": [],
        "result_schema": {
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        "signature": "(self) -> int"
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    ],
    "qualified_name": "FiniteOmegaCategory.depth",
    "releases_gil": false,
    "result_schema": {
      "type": "integer"
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    "signature": "(self) -> int",
    "signatures": [
      "(self) -> int"
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    "summary": "Return the depth recorded by this `FiniteOmegaCategory`.",
    "summary_source": "derived projection vocabulary"
  },
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    "canonical_name": "category.FiniteOmegaCategory.dimension",
    "determinism": "deterministic",
    "doc": "",
    "domain": "category",
    "file": "noether/category/__init__.pyi",
    "input_schema": {
      "additionalProperties": false,
      "properties": {},
      "type": "object"
    },
    "kind": "property",
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    "module": "noether.category",
    "name": "dimension",
    "overloads": [
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        "result_schema": {
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        "signature": "(self) -> int"
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    ],
    "qualified_name": "FiniteOmegaCategory.dimension",
    "releases_gil": false,
    "result_schema": {
      "type": "integer"
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    "signature": "(self) -> int",
    "signatures": [
      "(self) -> int"
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    "summary": "Return the dimension recorded by this `FiniteOmegaCategory`.",
    "summary_source": "derived projection vocabulary"
  },
  {
    "canonical_name": "category.FiniteOmegaCategory.exact_cell_counts",
    "determinism": "deterministic",
    "doc": "",
    "domain": "category",
    "file": "noether/category/__init__.pyi",
    "input_schema": {
      "additionalProperties": false,
      "properties": {},
      "type": "object"
    },
    "kind": "method",
    "line": 78,
    "module": "noether.category",
    "name": "exact_cell_counts",
    "overloads": [
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          "type": "array",
          "x-python-type": "list[int]"
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        "signature": "(self) -> list[int]"
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    ],
    "qualified_name": "FiniteOmegaCategory.exact_cell_counts",
    "releases_gil": true,
    "result_schema": {
      "items": {
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      "type": "array",
      "x-python-type": "list[int]"
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    "signature": "(self) -> list[int]",
    "signatures": [
      "(self) -> list[int]"
    ],
    "summary": "Compute exact cell counts for this `FiniteOmegaCategory`.",
    "summary_source": "derived projection vocabulary"
  },
  {
    "canonical_name": "category.FiniteOmegaCategory.exact_cells",
    "determinism": "deterministic",
    "doc": "",
    "domain": "category",
    "file": "noether/category/__init__.pyi",
    "input_schema": {
      "additionalProperties": false,
      "properties": {
        "dimension": {
          "type": "integer"
        }
      },
      "required": [
        "dimension"
      ],
      "type": "object"
    },
    "kind": "method",
    "line": 77,
    "module": "noether.category",
    "name": "exact_cells",
    "overloads": [
      {
        "input_schema": {
          "additionalProperties": false,
          "properties": {
            "dimension": {
              "type": "integer"
            }
          },
          "required": [
            "dimension"
          ],
          "type": "object"
        },
        "parameters": [
          {
            "kind": "positional_or_keyword",
            "name": "dimension",
            "required": true,
            "schema": {
              "type": "integer"
            }
          }
        ],
        "result_schema": {
          "items": {
            "x-python-type": "SteinerCell"
          },
          "type": "array",
          "x-python-type": "list[SteinerCell]"
        },
        "signature": "(self, dimension: int) -> list[SteinerCell]"
      }
    ],
    "qualified_name": "FiniteOmegaCategory.exact_cells",
    "releases_gil": true,
    "result_schema": {
      "items": {
        "x-python-type": "SteinerCell"
      },
      "type": "array",
      "x-python-type": "list[SteinerCell]"
    },
    "signature": "(self, dimension: int) -> list[SteinerCell]",
    "signatures": [
      "(self, dimension: int) -> list[SteinerCell]"
    ],
    "summary": "Compute exact cells for this `FiniteOmegaCategory`.",
    "summary_source": "derived projection vocabulary"
  },
  {
    "canonical_name": "category.FiniteOmegaCategory.mapping_category",
    "determinism": "deterministic",
    "doc": "",
    "domain": "category",
    "file": "noether/category/__init__.pyi",
    "input_schema": {
      "additionalProperties": false,
      "properties": {
        "source": {
          "x-python-type": "SteinerCell"
        },
        "target": {
          "x-python-type": "SteinerCell"
        }
      },
      "required": [
        "source",
        "target"
      ],
      "type": "object"
    },
    "kind": "method",
    "line": 80,
    "module": "noether.category",
    "name": "mapping_category",
    "overloads": [
      {
        "input_schema": {
          "additionalProperties": false,
          "properties": {
            "source": {
              "x-python-type": "SteinerCell"
            },
            "target": {
              "x-python-type": "SteinerCell"
            }
          },
          "required": [
            "source",
            "target"
          ],
          "type": "object"
        },
        "parameters": [
          {
            "kind": "positional_or_keyword",
            "name": "source",
            "required": true,
            "schema": {
              "x-python-type": "SteinerCell"
            }
          },
          {
            "kind": "positional_or_keyword",
            "name": "target",
            "required": true,
            "schema": {
              "x-python-type": "SteinerCell"
            }
          }
        ],
        "result_schema": {
          "x-python-type": "FiniteOmegaCategory"
        },
        "signature": "(self, source: SteinerCell, target: SteinerCell) -> FiniteOmegaCategory"
      }
    ],
    "qualified_name": "FiniteOmegaCategory.mapping_category",
    "releases_gil": true,
    "result_schema": {
      "x-python-type": "FiniteOmegaCategory"
    },
    "signature": "(self, source: SteinerCell, target: SteinerCell) -> FiniteOmegaCategory",
    "signatures": [
      "(self, source: SteinerCell, target: SteinerCell) -> FiniteOmegaCategory"
    ],
    "summary": "Compute mapping category for this `FiniteOmegaCategory`.",
    "summary_source": "derived projection vocabulary"
  },
  {
    "canonical_name": "category.FiniteOmegaCategory.objects",
    "determinism": "deterministic",
    "doc": "",
    "domain": "category",
    "file": "noether/category/__init__.pyi",
    "input_schema": {
      "additionalProperties": false,
      "properties": {},
      "type": "object"
    },
    "kind": "method",
    "line": 76,
    "module": "noether.category",
    "name": "objects",
    "overloads": [
      {
        "input_schema": {
          "additionalProperties": false,
          "properties": {},
          "type": "object"
        },
        "parameters": [],
        "result_schema": {
          "items": {
            "x-python-type": "SteinerCell"
          },
          "type": "array",
          "x-python-type": "list[SteinerCell]"
        },
        "signature": "(self) -> list[SteinerCell]"
      }
    ],
    "qualified_name": "FiniteOmegaCategory.objects",
    "releases_gil": false,
    "result_schema": {
      "items": {
        "x-python-type": "SteinerCell"
      },
      "type": "array",
      "x-python-type": "list[SteinerCell]"
    },
    "signature": "(self) -> list[SteinerCell]",
    "signatures": [
      "(self) -> list[SteinerCell]"
    ],
    "summary": "Compute objects for this `FiniteOmegaCategory`.",
    "summary_source": "derived projection vocabulary"
  },
  {
    "canonical_name": "category.FiniteOmegaCategory.strong_steiner_model",
    "determinism": "deterministic",
    "doc": "",
    "domain": "category",
    "file": "noether/category/__init__.pyi",
    "input_schema": {
      "additionalProperties": false,
      "properties": {},
      "type": "object"
    },
    "kind": "method",
    "line": 79,
    "module": "noether.category",
    "name": "strong_steiner_model",
    "overloads": [
      {
        "input_schema": {
          "additionalProperties": false,
          "properties": {},
          "type": "object"
        },
        "parameters": [],
        "result_schema": {
          "anyOf": [
            {
              "x-python-type": "SteinerCategory"
            },
            {
              "type": "null"
            }
          ],
          "x-python-type": "SteinerCategory | None"
        },
        "signature": "(self) -> SteinerCategory | None"
      }
    ],
    "qualified_name": "FiniteOmegaCategory.strong_steiner_model",
    "releases_gil": true,
    "result_schema": {
      "anyOf": [
        {
          "x-python-type": "SteinerCategory"
        },
        {
          "type": "null"
        }
      ],
      "x-python-type": "SteinerCategory | None"
    },
    "signature": "(self) -> SteinerCategory | None",
    "signatures": [
      "(self) -> SteinerCategory | None"
    ],
    "summary": "Compute strong steiner model for this `FiniteOmegaCategory`.",
    "summary_source": "derived projection vocabulary"
  },
  {
    "canonical_name": "category.OrientedBasepoint",
    "determinism": "deterministic",
    "doc": "One oriented globe-boundary basepoint.",
    "domain": "category",
    "file": "noether/category/__init__.pyi",
    "input_schema": {},
    "kind": "class",
    "line": 87,
    "module": "noether.category",
    "name": "OrientedBasepoint",
    "overloads": [],
    "qualified_name": "OrientedBasepoint",
    "releases_gil": false,
    "result_schema": {},
    "signature": "",
    "signatures": [],
    "summary": "One oriented globe-boundary basepoint.",
    "summary_source": "public docstring"
  },
  {
    "canonical_name": "category.OrientedBasepoint.dimension",
    "determinism": "deterministic",
    "doc": "",
    "domain": "category",
    "file": "noether/category/__init__.pyi",
    "input_schema": {
      "additionalProperties": false,
      "properties": {},
      "type": "object"
    },
    "kind": "property",
    "line": 94,
    "module": "noether.category",
    "name": "dimension",
    "overloads": [
      {
        "input_schema": {
          "additionalProperties": false,
          "properties": {},
          "type": "object"
        },
        "parameters": [],
        "result_schema": {
          "type": "integer"
        },
        "signature": "(self) -> int"
      }
    ],
    "qualified_name": "OrientedBasepoint.dimension",
    "releases_gil": false,
    "result_schema": {
      "type": "integer"
    },
    "signature": "(self) -> int",
    "signatures": [
      "(self) -> int"
    ],
    "summary": "Return the dimension recorded by this `OrientedBasepoint`.",
    "summary_source": "derived projection vocabulary"
  },
  {
    "canonical_name": "category.OrientedBasepoint.from_pairs",
    "determinism": "deterministic",
    "doc": "",
    "domain": "category",
    "file": "noether/category/__init__.pyi",
    "input_schema": {
      "additionalProperties": false,
      "properties": {
        "category": {
          "x-python-type": "SteinerCategory"
        },
        "pairs": {
          "items": {
            "maxItems": 2,
            "minItems": 2,
            "prefixItems": [
              {
                "x-python-type": "SteinerCell"
              },
              {
                "x-python-type": "SteinerCell"
              }
            ],
            "type": "array",
            "x-python-type": "tuple[SteinerCell, SteinerCell]"
          },
          "type": "array",
          "x-python-type": "Sequence[tuple[SteinerCell, SteinerCell]]"
        }
      },
      "required": [
        "category",
        "pairs"
      ],
      "type": "object"
    },
    "kind": "staticmethod",
    "line": 89,
    "module": "noether.category",
    "name": "from_pairs",
    "overloads": [
      {
        "input_schema": {
          "additionalProperties": false,
          "properties": {
            "category": {
              "x-python-type": "SteinerCategory"
            },
            "pairs": {
              "items": {
                "maxItems": 2,
                "minItems": 2,
                "prefixItems": [
                  {
                    "x-python-type": "SteinerCell"
                  },
                  {
                    "x-python-type": "SteinerCell"
                  }
                ],
                "type": "array",
                "x-python-type": "tuple[SteinerCell, SteinerCell]"
              },
              "type": "array",
              "x-python-type": "Sequence[tuple[SteinerCell, SteinerCell]]"
            }
          },
          "required": [
            "category",
            "pairs"
          ],
          "type": "object"
        },
        "parameters": [
          {
            "kind": "positional_or_keyword",
            "name": "category",
            "required": true,
            "schema": {
              "x-python-type": "SteinerCategory"
            }
          },
          {
            "kind": "positional_or_keyword",
            "name": "pairs",
            "required": true,
            "schema": {
              "items": {
                "maxItems": 2,
                "minItems": 2,
                "prefixItems": [
                  {
                    "x-python-type": "SteinerCell"
                  },
                  {
                    "x-python-type": "SteinerCell"
                  }
                ],
                "type": "array",
                "x-python-type": "tuple[SteinerCell, SteinerCell]"
              },
              "type": "array",
              "x-python-type": "Sequence[tuple[SteinerCell, SteinerCell]]"
            }
          }
        ],
        "result_schema": {
          "x-python-type": "OrientedBasepoint"
        },
        "signature": "(category: SteinerCategory, pairs: Sequence[tuple[SteinerCell, SteinerCell]]) -> OrientedBasepoint"
      }
    ],
    "qualified_name": "OrientedBasepoint.from_pairs",
    "releases_gil": false,
    "result_schema": {
      "x-python-type": "OrientedBasepoint"
    },
    "signature": "(category: SteinerCategory, pairs: Sequence[tuple[SteinerCell, SteinerCell]]) -> OrientedBasepoint",
    "signatures": [
      "(category: SteinerCategory, pairs: Sequence[tuple[SteinerCell, SteinerCell]]) -> OrientedBasepoint"
    ],
    "summary": "Construct a `OrientedBasepoint` using pairs.",
    "summary_source": "derived projection vocabulary"
  },
  {
    "canonical_name": "category.OrientedBasepoint.homotopy_degree",
    "determinism": "deterministic",
    "doc": "",
    "domain": "category",
    "file": "noether/category/__init__.pyi",
    "input_schema": {
      "additionalProperties": false,
      "properties": {},
      "type": "object"
    },
    "kind": "property",
    "line": 96,
    "module": "noether.category",
    "name": "homotopy_degree",
    "overloads": [
      {
        "input_schema": {
          "additionalProperties": false,
          "properties": {},
          "type": "object"
        },
        "parameters": [],
        "result_schema": {
          "type": "integer"
        },
        "signature": "(self) -> int"
      }
    ],
    "qualified_name": "OrientedBasepoint.homotopy_degree",
    "releases_gil": false,
    "result_schema": {
      "type": "integer"
    },
    "signature": "(self) -> int",
    "signatures": [
      "(self) -> int"
    ],
    "summary": "Return the homotopy degree recorded by this `OrientedBasepoint`.",
    "summary_source": "derived projection vocabulary"
  },
  {
    "canonical_name": "category.OrientedBasepoint.homotopy_poset",
    "determinism": "deterministic",
    "doc": "",
    "domain": "category",
    "file": "noether/category/__init__.pyi",
    "input_schema": {
      "additionalProperties": false,
      "properties": {
        "category": {
          "x-python-type": "SteinerCategory"
        }
      },
      "required": [
        "category"
      ],
      "type": "object"
    },
    "kind": "method",
    "line": 105,
    "module": "noether.category",
    "name": "homotopy_poset",
    "overloads": [
      {
        "input_schema": {
          "additionalProperties": false,
          "properties": {
            "category": {
              "x-python-type": "SteinerCategory"
            }
          },
          "required": [
            "category"
          ],
          "type": "object"
        },
        "parameters": [
          {
            "kind": "positional_or_keyword",
            "name": "category",
            "required": true,
            "schema": {
              "x-python-type": "SteinerCategory"
            }
          }
        ],
        "result_schema": {
          "x-python-type": "Poset"
        },
        "signature": "(self, category: SteinerCategory) -> Poset"
      }
    ],
    "qualified_name": "OrientedBasepoint.homotopy_poset",
    "releases_gil": true,
    "result_schema": {
      "x-python-type": "Poset"
    },
    "signature": "(self, category: SteinerCategory) -> Poset",
    "signatures": [
      "(self, category: SteinerCategory) -> Poset"
    ],
    "summary": "Compute homotopy poset for this `OrientedBasepoint`.",
    "summary_source": "derived projection vocabulary"
  },
  {
    "canonical_name": "category.OrientedBasepoint.is_monomorphic",
    "determinism": "deterministic",
    "doc": "",
    "domain": "category",
    "file": "noether/category/__init__.pyi",
    "input_schema": {
      "additionalProperties": false,
      "properties": {},
      "type": "object"
    },
    "kind": "property",
    "line": 100,
    "module": "noether.category",
    "name": "is_monomorphic",
    "overloads": [
      {
        "input_schema": {
          "additionalProperties": false,
          "properties": {},
          "type": "object"
        },
        "parameters": [],
        "result_schema": {
          "type": "boolean"
        },
        "signature": "(self) -> bool"
      }
    ],
    "qualified_name": "OrientedBasepoint.is_monomorphic",
    "releases_gil": false,
    "result_schema": {
      "type": "boolean"
    },
    "signature": "(self) -> bool",
    "signatures": [
      "(self) -> bool"
    ],
    "summary": "Test whether this `OrientedBasepoint` is monomorphic.",
    "summary_source": "derived projection vocabulary"
  },
  {
    "canonical_name": "category.OrientedBasepoint.iterated_mapping_category",
    "determinism": "deterministic",
    "doc": "",
    "domain": "category",
    "file": "noether/category/__init__.pyi",
    "input_schema": {
      "additionalProperties": false,
      "properties": {
        "category": {
          "x-python-type": "SteinerCategory"
        }
      },
      "required": [
        "category"
      ],
      "type": "object"
    },
    "kind": "method",
    "line": 101,
    "module": "noether.category",
    "name": "iterated_mapping_category",
    "overloads": [
      {
        "input_schema": {
          "additionalProperties": false,
          "properties": {
            "category": {
              "x-python-type": "SteinerCategory"
            }
          },
          "required": [
            "category"
          ],
          "type": "object"
        },
        "parameters": [
          {
            "kind": "positional_or_keyword",
            "name": "category",
            "required": true,
            "schema": {
              "x-python-type": "SteinerCategory"
            }
          }
        ],
        "result_schema": {
          "x-python-type": "FiniteOmegaCategory"
        },
        "signature": "(self, category: SteinerCategory) -> FiniteOmegaCategory"
      }
    ],
    "qualified_name": "OrientedBasepoint.iterated_mapping_category",
    "releases_gil": true,
    "result_schema": {
      "x-python-type": "FiniteOmegaCategory"
    },
    "signature": "(self, category: SteinerCategory) -> FiniteOmegaCategory",
    "signatures": [
      "(self, category: SteinerCategory) -> FiniteOmegaCategory"
    ],
    "summary": "Compute iterated mapping category for this `OrientedBasepoint`.",
    "summary_source": "derived projection vocabulary"
  },
  {
    "canonical_name": "category.OrientedBasepoint.pairs",
    "determinism": "deterministic",
    "doc": "",
    "domain": "category",
    "file": "noether/category/__init__.pyi",
    "input_schema": {
      "additionalProperties": false,
      "properties": {},
      "type": "object"
    },
    "kind": "property",
    "line": 98,
    "module": "noether.category",
    "name": "pairs",
    "overloads": [
      {
        "input_schema": {
          "additionalProperties": false,
          "properties": {},
          "type": "object"
        },
        "parameters": [],
        "result_schema": {
          "items": {
            "maxItems": 2,
            "minItems": 2,
            "prefixItems": [
              {
                "x-python-type": "SteinerCell"
              },
              {
                "x-python-type": "SteinerCell"
              }
            ],
            "type": "array",
            "x-python-type": "tuple[SteinerCell, SteinerCell]"
          },
          "type": "array",
          "x-python-type": "list[tuple[SteinerCell, SteinerCell]]"
        },
        "signature": "(self) -> list[tuple[SteinerCell, SteinerCell]]"
      }
    ],
    "qualified_name": "OrientedBasepoint.pairs",
    "releases_gil": false,
    "result_schema": {
      "items": {
        "maxItems": 2,
        "minItems": 2,
        "prefixItems": [
          {
            "x-python-type": "SteinerCell"
          },
          {
            "x-python-type": "SteinerCell"
          }
        ],
        "type": "array",
        "x-python-type": "tuple[SteinerCell, SteinerCell]"
      },
      "type": "array",
      "x-python-type": "list[tuple[SteinerCell, SteinerCell]]"
    },
    "signature": "(self) -> list[tuple[SteinerCell, SteinerCell]]",
    "signatures": [
      "(self) -> list[tuple[SteinerCell, SteinerCell]]"
    ],
    "summary": "Return the pairs recorded by this `OrientedBasepoint`.",
    "summary_source": "derived projection vocabulary"
  },
  {
    "canonical_name": "category.SignedBoundary",
    "determinism": "deterministic",
    "doc": "Disjoint signed support of one atomic boundary.",
    "domain": "category",
    "file": "noether/category/__init__.pyi",
    "input_schema": {
      "additionalProperties": false,
      "properties": {
        "negative": {
          "items": {
            "type": "integer"
          },
          "type": "array",
          "x-python-type": "Sequence[int]"
        },
        "positive": {
          "items": {
            "type": "integer"
          },
          "type": "array",
          "x-python-type": "Sequence[int]"
        }
      },
      "type": "object"
    },
    "kind": "class",
    "line": 19,
    "module": "noether.category",
    "name": "SignedBoundary",
    "overloads": [
      {
        "input_schema": {
          "additionalProperties": false,
          "properties": {
            "negative": {
              "items": {
                "type": "integer"
              },
              "type": "array",
              "x-python-type": "Sequence[int]"
            },
            "positive": {
              "items": {
                "type": "integer"
              },
              "type": "array",
              "x-python-type": "Sequence[int]"
            }
          },
          "type": "object"
        },
        "parameters": [
          {
            "default": "...",
            "kind": "positional_or_keyword",
            "name": "negative",
            "required": false,
            "schema": {
              "items": {
                "type": "integer"
              },
              "type": "array",
              "x-python-type": "Sequence[int]"
            }
          },
          {
            "default": "...",
            "kind": "positional_or_keyword",
            "name": "positive",
            "required": false,
            "schema": {
              "items": {
                "type": "integer"
              },
              "type": "array",
              "x-python-type": "Sequence[int]"
            }
          }
        ],
        "result_schema": {
          "type": "null"
        },
        "signature": "(self, negative: Sequence[int]=..., positive: Sequence[int]=...) -> None"
      }
    ],
    "qualified_name": "SignedBoundary",
    "releases_gil": false,
    "result_schema": {
      "type": "null"
    },
    "signature": "(self, negative: Sequence[int]=..., positive: Sequence[int]=...) -> None",
    "signatures": [
      "(self, negative: Sequence[int]=..., positive: Sequence[int]=...) -> None"
    ],
    "summary": "Disjoint signed support of one atomic boundary.",
    "summary_source": "public docstring"
  },
  {
    "canonical_name": "category.SignedBoundary.negative",
    "determinism": "deterministic",
    "doc": "",
    "domain": "category",
    "file": "noether/category/__init__.pyi",
    "input_schema": {
      "additionalProperties": false,
      "properties": {},
      "type": "object"
    },
    "kind": "property",
    "line": 26,
    "module": "noether.category",
    "name": "negative",
    "overloads": [
      {
        "input_schema": {
          "additionalProperties": false,
          "properties": {},
          "type": "object"
        },
        "parameters": [],
        "result_schema": {
          "items": {
            "type": "integer"
          },
          "type": "array",
          "x-python-type": "list[int]"
        },
        "signature": "(self) -> list[int]"
      }
    ],
    "qualified_name": "SignedBoundary.negative",
    "releases_gil": false,
    "result_schema": {
      "items": {
        "type": "integer"
      },
      "type": "array",
      "x-python-type": "list[int]"
    },
    "signature": "(self) -> list[int]",
    "signatures": [
      "(self) -> list[int]"
    ],
    "summary": "Return the negative recorded by this `SignedBoundary`.",
    "summary_source": "derived projection vocabulary"
  },
  {
    "canonical_name": "category.SignedBoundary.positive",
    "determinism": "deterministic",
    "doc": "",
    "domain": "category",
    "file": "noether/category/__init__.pyi",
    "input_schema": {
      "additionalProperties": false,
      "properties": {},
      "type": "object"
    },
    "kind": "property",
    "line": 28,
    "module": "noether.category",
    "name": "positive",
    "overloads": [
      {
        "input_schema": {
          "additionalProperties": false,
          "properties": {},
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    "summary": "Exact category canonicalization and symmetry evidence.",
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    "domain": "category",
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    "summary": "Return the automorphism group recorded by this `SteinerCanonicalLabel`.",
    "summary_source": "derived projection vocabulary"
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    "canonical_name": "category.SteinerCanonicalLabel.category",
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    "doc": "",
    "domain": "category",
    "file": "noether/category/__init__.pyi",
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    "kind": "property",
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    "signatures": [
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    "summary": "Return the category recorded by this `SteinerCanonicalLabel`.",
    "summary_source": "derived projection vocabulary"
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    "canonical_name": "category.SteinerCanonicalLabel.degreewise_labeling",
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    "doc": "",
    "domain": "category",
    "file": "noether/category/__init__.pyi",
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      "properties": {},
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    "canonical_name": "category.SteinerCanonicalLabel.encoding",
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    "doc": "",
    "domain": "category",
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    "signatures": [
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    "summary": "Return the encoding recorded by this `SteinerCanonicalLabel`.",
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    "canonical_name": "category.SteinerCanonicalLabel.labeling",
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    "domain": "category",
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      "type": "object"
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    "qualified_name": "SteinerCanonicalLabel.labeling",
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    "summary": "Return the labeling recorded by this `SteinerCanonicalLabel`.",
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    "canonical_name": "category.SteinerCanonicalLabel.transport_cell",
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    "domain": "category",
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    "summary": "Test mathematical equality with another value.",
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    "summary": "Compute atom for this `SteinerCategory`.",
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    "canonical_name": "category.SteinerCategory.atom_count",
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    "domain": "category",
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    "kind": "property",
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    "summary": "Return the atom count recorded by this `SteinerCategory`.",
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    "summary": "Compute atom count in degree for this `SteinerCategory`.",
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    "summary": "Return the atom counts recorded by this `SteinerCategory`.",
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    "summary": "Compute automorphism group for this `SteinerCategory`.",
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      "required": [
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    "result_schema": {
      "x-python-type": "SignedBoundary"
    },
    "signature": "(self, atom: AtomId) -> SignedBoundary",
    "signatures": [
      "(self, atom: AtomId) -> SignedBoundary"
    ],
    "summary": "Compute boundary for this `SteinerCategory`.",
    "summary_source": "derived projection vocabulary",
    "type_id": "noether/category/SteinerCategory@1"
  },
  {
    "canonical_name": "category.SteinerCategory.canonical_form",
    "determinism": "deterministic",
    "doc": "",
    "domain": "category",
    "file": "noether/category/__init__.pyi",
    "input_schema": {
      "additionalProperties": false,
      "properties": {},
      "type": "object"
    },
    "kind": "method",
    "line": 227,
    "module": "noether.category",
    "name": "canonical_form",
    "overloads": [
      {
        "input_schema": {
          "additionalProperties": false,
          "properties": {},
          "type": "object"
        },
        "parameters": [],
        "result_schema": {
          "x-python-type": "SteinerCategory"
        },
        "signature": "(self) -> SteinerCategory"
      }
    ],
    "qualified_name": "SteinerCategory.canonical_form",
    "releases_gil": true,
    "result_schema": {
      "x-python-type": "SteinerCategory"
    },
    "signature": "(self) -> SteinerCategory",
    "signatures": [
      "(self) -> SteinerCategory"
    ],
    "summary": "Compute canonical form for this `SteinerCategory`.",
    "summary_source": "derived projection vocabulary",
    "type_id": "noether/category/SteinerCategory@1"
  },
  {
    "canonical_name": "category.SteinerCategory.canonical_label",
    "determinism": "deterministic",
    "doc": "",
    "domain": "category",
    "file": "noether/category/__init__.pyi",
    "input_schema": {
      "additionalProperties": false,
      "properties": {},
      "type": "object"
    },
    "kind": "method",
    "line": 226,
    "module": "noether.category",
    "name": "canonical_label",
    "overloads": [
      {
        "input_schema": {
          "additionalProperties": false,
          "properties": {},
          "type": "object"
        },
        "parameters": [],
        "result_schema": {
          "x-python-type": "SteinerCanonicalLabel"
        },
        "signature": "(self) -> SteinerCanonicalLabel"
      }
    ],
    "qualified_name": "SteinerCategory.canonical_label",
    "releases_gil": true,
    "result_schema": {
      "x-python-type": "SteinerCanonicalLabel"
    },
    "signature": "(self) -> SteinerCanonicalLabel",
    "signatures": [
      "(self) -> SteinerCanonicalLabel"
    ],
    "summary": "Compute canonical label for this `SteinerCategory`.",
    "summary_source": "derived projection vocabulary",
    "type_id": "noether/category/SteinerCategory@1"
  },
  {
    "canonical_name": "category.SteinerCategory.categorical_homology",
    "determinism": "deterministic",
    "doc": "",
    "domain": "category",
    "file": "noether/category/__init__.pyi",
    "input_schema": {
      "additionalProperties": false,
      "properties": {},
      "type": "object"
    },
    "kind": "method",
    "line": 237,
    "module": "noether.category",
    "name": "categorical_homology",
    "overloads": [
      {
        "input_schema": {
          "additionalProperties": false,
          "properties": {},
          "type": "object"
        },
        "parameters": [],
        "result_schema": {
          "x-python-type": "CategoricalHomologyPresentation"
        },
        "signature": "(self) -> CategoricalHomologyPresentation"
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    ],
    "qualified_name": "SteinerCategory.categorical_homology",
    "releases_gil": true,
    "result_schema": {
      "x-python-type": "CategoricalHomologyPresentation"
    },
    "signature": "(self) -> CategoricalHomologyPresentation",
    "signatures": [
      "(self) -> CategoricalHomologyPresentation"
    ],
    "summary": "Compute categorical homology for this `SteinerCategory`.",
    "summary_source": "derived projection vocabulary",
    "type_id": "noether/category/SteinerCategory@1"
  },
  {
    "canonical_name": "category.SteinerCategory.cell",
    "determinism": "deterministic",
    "doc": "",
    "domain": "category",
    "file": "noether/category/__init__.pyi",
    "input_schema": {
      "additionalProperties": false,
      "properties": {
        "dimension": {
          "type": "integer"
        },
        "top_atoms": {
          "items": {
            "type": "integer"
          },
          "type": "array",
          "x-python-type": "Sequence[int]"
        }
      },
      "required": [
        "dimension",
        "top_atoms"
      ],
      "type": "object"
    },
    "kind": "method",
    "line": 199,
    "module": "noether.category",
    "name": "cell",
    "overloads": [
      {
        "input_schema": {
          "additionalProperties": false,
          "properties": {
            "dimension": {
              "type": "integer"
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            "top_atoms": {
              "items": {
                "type": "integer"
              },
              "type": "array",
              "x-python-type": "Sequence[int]"
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          },
          "required": [
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            "top_atoms"
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          "type": "object"
        },
        "parameters": [
          {
            "kind": "positional_or_keyword",
            "name": "dimension",
            "required": true,
            "schema": {
              "type": "integer"
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          },
          {
            "kind": "positional_or_keyword",
            "name": "top_atoms",
            "required": true,
            "schema": {
              "items": {
                "type": "integer"
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              "type": "array",
              "x-python-type": "Sequence[int]"
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          }
        ],
        "result_schema": {
          "x-python-type": "SteinerCell"
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        "signature": "(self, dimension: int, top_atoms: Sequence[int]) -> SteinerCell"
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    ],
    "qualified_name": "SteinerCategory.cell",
    "releases_gil": false,
    "result_schema": {
      "x-python-type": "SteinerCell"
    },
    "signature": "(self, dimension: int, top_atoms: Sequence[int]) -> SteinerCell",
    "signatures": [
      "(self, dimension: int, top_atoms: Sequence[int]) -> SteinerCell"
    ],
    "summary": "Compute cell for this `SteinerCategory`.",
    "summary_source": "derived projection vocabulary",
    "type_id": "noether/category/SteinerCategory@1"
  },
  {
    "canonical_name": "category.SteinerCategory.cell_from_supports",
    "determinism": "deterministic",
    "doc": "",
    "domain": "category",
    "file": "noether/category/__init__.pyi",
    "input_schema": {
      "additionalProperties": false,
      "properties": {
        "negative_supports": {
          "items": {
            "items": {
              "type": "integer"
            },
            "type": "array",
            "x-python-type": "Sequence[int]"
          },
          "type": "array",
          "x-python-type": "Sequence[Sequence[int]]"
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        "positive_supports": {
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            "items": {
              "type": "integer"
            },
            "type": "array",
            "x-python-type": "Sequence[int]"
          },
          "type": "array",
          "x-python-type": "Sequence[Sequence[int]]"
        }
      },
      "required": [
        "negative_supports",
        "positive_supports"
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      "type": "object"
    },
    "kind": "method",
    "line": 200,
    "module": "noether.category",
    "name": "cell_from_supports",
    "overloads": [
      {
        "input_schema": {
          "additionalProperties": false,
          "properties": {
            "negative_supports": {
              "items": {
                "items": {
                  "type": "integer"
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                "type": "array",
                "x-python-type": "Sequence[int]"
              },
              "type": "array",
              "x-python-type": "Sequence[Sequence[int]]"
            },
            "positive_supports": {
              "items": {
                "items": {
                  "type": "integer"
                },
                "type": "array",
                "x-python-type": "Sequence[int]"
              },
              "type": "array",
              "x-python-type": "Sequence[Sequence[int]]"
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          },
          "required": [
            "negative_supports",
            "positive_supports"
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          "type": "object"
        },
        "parameters": [
          {
            "kind": "positional_or_keyword",
            "name": "negative_supports",
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            "schema": {
              "items": {
                "items": {
                  "type": "integer"
                },
                "type": "array",
                "x-python-type": "Sequence[int]"
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              "type": "array",
              "x-python-type": "Sequence[Sequence[int]]"
            }
          },
          {
            "kind": "positional_or_keyword",
            "name": "positive_supports",
            "required": true,
            "schema": {
              "items": {
                "items": {
                  "type": "integer"
                },
                "type": "array",
                "x-python-type": "Sequence[int]"
              },
              "type": "array",
              "x-python-type": "Sequence[Sequence[int]]"
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          }
        ],
        "result_schema": {
          "x-python-type": "SteinerCell"
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        "signature": "(self, negative_supports: Sequence[Sequence[int]], positive_supports: Sequence[Sequence[int]]) -> SteinerCell"
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    ],
    "qualified_name": "SteinerCategory.cell_from_supports",
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    "result_schema": {
      "x-python-type": "SteinerCell"
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    "signature": "(self, negative_supports: Sequence[Sequence[int]], positive_supports: Sequence[Sequence[int]]) -> SteinerCell",
    "signatures": [
      "(self, negative_supports: Sequence[Sequence[int]], positive_supports: Sequence[Sequence[int]]) -> SteinerCell"
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    "summary": "Compute cell from supports for this `SteinerCategory`.",
    "summary_source": "derived projection vocabulary",
    "type_id": "noether/category/SteinerCategory@1"
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  {
    "canonical_name": "category.SteinerCategory.cells",
    "determinism": "deterministic",
    "doc": "",
    "domain": "category",
    "file": "noether/category/__init__.pyi",
    "input_schema": {
      "additionalProperties": false,
      "properties": {
        "dimension": {
          "type": "integer"
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      },
      "required": [
        "dimension"
      ],
      "type": "object"
    },
    "kind": "method",
    "line": 206,
    "module": "noether.category",
    "name": "cells",
    "overloads": [
      {
        "input_schema": {
          "additionalProperties": false,
          "properties": {
            "dimension": {
              "type": "integer"
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          },
          "required": [
            "dimension"
          ],
          "type": "object"
        },
        "parameters": [
          {
            "kind": "positional_or_keyword",
            "name": "dimension",
            "required": true,
            "schema": {
              "type": "integer"
            }
          }
        ],
        "result_schema": {
          "items": {
            "x-python-type": "SteinerCell"
          },
          "type": "array",
          "x-python-type": "list[SteinerCell]"
        },
        "signature": "(self, dimension: int) -> list[SteinerCell]"
      }
    ],
    "qualified_name": "SteinerCategory.cells",
    "releases_gil": true,
    "result_schema": {
      "items": {
        "x-python-type": "SteinerCell"
      },
      "type": "array",
      "x-python-type": "list[SteinerCell]"
    },
    "signature": "(self, dimension: int) -> list[SteinerCell]",
    "signatures": [
      "(self, dimension: int) -> list[SteinerCell]"
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    "summary": "Compute cells for this `SteinerCategory`.",
    "summary_source": "derived projection vocabulary",
    "type_id": "noether/category/SteinerCategory@1"
  },
  {
    "canonical_name": "category.SteinerCategory.component_poset",
    "determinism": "deterministic",
    "doc": "",
    "domain": "category",
    "file": "noether/category/__init__.pyi",
    "input_schema": {
      "additionalProperties": false,
      "properties": {},
      "type": "object"
    },
    "kind": "method",
    "line": 213,
    "module": "noether.category",
    "name": "component_poset",
    "overloads": [
      {
        "input_schema": {
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          "properties": {},
          "type": "object"
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        "parameters": [],
        "result_schema": {
          "x-python-type": "Poset"
        },
        "signature": "(self) -> Poset"
      }
    ],
    "qualified_name": "SteinerCategory.component_poset",
    "releases_gil": true,
    "result_schema": {
      "x-python-type": "Poset"
    },
    "signature": "(self) -> Poset",
    "signatures": [
      "(self) -> Poset"
    ],
    "summary": "Compute component poset for this `SteinerCategory`.",
    "summary_source": "derived projection vocabulary",
    "type_id": "noether/category/SteinerCategory@1"
  },
  {
    "canonical_name": "category.SteinerCategory.cube",
    "determinism": "deterministic",
    "doc": "",
    "domain": "category",
    "file": "noether/category/__init__.pyi",
    "input_schema": {
      "additionalProperties": false,
      "properties": {
        "dimension": {
          "type": "integer"
        }
      },
      "required": [
        "dimension"
      ],
      "type": "object"
    },
    "kind": "staticmethod",
    "line": 183,
    "module": "noether.category",
    "name": "cube",
    "overloads": [
      {
        "input_schema": {
          "additionalProperties": false,
          "properties": {
            "dimension": {
              "type": "integer"
            }
          },
          "required": [
            "dimension"
          ],
          "type": "object"
        },
        "parameters": [
          {
            "kind": "positional_or_keyword",
            "name": "dimension",
            "required": true,
            "schema": {
              "type": "integer"
            }
          }
        ],
        "result_schema": {
          "x-python-type": "SteinerCategory"
        },
        "signature": "(dimension: int) -> SteinerCategory"
      }
    ],
    "qualified_name": "SteinerCategory.cube",
    "releases_gil": false,
    "result_schema": {
      "x-python-type": "SteinerCategory"
    },
    "signature": "(dimension: int) -> SteinerCategory",
    "signatures": [
      "(dimension: int) -> SteinerCategory"
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    "summary": "Construct a `SteinerCategory` using cube.",
    "summary_source": "derived projection vocabulary",
    "type_id": "noether/category/SteinerCategory@1"
  },
  {
    "canonical_name": "category.SteinerCategory.dimension",
    "determinism": "deterministic",
    "doc": "",
    "domain": "category",
    "file": "noether/category/__init__.pyi",
    "input_schema": {
      "additionalProperties": false,
      "properties": {},
      "type": "object"
    },
    "kind": "property",
    "line": 189,
    "module": "noether.category",
    "name": "dimension",
    "overloads": [
      {
        "input_schema": {
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          "properties": {},
          "type": "object"
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        "parameters": [],
        "result_schema": {
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        "signature": "(self) -> int"
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    ],
    "qualified_name": "SteinerCategory.dimension",
    "releases_gil": false,
    "result_schema": {
      "type": "integer"
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    "signature": "(self) -> int",
    "signatures": [
      "(self) -> int"
    ],
    "summary": "Return the dimension recorded by this `SteinerCategory`.",
    "summary_source": "derived projection vocabulary",
    "type_id": "noether/category/SteinerCategory@1"
  },
  {
    "canonical_name": "category.SteinerCategory.disjoint_union",
    "determinism": "deterministic",
    "doc": "",
    "domain": "category",
    "file": "noether/category/__init__.pyi",
    "input_schema": {
      "additionalProperties": false,
      "properties": {
        "other": {
          "x-python-type": "SteinerCategory"
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      },
      "required": [
        "other"
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      "type": "object"
    },
    "kind": "method",
    "line": 184,
    "module": "noether.category",
    "name": "disjoint_union",
    "overloads": [
      {
        "input_schema": {
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          "properties": {
            "other": {
              "x-python-type": "SteinerCategory"
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          "required": [
            "other"
          ],
          "type": "object"
        },
        "parameters": [
          {
            "kind": "positional_or_keyword",
            "name": "other",
            "required": true,
            "schema": {
              "x-python-type": "SteinerCategory"
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          }
        ],
        "result_schema": {
          "x-python-type": "SteinerCategory"
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        "signature": "(self, other: SteinerCategory) -> SteinerCategory"
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    ],
    "qualified_name": "SteinerCategory.disjoint_union",
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    "result_schema": {
      "x-python-type": "SteinerCategory"
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    "signature": "(self, other: SteinerCategory) -> SteinerCategory",
    "signatures": [
      "(self, other: SteinerCategory) -> SteinerCategory"
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    "summary": "Compute disjoint union for this `SteinerCategory`.",
    "summary_source": "derived projection vocabulary",
    "type_id": "noether/category/SteinerCategory@1"
  },
  {
    "canonical_name": "category.SteinerCategory.dual",
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    "doc": "",
    "domain": "category",
    "file": "noether/category/__init__.pyi",
    "input_schema": {
      "additionalProperties": false,
      "properties": {
        "dimensions": {
          "items": {
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          },
          "type": "array",
          "x-python-type": "Sequence[int]"
        }
      },
      "required": [
        "dimensions"
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      "type": "object"
    },
    "kind": "method",
    "line": 225,
    "module": "noether.category",
    "name": "dual",
    "overloads": [
      {
        "input_schema": {
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          "properties": {
            "dimensions": {
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              "x-python-type": "Sequence[int]"
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          },
          "required": [
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          ],
          "type": "object"
        },
        "parameters": [
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            "kind": "positional_or_keyword",
            "name": "dimensions",
            "required": true,
            "schema": {
              "items": {
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              },
              "type": "array",
              "x-python-type": "Sequence[int]"
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          }
        ],
        "result_schema": {
          "x-python-type": "SteinerCategory"
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        "signature": "(self, dimensions: Sequence[int]) -> SteinerCategory"
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    ],
    "qualified_name": "SteinerCategory.dual",
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    "result_schema": {
      "x-python-type": "SteinerCategory"
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    "signature": "(self, dimensions: Sequence[int]) -> SteinerCategory",
    "signatures": [
      "(self, dimensions: Sequence[int]) -> SteinerCategory"
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    "summary": "Compute dual for this `SteinerCategory`.",
    "summary_source": "derived projection vocabulary",
    "type_id": "noether/category/SteinerCategory@1"
  },
  {
    "canonical_name": "category.SteinerCategory.empty",
    "determinism": "deterministic",
    "doc": "",
    "domain": "category",
    "file": "noether/category/__init__.pyi",
    "input_schema": {
      "additionalProperties": false,
      "properties": {},
      "type": "object"
    },
    "kind": "staticmethod",
    "line": 175,
    "module": "noether.category",
    "name": "empty",
    "overloads": [
      {
        "input_schema": {
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          "properties": {},
          "type": "object"
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        "parameters": [],
        "result_schema": {
          "x-python-type": "SteinerCategory"
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        "signature": "() -> SteinerCategory"
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    ],
    "qualified_name": "SteinerCategory.empty",
    "releases_gil": false,
    "result_schema": {
      "x-python-type": "SteinerCategory"
    },
    "signature": "() -> SteinerCategory",
    "signatures": [
      "() -> SteinerCategory"
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    "summary": "Construct a `SteinerCategory` using empty.",
    "summary_source": "derived projection vocabulary",
    "type_id": "noether/category/SteinerCategory@1"
  },
  {
    "canonical_name": "category.SteinerCategory.exact_cell_counts",
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    "doc": "",
    "domain": "category",
    "file": "noether/category/__init__.pyi",
    "input_schema": {
      "additionalProperties": false,
      "properties": {},
      "type": "object"
    },
    "kind": "method",
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    "module": "noether.category",
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      {
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          "type": "array",
          "x-python-type": "list[int]"
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        "signature": "(self) -> list[int]"
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    ],
    "qualified_name": "SteinerCategory.exact_cell_counts",
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    "result_schema": {
      "items": {
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      "type": "array",
      "x-python-type": "list[int]"
    },
    "signature": "(self) -> list[int]",
    "signatures": [
      "(self) -> list[int]"
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    "summary": "Compute exact cell counts for this `SteinerCategory`.",
    "summary_source": "derived projection vocabulary",
    "type_id": "noether/category/SteinerCategory@1"
  },
  {
    "canonical_name": "category.SteinerCategory.exact_cells",
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    "doc": "",
    "domain": "category",
    "file": "noether/category/__init__.pyi",
    "input_schema": {
      "additionalProperties": false,
      "properties": {
        "dimension": {
          "type": "integer"
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      },
      "required": [
        "dimension"
      ],
      "type": "object"
    },
    "kind": "method",
    "line": 205,
    "module": "noether.category",
    "name": "exact_cells",
    "overloads": [
      {
        "input_schema": {
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          "properties": {
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          "type": "object"
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        "parameters": [
          {
            "kind": "positional_or_keyword",
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            "schema": {
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          }
        ],
        "result_schema": {
          "items": {
            "x-python-type": "SteinerCell"
          },
          "type": "array",
          "x-python-type": "list[SteinerCell]"
        },
        "signature": "(self, dimension: int) -> list[SteinerCell]"
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    ],
    "qualified_name": "SteinerCategory.exact_cells",
    "releases_gil": true,
    "result_schema": {
      "items": {
        "x-python-type": "SteinerCell"
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      "type": "array",
      "x-python-type": "list[SteinerCell]"
    },
    "signature": "(self, dimension: int) -> list[SteinerCell]",
    "signatures": [
      "(self, dimension: int) -> list[SteinerCell]"
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    "summary": "Compute exact cells for this `SteinerCategory`.",
    "summary_source": "derived projection vocabulary",
    "type_id": "noether/category/SteinerCategory@1"
  },
  {
    "canonical_name": "category.SteinerCategory.from_json",
    "determinism": "deterministic",
    "doc": "",
    "domain": "category",
    "file": "noether/category/__init__.pyi",
    "input_schema": {
      "additionalProperties": false,
      "properties": {
        "value": {
          "type": "string"
        }
      },
      "required": [
        "value"
      ],
      "type": "object"
    },
    "kind": "staticmethod",
    "line": 186,
    "module": "noether.category",
    "name": "from_json",
    "overloads": [
      {
        "input_schema": {
          "additionalProperties": false,
          "properties": {
            "value": {
              "type": "string"
            }
          },
          "required": [
            "value"
          ],
          "type": "object"
        },
        "parameters": [
          {
            "kind": "positional_or_keyword",
            "name": "value",
            "required": true,
            "schema": {
              "type": "string"
            }
          }
        ],
        "result_schema": {
          "x-python-type": "SteinerCategory"
        },
        "signature": "(value: str) -> SteinerCategory"
      }
    ],
    "qualified_name": "SteinerCategory.from_json",
    "releases_gil": false,
    "result_schema": {
      "x-python-type": "SteinerCategory"
    },
    "signature": "(value: str) -> SteinerCategory",
    "signatures": [
      "(value: str) -> SteinerCategory"
    ],
    "summary": "Deserialize and revalidate a `SteinerCategory` from typed JSON.",
    "summary_source": "derived projection vocabulary",
    "type_id": "noether/category/SteinerCategory@1"
  },
  {
    "canonical_name": "category.SteinerCategory.fundamental_poset",
    "determinism": "deterministic",
    "doc": "",
    "domain": "category",
    "file": "noether/category/__init__.pyi",
    "input_schema": {
      "additionalProperties": false,
      "properties": {
        "source": {
          "x-python-type": "SteinerCell"
        },
        "target": {
          "x-python-type": "SteinerCell"
        }
      },
      "required": [
        "source",
        "target"
      ],
      "type": "object"
    },
    "kind": "method",
    "line": 214,
    "module": "noether.category",
    "name": "fundamental_poset",
    "overloads": [
      {
        "input_schema": {
          "additionalProperties": false,
          "properties": {
            "source": {
              "x-python-type": "SteinerCell"
            },
            "target": {
              "x-python-type": "SteinerCell"
            }
          },
          "required": [
            "source",
            "target"
          ],
          "type": "object"
        },
        "parameters": [
          {
            "kind": "positional_or_keyword",
            "name": "source",
            "required": true,
            "schema": {
              "x-python-type": "SteinerCell"
            }
          },
          {
            "kind": "positional_or_keyword",
            "name": "target",
            "required": true,
            "schema": {
              "x-python-type": "SteinerCell"
            }
          }
        ],
        "result_schema": {
          "x-python-type": "Poset"
        },
        "signature": "(self, source: SteinerCell, target: SteinerCell) -> Poset"
      }
    ],
    "qualified_name": "SteinerCategory.fundamental_poset",
    "releases_gil": true,
    "result_schema": {
      "x-python-type": "Poset"
    },
    "signature": "(self, source: SteinerCell, target: SteinerCell) -> Poset",
    "signatures": [
      "(self, source: SteinerCell, target: SteinerCell) -> Poset"
    ],
    "summary": "Compute fundamental poset for this `SteinerCategory`.",
    "summary_source": "derived projection vocabulary",
    "type_id": "noether/category/SteinerCategory@1"
  },
  {
    "canonical_name": "category.SteinerCategory.globe",
    "determinism": "deterministic",
    "doc": "",
    "domain": "category",
    "file": "noether/category/__init__.pyi",
    "input_schema": {
      "additionalProperties": false,
      "properties": {
        "dimension": {
          "type": "integer"
        }
      },
      "required": [
        "dimension"
      ],
      "type": "object"
    },
    "kind": "staticmethod",
    "line": 179,
    "module": "noether.category",
    "name": "globe",
    "overloads": [
      {
        "input_schema": {
          "additionalProperties": false,
          "properties": {
            "dimension": {
              "type": "integer"
            }
          },
          "required": [
            "dimension"
          ],
          "type": "object"
        },
        "parameters": [
          {
            "kind": "positional_or_keyword",
            "name": "dimension",
            "required": true,
            "schema": {
              "type": "integer"
            }
          }
        ],
        "result_schema": {
          "x-python-type": "SteinerCategory"
        },
        "signature": "(dimension: int) -> SteinerCategory"
      }
    ],
    "qualified_name": "SteinerCategory.globe",
    "releases_gil": false,
    "result_schema": {
      "x-python-type": "SteinerCategory"
    },
    "signature": "(dimension: int) -> SteinerCategory",
    "signatures": [
      "(dimension: int) -> SteinerCategory"
    ],
    "summary": "Construct a `SteinerCategory` using globe.",
    "summary_source": "derived projection vocabulary",
    "type_id": "noether/category/SteinerCategory@1"
  },
  {
    "canonical_name": "category.SteinerCategory.gray_tensor",
    "determinism": "deterministic",
    "doc": "",
    "domain": "category",
    "file": "noether/category/__init__.pyi",
    "input_schema": {
      "additionalProperties": false,
      "properties": {
        "other": {
          "x-python-type": "SteinerCategory"
        }
      },
      "required": [
        "other"
      ],
      "type": "object"
    },
    "kind": "method",
    "line": 235,
    "module": "noether.category",
    "name": "gray_tensor",
    "overloads": [
      {
        "input_schema": {
          "additionalProperties": false,
          "properties": {
            "other": {
              "x-python-type": "SteinerCategory"
            }
          },
          "required": [
            "other"
          ],
          "type": "object"
        },
        "parameters": [
          {
            "kind": "positional_or_keyword",
            "name": "other",
            "required": true,
            "schema": {
              "x-python-type": "SteinerCategory"
            }
          }
        ],
        "result_schema": {
          "x-python-type": "SteinerCategory"
        },
        "signature": "(self, other: SteinerCategory) -> SteinerCategory"
      }
    ],
    "qualified_name": "SteinerCategory.gray_tensor",
    "releases_gil": true,
    "result_schema": {
      "x-python-type": "SteinerCategory"
    },
    "signature": "(self, other: SteinerCategory) -> SteinerCategory",
    "signatures": [
      "(self, other: SteinerCategory) -> SteinerCategory"
    ],
    "summary": "Compute gray tensor for this `SteinerCategory`.",
    "summary_source": "derived projection vocabulary",
    "type_id": "noether/category/SteinerCategory@1"
  },
  {
    "canonical_name": "category.SteinerCategory.homotopy_poset",
    "determinism": "deterministic",
    "doc": "",
    "domain": "category",
    "file": "noether/category/__init__.pyi",
    "input_schema": {
      "additionalProperties": false,
      "properties": {
        "basepoint": {
          "x-python-type": "OrientedBasepoint"
        }
      },
      "required": [
        "basepoint"
      ],
      "type": "object"
    },
    "kind": "method",
    "line": 223,
    "module": "noether.category",
    "name": "homotopy_poset",
    "overloads": [
      {
        "input_schema": {
          "additionalProperties": false,
          "properties": {
            "basepoint": {
              "x-python-type": "OrientedBasepoint"
            }
          },
          "required": [
            "basepoint"
          ],
          "type": "object"
        },
        "parameters": [
          {
            "kind": "positional_or_keyword",
            "name": "basepoint",
            "required": true,
            "schema": {
              "x-python-type": "OrientedBasepoint"
            }
          }
        ],
        "result_schema": {
          "x-python-type": "Poset"
        },
        "signature": "(self, basepoint: OrientedBasepoint) -> Poset"
      }
    ],
    "qualified_name": "SteinerCategory.homotopy_poset",
    "releases_gil": true,
    "result_schema": {
      "x-python-type": "Poset"
    },
    "signature": "(self, basepoint: OrientedBasepoint) -> Poset",
    "signatures": [
      "(self, basepoint: OrientedBasepoint) -> Poset"
    ],
    "summary": "Compute homotopy poset for this `SteinerCategory`.",
    "summary_source": "derived projection vocabulary",
    "type_id": "noether/category/SteinerCategory@1"
  },
  {
    "canonical_name": "category.SteinerCategory.is_isomorphic",
    "determinism": "deterministic",
    "doc": "",
    "domain": "category",
    "file": "noether/category/__init__.pyi",
    "input_schema": {
      "additionalProperties": false,
      "properties": {
        "other": {
          "x-python-type": "SteinerCategory"
        }
      },
      "required": [
        "other"
      ],
      "type": "object"
    },
    "kind": "method",
    "line": 228,
    "module": "noether.category",
    "name": "is_isomorphic",
    "overloads": [
      {
        "input_schema": {
          "additionalProperties": false,
          "properties": {
            "other": {
              "x-python-type": "SteinerCategory"
            }
          },
          "required": [
            "other"
          ],
          "type": "object"
        },
        "parameters": [
          {
            "kind": "positional_or_keyword",
            "name": "other",
            "required": true,
            "schema": {
              "x-python-type": "SteinerCategory"
            }
          }
        ],
        "result_schema": {
          "type": "boolean"
        },
        "signature": "(self, other: SteinerCategory) -> bool"
      }
    ],
    "qualified_name": "SteinerCategory.is_isomorphic",
    "releases_gil": true,
    "result_schema": {
      "type": "boolean"
    },
    "signature": "(self, other: SteinerCategory) -> bool",
    "signatures": [
      "(self, other: SteinerCategory) -> bool"
    ],
    "summary": "Test whether this `SteinerCategory` is isomorphic.",
    "summary_source": "derived projection vocabulary",
    "type_id": "noether/category/SteinerCategory@1"
  },
  {
    "canonical_name": "category.SteinerCategory.mapping_category",
    "determinism": "deterministic",
    "doc": "",
    "domain": "category",
    "file": "noether/category/__init__.pyi",
    "input_schema": {
      "additionalProperties": false,
      "properties": {
        "source": {
          "x-python-type": "SteinerCell"
        },
        "target": {
          "x-python-type": "SteinerCell"
        }
      },
      "required": [
        "source",
        "target"
      ],
      "type": "object"
    },
    "kind": "method",
    "line": 208,
    "module": "noether.category",
    "name": "mapping_category",
    "overloads": [
      {
        "input_schema": {
          "additionalProperties": false,
          "properties": {
            "source": {
              "x-python-type": "SteinerCell"
            },
            "target": {
              "x-python-type": "SteinerCell"
            }
          },
          "required": [
            "source",
            "target"
          ],
          "type": "object"
        },
        "parameters": [
          {
            "kind": "positional_or_keyword",
            "name": "source",
            "required": true,
            "schema": {
              "x-python-type": "SteinerCell"
            }
          },
          {
            "kind": "positional_or_keyword",
            "name": "target",
            "required": true,
            "schema": {
              "x-python-type": "SteinerCell"
            }
          }
        ],
        "result_schema": {
          "x-python-type": "FiniteOmegaCategory"
        },
        "signature": "(self, source: SteinerCell, target: SteinerCell) -> FiniteOmegaCategory"
      }
    ],
    "qualified_name": "SteinerCategory.mapping_category",
    "releases_gil": true,
    "result_schema": {
      "x-python-type": "FiniteOmegaCategory"
    },
    "signature": "(self, source: SteinerCell, target: SteinerCell) -> FiniteOmegaCategory",
    "signatures": [
      "(self, source: SteinerCell, target: SteinerCell) -> FiniteOmegaCategory"
    ],
    "summary": "Compute mapping category for this `SteinerCategory`.",
    "summary_source": "derived projection vocabulary",
    "type_id": "noether/category/SteinerCategory@1"
  },
  {
    "canonical_name": "category.SteinerCategory.monomorphic_oriented_basepoints",
    "determinism": "deterministic",
    "doc": "",
    "domain": "category",
    "file": "noether/category/__init__.pyi",
    "input_schema": {
      "additionalProperties": false,
      "properties": {
        "dimension": {
          "type": "integer"
        }
      },
      "required": [
        "dimension"
      ],
      "type": "object"
    },
    "kind": "method",
    "line": 219,
    "module": "noether.category",
    "name": "monomorphic_oriented_basepoints",
    "overloads": [
      {
        "input_schema": {
          "additionalProperties": false,
          "properties": {
            "dimension": {
              "type": "integer"
            }
          },
          "required": [
            "dimension"
          ],
          "type": "object"
        },
        "parameters": [
          {
            "kind": "positional_or_keyword",
            "name": "dimension",
            "required": true,
            "schema": {
              "type": "integer"
            }
          }
        ],
        "result_schema": {
          "items": {
            "x-python-type": "OrientedBasepoint"
          },
          "type": "array",
          "x-python-type": "list[OrientedBasepoint]"
        },
        "signature": "(self, dimension: int) -> list[OrientedBasepoint]"
      }
    ],
    "qualified_name": "SteinerCategory.monomorphic_oriented_basepoints",
    "releases_gil": true,
    "result_schema": {
      "items": {
        "x-python-type": "OrientedBasepoint"
      },
      "type": "array",
      "x-python-type": "list[OrientedBasepoint]"
    },
    "signature": "(self, dimension: int) -> list[OrientedBasepoint]",
    "signatures": [
      "(self, dimension: int) -> list[OrientedBasepoint]"
    ],
    "summary": "Compute monomorphic oriented basepoints for this `SteinerCategory`.",
    "summary_source": "derived projection vocabulary",
    "type_id": "noether/category/SteinerCategory@1"
  },
  {
    "algorithms": [
      "Street oriental construction"
    ],
    "backends": [
      "noether"
    ],
    "canonical_name": "category.SteinerCategory.oriental",
    "determinism": "deterministic",
    "doc": "Construct the Street oriental of the requested dimension.",
    "domain": "category",
    "errors": [
      "dimension exceeds the bounded construction limit"
    ],
    "evidence": [
      "SteinerCategory.verify"
    ],
    "file": "noether/category/__init__.pyi",
    "input_schema": {
      "additionalProperties": false,
      "properties": {
        "dimension": {
          "type": "integer"
        }
      },
      "required": [
        "dimension"
      ],
      "type": "object"
    },
    "kind": "staticmethod",
    "line": 181,
    "module": "noether.category",
    "name": "oriental",
    "overloads": [
      {
        "input_schema": {
          "additionalProperties": false,
          "properties": {
            "dimension": {
              "type": "integer"
            }
          },
          "required": [
            "dimension"
          ],
          "type": "object"
        },
        "parameters": [
          {
            "kind": "positional_or_keyword",
            "name": "dimension",
            "required": true,
            "schema": {
              "type": "integer"
            }
          }
        ],
        "result_schema": {
          "x-python-type": "SteinerCategory"
        },
        "signature": "(dimension: int) -> SteinerCategory"
      }
    ],
    "qualified_name": "SteinerCategory.oriental",
    "releases_gil": false,
    "result_schema": {
      "x-python-type": "SteinerCategory"
    },
    "signature": "(dimension: int) -> SteinerCategory",
    "signatures": [
      "(dimension: int) -> SteinerCategory"
    ],
    "summary": "Construct the Street oriental of the requested dimension.",
    "summary_source": "public docstring",
    "type_id": "noether/category/SteinerCategory@1"
  },
  {
    "canonical_name": "category.SteinerCategory.point",
    "determinism": "deterministic",
    "doc": "",
    "domain": "category",
    "example": "from noether.category import SteinerCategory\npoint = SteinerCategory.point()\nassert point.dimension == 0\nassert point.atom_counts == [1]\npoint.verify()",
    "file": "noether/category/__init__.pyi",
    "input_schema": {
      "additionalProperties": false,
      "properties": {},
      "type": "object"
    },
    "kind": "staticmethod",
    "line": 177,
    "module": "noether.category",
    "name": "point",
    "overloads": [
      {
        "input_schema": {
          "additionalProperties": false,
          "properties": {},
          "type": "object"
        },
        "parameters": [],
        "result_schema": {
          "x-python-type": "SteinerCategory"
        },
        "signature": "() -> SteinerCategory"
      }
    ],
    "qualified_name": "SteinerCategory.point",
    "releases_gil": false,
    "result_schema": {
      "x-python-type": "SteinerCategory"
    },
    "signature": "() -> SteinerCategory",
    "signatures": [
      "() -> SteinerCategory"
    ],
    "summary": "Construct a `SteinerCategory` using point.",
    "summary_source": "derived projection vocabulary",
    "type_id": "noether/category/SteinerCategory@1"
  },
  {
    "canonical_name": "category.SteinerCategory.polygraphic_homology",
    "determinism": "deterministic",
    "doc": "",
    "domain": "category",
    "file": "noether/category/__init__.pyi",
    "input_schema": {
      "additionalProperties": false,
      "properties": {},
      "type": "object"
    },
    "kind": "method",
    "line": 231,
    "module": "noether.category",
    "name": "polygraphic_homology",
    "overloads": [
      {
        "input_schema": {
          "additionalProperties": false,
          "properties": {},
          "type": "object"
        },
        "parameters": [],
        "result_schema": {
          "x-python-type": "GradedHomology[IntegerRing]"
        },
        "signature": "(self) -> GradedHomology[IntegerRing]"
      }
    ],
    "qualified_name": "SteinerCategory.polygraphic_homology",
    "releases_gil": true,
    "result_schema": {
      "x-python-type": "GradedHomology[IntegerRing]"
    },
    "signature": "(self) -> GradedHomology[IntegerRing]",
    "signatures": [
      "(self) -> GradedHomology[IntegerRing]"
    ],
    "summary": "Compute polygraphic homology for this `SteinerCategory`.",
    "summary_source": "derived projection vocabulary",
    "type_id": "noether/category/SteinerCategory@1"
  },
  {
    "canonical_name": "category.SteinerCategory.polygraphic_homology_invariants",
    "determinism": "deterministic",
    "doc": "",
    "domain": "category",
    "file": "noether/category/__init__.pyi",
    "input_schema": {
      "additionalProperties": false,
      "properties": {},
      "type": "object"
    },
    "kind": "method",
    "line": 232,
    "module": "noether.category",
    "name": "polygraphic_homology_invariants",
    "overloads": [
      {
        "input_schema": {
          "additionalProperties": false,
          "properties": {},
          "type": "object"
        },
        "parameters": [],
        "result_schema": {
          "additionalProperties": {
            "x-python-type": "FinitelyGeneratedAbelianGroup"
          },
          "type": "object",
          "x-python-type": "dict[int, FinitelyGeneratedAbelianGroup]"
        },
        "signature": "(self) -> dict[int, FinitelyGeneratedAbelianGroup]"
      }
    ],
    "qualified_name": "SteinerCategory.polygraphic_homology_invariants",
    "releases_gil": true,
    "result_schema": {
      "additionalProperties": {
        "x-python-type": "FinitelyGeneratedAbelianGroup"
      },
      "type": "object",
      "x-python-type": "dict[int, FinitelyGeneratedAbelianGroup]"
    },
    "signature": "(self) -> dict[int, FinitelyGeneratedAbelianGroup]",
    "signatures": [
      "(self) -> dict[int, FinitelyGeneratedAbelianGroup]"
    ],
    "summary": "Compute polygraphic homology invariants for this `SteinerCategory`.",
    "summary_source": "derived projection vocabulary",
    "type_id": "noether/category/SteinerCategory@1"
  },
  {
    "canonical_name": "category.SteinerCategory.precedence_poset",
    "determinism": "deterministic",
    "doc": "",
    "domain": "category",
    "file": "noether/category/__init__.pyi",
    "input_schema": {
      "additionalProperties": false,
      "properties": {},
      "type": "object"
    },
    "kind": "method",
    "line": 197,
    "module": "noether.category",
    "name": "precedence_poset",
    "overloads": [
      {
        "input_schema": {
          "additionalProperties": false,
          "properties": {},
          "type": "object"
        },
        "parameters": [],
        "result_schema": {
          "x-python-type": "Poset"
        },
        "signature": "(self) -> Poset"
      }
    ],
    "qualified_name": "SteinerCategory.precedence_poset",
    "releases_gil": false,
    "result_schema": {
      "x-python-type": "Poset"
    },
    "signature": "(self) -> Poset",
    "signatures": [
      "(self) -> Poset"
    ],
    "summary": "Compute precedence poset for this `SteinerCategory`.",
    "summary_source": "derived projection vocabulary",
    "type_id": "noether/category/SteinerCategory@1"
  },
  {
    "canonical_name": "category.SteinerCategory.reduced_categorical_homology",
    "determinism": "deterministic",
    "doc": "",
    "domain": "category",
    "file": "noether/category/__init__.pyi",
    "input_schema": {
      "additionalProperties": false,
      "properties": {
        "object": {
          "x-python-type": "SteinerCell"
        }
      },
      "required": [
        "object"
      ],
      "type": "object"
    },
    "kind": "method",
    "line": 238,
    "module": "noether.category",
    "name": "reduced_categorical_homology",
    "overloads": [
      {
        "input_schema": {
          "additionalProperties": false,
          "properties": {
            "object": {
              "x-python-type": "SteinerCell"
            }
          },
          "required": [
            "object"
          ],
          "type": "object"
        },
        "parameters": [
          {
            "kind": "positional_or_keyword",
            "name": "object",
            "required": true,
            "schema": {
              "x-python-type": "SteinerCell"
            }
          }
        ],
        "result_schema": {
          "x-python-type": "CategoricalHomologyPresentation"
        },
        "signature": "(self, object: SteinerCell) -> CategoricalHomologyPresentation"
      }
    ],
    "qualified_name": "SteinerCategory.reduced_categorical_homology",
    "releases_gil": true,
    "result_schema": {
      "x-python-type": "CategoricalHomologyPresentation"
    },
    "signature": "(self, object: SteinerCell) -> CategoricalHomologyPresentation",
    "signatures": [
      "(self, object: SteinerCell) -> CategoricalHomologyPresentation"
    ],
    "summary": "Compute reduced categorical homology for this `SteinerCategory`.",
    "summary_source": "derived projection vocabulary",
    "type_id": "noether/category/SteinerCategory@1"
  },
  {
    "canonical_name": "category.SteinerCategory.street_nerve_skeleton",
    "determinism": "deterministic",
    "doc": "",
    "domain": "category",
    "file": "noether/category/__init__.pyi",
    "input_schema": {
      "additionalProperties": false,
      "properties": {
        "max_degree": {
          "type": "integer"
        }
      },
      "required": [
        "max_degree"
      ],
      "type": "object"
    },
    "kind": "method",
    "line": 236,
    "module": "noether.category",
    "name": "street_nerve_skeleton",
    "overloads": [
      {
        "input_schema": {
          "additionalProperties": false,
          "properties": {
            "max_degree": {
              "type": "integer"
            }
          },
          "required": [
            "max_degree"
          ],
          "type": "object"
        },
        "parameters": [
          {
            "kind": "positional_or_keyword",
            "name": "max_degree",
            "required": true,
            "schema": {
              "type": "integer"
            }
          }
        ],
        "result_schema": {
          "x-python-type": "StreetNerveSkeleton"
        },
        "signature": "(self, max_degree: int) -> StreetNerveSkeleton"
      }
    ],
    "qualified_name": "SteinerCategory.street_nerve_skeleton",
    "releases_gil": true,
    "result_schema": {
      "x-python-type": "StreetNerveSkeleton"
    },
    "signature": "(self, max_degree: int) -> StreetNerveSkeleton",
    "signatures": [
      "(self, max_degree: int) -> StreetNerveSkeleton"
    ],
    "summary": "Compute street nerve skeleton for this `SteinerCategory`.",
    "summary_source": "derived projection vocabulary",
    "type_id": "noether/category/SteinerCategory@1"
  },
  {
    "canonical_name": "category.SteinerCategory.to_json",
    "determinism": "deterministic",
    "doc": "",
    "domain": "category",
    "file": "noether/category/__init__.pyi",
    "input_schema": {
      "additionalProperties": false,
      "properties": {},
      "type": "object"
    },
    "kind": "method",
    "line": 187,
    "module": "noether.category",
    "name": "to_json",
    "overloads": [
      {
        "input_schema": {
          "additionalProperties": false,
          "properties": {},
          "type": "object"
        },
        "parameters": [],
        "result_schema": {
          "type": "string"
        },
        "signature": "(self) -> str"
      }
    ],
    "qualified_name": "SteinerCategory.to_json",
    "releases_gil": false,
    "result_schema": {
      "type": "string"
    },
    "signature": "(self) -> str",
    "signatures": [
      "(self) -> str"
    ],
    "summary": "Serialize this `SteinerCategory` to its canonical typed JSON representation.",
    "summary_source": "derived projection vocabulary",
    "type_id": "noether/category/SteinerCategory@1"
  },
  {
    "canonical_name": "category.SteinerCategory.truncate",
    "determinism": "deterministic",
    "doc": "",
    "domain": "category",
    "file": "noether/category/__init__.pyi",
    "input_schema": {
      "additionalProperties": false,
      "properties": {
        "dimension": {
          "type": "integer"
        }
      },
      "required": [
        "dimension"
      ],
      "type": "object"
    },
    "kind": "method",
    "line": 224,
    "module": "noether.category",
    "name": "truncate",
    "overloads": [
      {
        "input_schema": {
          "additionalProperties": false,
          "properties": {
            "dimension": {
              "type": "integer"
            }
          },
          "required": [
            "dimension"
          ],
          "type": "object"
        },
        "parameters": [
          {
            "kind": "positional_or_keyword",
            "name": "dimension",
            "required": true,
            "schema": {
              "type": "integer"
            }
          }
        ],
        "result_schema": {
          "x-python-type": "SteinerCategory"
        },
        "signature": "(self, dimension: int) -> SteinerCategory"
      }
    ],
    "qualified_name": "SteinerCategory.truncate",
    "releases_gil": true,
    "result_schema": {
      "x-python-type": "SteinerCategory"
    },
    "signature": "(self, dimension: int) -> SteinerCategory",
    "signatures": [
      "(self, dimension: int) -> SteinerCategory"
    ],
    "summary": "Compute truncate for this `SteinerCategory`.",
    "summary_source": "derived projection vocabulary",
    "type_id": "noether/category/SteinerCategory@1"
  },
  {
    "canonical_name": "category.SteinerCategory.underlying_chain_complex",
    "determinism": "deterministic",
    "doc": "",
    "domain": "category",
    "file": "noether/category/__init__.pyi",
    "input_schema": {
      "additionalProperties": false,
      "properties": {},
      "type": "object"
    },
    "kind": "method",
    "line": 230,
    "module": "noether.category",
    "name": "underlying_chain_complex",
    "overloads": [
      {
        "input_schema": {
          "additionalProperties": false,
          "properties": {},
          "type": "object"
        },
        "parameters": [],
        "result_schema": {
          "x-python-type": "ChainComplex[IntegerRing]"
        },
        "signature": "(self) -> ChainComplex[IntegerRing]"
      }
    ],
    "qualified_name": "SteinerCategory.underlying_chain_complex",
    "releases_gil": true,
    "result_schema": {
      "x-python-type": "ChainComplex[IntegerRing]"
    },
    "signature": "(self) -> ChainComplex[IntegerRing]",
    "signatures": [
      "(self) -> ChainComplex[IntegerRing]"
    ],
    "summary": "Compute underlying chain complex for this `SteinerCategory`.",
    "summary_source": "derived projection vocabulary",
    "type_id": "noether/category/SteinerCategory@1"
  },
  {
    "canonical_name": "category.SteinerCategory.verify",
    "determinism": "deterministic",
    "doc": "",
    "domain": "category",
    "file": "noether/category/__init__.pyi",
    "input_schema": {
      "additionalProperties": false,
      "properties": {},
      "type": "object"
    },
    "kind": "method",
    "line": 196,
    "module": "noether.category",
    "name": "verify",
    "overloads": [
      {
        "input_schema": {
          "additionalProperties": false,
          "properties": {},
          "type": "object"
        },
        "parameters": [],
        "result_schema": {
          "type": "null"
        },
        "signature": "(self) -> None"
      }
    ],
    "qualified_name": "SteinerCategory.verify",
    "releases_gil": false,
    "result_schema": {
      "type": "null"
    },
    "signature": "(self) -> None",
    "signatures": [
      "(self) -> None"
    ],
    "summary": "Replay and verify the mathematical evidence carried by this `SteinerCategory`.",
    "summary_source": "derived projection vocabulary",
    "type_id": "noether/category/SteinerCategory@1"
  },
  {
    "canonical_name": "category.SteinerCell",
    "determinism": "deterministic",
    "doc": "One exact cell in a Steiner-category context.",
    "domain": "category",
    "file": "noether/category/__init__.pyi",
    "input_schema": {},
    "kind": "class",
    "line": 30,
    "module": "noether.category",
    "name": "SteinerCell",
    "overloads": [],
    "qualified_name": "SteinerCell",
    "releases_gil": false,
    "result_schema": {},
    "signature": "",
    "signatures": [],
    "summary": "One exact cell in a Steiner-category context.",
    "summary_source": "public docstring"
  },
  {
    "canonical_name": "category.SteinerCell.apply_automorphism",
    "determinism": "deterministic",
    "doc": "",
    "domain": "category",
    "file": "noether/category/__init__.pyi",
    "input_schema": {
      "additionalProperties": false,
      "properties": {
        "automorphism": {
          "x-python-type": "Permutation"
        }
      },
      "required": [
        "automorphism"
      ],
      "type": "object"
    },
    "kind": "method",
    "line": 31,
    "module": "noether.category",
    "name": "apply_automorphism",
    "overloads": [
      {
        "input_schema": {
          "additionalProperties": false,
          "properties": {
            "automorphism": {
              "x-python-type": "Permutation"
            }
          },
          "required": [
            "automorphism"
          ],
          "type": "object"
        },
        "parameters": [
          {
            "kind": "positional_or_keyword",
            "name": "automorphism",
            "required": true,
            "schema": {
              "x-python-type": "Permutation"
            }
          }
        ],
        "result_schema": {
          "x-python-type": "SteinerCell"
        },
        "signature": "(self, automorphism: Permutation) -> SteinerCell"
      }
    ],
    "qualified_name": "SteinerCell.apply_automorphism",
    "releases_gil": false,
    "result_schema": {
      "x-python-type": "SteinerCell"
    },
    "signature": "(self, automorphism: Permutation) -> SteinerCell",
    "signatures": [
      "(self, automorphism: Permutation) -> SteinerCell"
    ],
    "summary": "Compute apply automorphism for this `SteinerCell`.",
    "summary_source": "derived projection vocabulary"
  },
  {
    "canonical_name": "category.SteinerCell.category",
    "determinism": "deterministic",
    "doc": "",
    "domain": "category",
    "file": "noether/category/__init__.pyi",
    "input_schema": {
      "additionalProperties": false,
      "properties": {},
      "type": "object"
    },
    "kind": "property",
    "line": 33,
    "module": "noether.category",
    "name": "category",
    "overloads": [
      {
        "input_schema": {
          "additionalProperties": false,
          "properties": {},
          "type": "object"
        },
        "parameters": [],
        "result_schema": {
          "x-python-type": "SteinerCategory"
        },
        "signature": "(self) -> SteinerCategory"
      }
    ],
    "qualified_name": "SteinerCell.category",
    "releases_gil": false,
    "result_schema": {
      "x-python-type": "SteinerCategory"
    },
    "signature": "(self) -> SteinerCategory",
    "signatures": [
      "(self) -> SteinerCategory"
    ],
    "summary": "Return the category recorded by this `SteinerCell`.",
    "summary_source": "derived projection vocabulary"
  },
  {
    "canonical_name": "category.SteinerCell.compose_at",
    "determinism": "deterministic",
    "doc": "",
    "domain": "category",
    "file": "noether/category/__init__.pyi",
    "input_schema": {
      "additionalProperties": false,
      "properties": {
        "dimension": {
          "type": "integer"
        },
        "next": {
          "x-python-type": "SteinerCell"
        }
      },
      "required": [
        "next",
        "dimension"
      ],
      "type": "object"
    },
    "kind": "method",
    "line": 53,
    "module": "noether.category",
    "name": "compose_at",
    "overloads": [
      {
        "input_schema": {
          "additionalProperties": false,
          "properties": {
            "dimension": {
              "type": "integer"
            },
            "next": {
              "x-python-type": "SteinerCell"
            }
          },
          "required": [
            "next",
            "dimension"
          ],
          "type": "object"
        },
        "parameters": [
          {
            "kind": "positional_or_keyword",
            "name": "next",
            "required": true,
            "schema": {
              "x-python-type": "SteinerCell"
            }
          },
          {
            "kind": "positional_or_keyword",
            "name": "dimension",
            "required": true,
            "schema": {
              "type": "integer"
            }
          }
        ],
        "result_schema": {
          "x-python-type": "SteinerCell"
        },
        "signature": "(self, next: SteinerCell, dimension: int) -> SteinerCell"
      }
    ],
    "qualified_name": "SteinerCell.compose_at",
    "releases_gil": false,
    "result_schema": {
      "x-python-type": "SteinerCell"
    },
    "signature": "(self, next: SteinerCell, dimension: int) -> SteinerCell",
    "signatures": [
      "(self, next: SteinerCell, dimension: int) -> SteinerCell"
    ],
    "summary": "Compute compose at for this `SteinerCell`.",
    "summary_source": "derived projection vocabulary"
  },
  {
    "canonical_name": "category.SteinerCell.dimension",
    "determinism": "deterministic",
    "doc": "",
    "domain": "category",
    "file": "noether/category/__init__.pyi",
    "input_schema": {
      "additionalProperties": false,
      "properties": {},
      "type": "object"
    },
    "kind": "property",
    "line": 35,
    "module": "noether.category",
    "name": "dimension",
    "overloads": [
      {
        "input_schema": {
          "additionalProperties": false,
          "properties": {},
          "type": "object"
        },
        "parameters": [],
        "result_schema": {
          "type": "integer"
        },
        "signature": "(self) -> int"
      }
    ],
    "qualified_name": "SteinerCell.dimension",
    "releases_gil": false,
    "result_schema": {
      "type": "integer"
    },
    "signature": "(self) -> int",
    "signatures": [
      "(self) -> int"
    ],
    "summary": "Return the dimension recorded by this `SteinerCell`.",
    "summary_source": "derived projection vocabulary"
  },
  {
    "canonical_name": "category.SteinerCell.identity",
    "determinism": "deterministic",
    "doc": "",
    "domain": "category",
    "file": "noether/category/__init__.pyi",
    "input_schema": {
      "additionalProperties": false,
      "properties": {},
      "type": "object"
    },
    "kind": "method",
    "line": 50,
    "module": "noether.category",
    "name": "identity",
    "overloads": [
      {
        "input_schema": {
          "additionalProperties": false,
          "properties": {},
          "type": "object"
        },
        "parameters": [],
        "result_schema": {
          "x-python-type": "SteinerCell"
        },
        "signature": "(self) -> SteinerCell"
      }
    ],
    "qualified_name": "SteinerCell.identity",
    "releases_gil": false,
    "result_schema": {
      "x-python-type": "SteinerCell"
    },
    "signature": "(self) -> SteinerCell",
    "signatures": [
      "(self) -> SteinerCell"
    ],
    "summary": "Compute identity for this `SteinerCell`.",
    "summary_source": "derived projection vocabulary"
  },
  {
    "canonical_name": "category.SteinerCell.intrinsic_dimension",
    "determinism": "deterministic",
    "doc": "",
    "domain": "category",
    "file": "noether/category/__init__.pyi",
    "input_schema": {
      "additionalProperties": false,
      "properties": {},
      "type": "object"
    },
    "kind": "property",
    "line": 37,
    "module": "noether.category",
    "name": "intrinsic_dimension",
    "overloads": [
      {
        "input_schema": {
          "additionalProperties": false,
          "properties": {},
          "type": "object"
        },
        "parameters": [],
        "result_schema": {
          "type": "integer"
        },
        "signature": "(self) -> int"
      }
    ],
    "qualified_name": "SteinerCell.intrinsic_dimension",
    "releases_gil": false,
    "result_schema": {
      "type": "integer"
    },
    "signature": "(self) -> int",
    "signatures": [
      "(self) -> int"
    ],
    "summary": "Return the intrinsic dimension recorded by this `SteinerCell`.",
    "summary_source": "derived projection vocabulary"
  },
  {
    "canonical_name": "category.SteinerCell.is_identity",
    "determinism": "deterministic",
    "doc": "",
    "domain": "category",
    "file": "noether/category/__init__.pyi",
    "input_schema": {
      "additionalProperties": false,
      "properties": {},
      "type": "object"
    },
    "kind": "property",
    "line": 39,
    "module": "noether.category",
    "name": "is_identity",
    "overloads": [
      {
        "input_schema": {
          "additionalProperties": false,
          "properties": {},
          "type": "object"
        },
        "parameters": [],
        "result_schema": {
          "type": "boolean"
        },
        "signature": "(self) -> bool"
      }
    ],
    "qualified_name": "SteinerCell.is_identity",
    "releases_gil": false,
    "result_schema": {
      "type": "boolean"
    },
    "signature": "(self) -> bool",
    "signatures": [
      "(self) -> bool"
    ],
    "summary": "Test whether this `SteinerCell` is identity.",
    "summary_source": "derived projection vocabulary"
  },
  {
    "canonical_name": "category.SteinerCell.negative_supports",
    "determinism": "deterministic",
    "doc": "",
    "domain": "category",
    "file": "noether/category/__init__.pyi",
    "input_schema": {
      "additionalProperties": false,
      "properties": {},
      "type": "object"
    },
    "kind": "property",
    "line": 43,
    "module": "noether.category",
    "name": "negative_supports",
    "overloads": [
      {
        "input_schema": {
          "additionalProperties": false,
          "properties": {},
          "type": "object"
        },
        "parameters": [],
        "result_schema": {
          "items": {
            "items": {
              "type": "integer"
            },
            "type": "array",
            "x-python-type": "list[int]"
          },
          "type": "array",
          "x-python-type": "list[list[int]]"
        },
        "signature": "(self) -> list[list[int]]"
      }
    ],
    "qualified_name": "SteinerCell.negative_supports",
    "releases_gil": false,
    "result_schema": {
      "items": {
        "items": {
          "type": "integer"
        },
        "type": "array",
        "x-python-type": "list[int]"
      },
      "type": "array",
      "x-python-type": "list[list[int]]"
    },
    "signature": "(self) -> list[list[int]]",
    "signatures": [
      "(self) -> list[list[int]]"
    ],
    "summary": "Return the negative supports recorded by this `SteinerCell`.",
    "summary_source": "derived projection vocabulary"
  },
  {
    "canonical_name": "category.SteinerCell.positive_supports",
    "determinism": "deterministic",
    "doc": "",
    "domain": "category",
    "file": "noether/category/__init__.pyi",
    "input_schema": {
      "additionalProperties": false,
      "properties": {},
      "type": "object"
    },
    "kind": "property",
    "line": 45,
    "module": "noether.category",
    "name": "positive_supports",
    "overloads": [
      {
        "input_schema": {
          "additionalProperties": false,
          "properties": {},
          "type": "object"
        },
        "parameters": [],
        "result_schema": {
          "items": {
            "items": {
              "type": "integer"
            },
            "type": "array",
            "x-python-type": "list[int]"
          },
          "type": "array",
          "x-python-type": "list[list[int]]"
        },
        "signature": "(self) -> list[list[int]]"
      }
    ],
    "qualified_name": "SteinerCell.positive_supports",
    "releases_gil": false,
    "result_schema": {
      "items": {
        "items": {
          "type": "integer"
        },
        "type": "array",
        "x-python-type": "list[int]"
      },
      "type": "array",
      "x-python-type": "list[list[int]]"
    },
    "signature": "(self) -> list[list[int]]",
    "signatures": [
      "(self) -> list[list[int]]"
    ],
    "summary": "Return the positive supports recorded by this `SteinerCell`.",
    "summary_source": "derived projection vocabulary"
  },
  {
    "canonical_name": "category.SteinerCell.source",
    "determinism": "deterministic",
    "doc": "",
    "domain": "category",
    "file": "noether/category/__init__.pyi",
    "input_schema": {
      "additionalProperties": false,
      "properties": {},
      "type": "object"
    },
    "kind": "property",
    "line": 47,
    "module": "noether.category",
    "name": "source",
    "overloads": [
      {
        "input_schema": {
          "additionalProperties": false,
          "properties": {},
          "type": "object"
        },
        "parameters": [],
        "result_schema": {
          "x-python-type": "SteinerCell"
        },
        "signature": "(self) -> SteinerCell"
      }
    ],
    "qualified_name": "SteinerCell.source",
    "releases_gil": false,
    "result_schema": {
      "x-python-type": "SteinerCell"
    },
    "signature": "(self) -> SteinerCell",
    "signatures": [
      "(self) -> SteinerCell"
    ],
    "summary": "Return the source recorded by this `SteinerCell`.",
    "summary_source": "derived projection vocabulary"
  },
  {
    "canonical_name": "category.SteinerCell.source_at",
    "determinism": "deterministic",
    "doc": "",
    "domain": "category",
    "file": "noether/category/__init__.pyi",
    "input_schema": {
      "additionalProperties": false,
      "properties": {
        "dimension": {
          "type": "integer"
        }
      },
      "required": [
        "dimension"
      ],
      "type": "object"
    },
    "kind": "method",
    "line": 51,
    "module": "noether.category",
    "name": "source_at",
    "overloads": [
      {
        "input_schema": {
          "additionalProperties": false,
          "properties": {
            "dimension": {
              "type": "integer"
            }
          },
          "required": [
            "dimension"
          ],
          "type": "object"
        },
        "parameters": [
          {
            "kind": "positional_or_keyword",
            "name": "dimension",
            "required": true,
            "schema": {
              "type": "integer"
            }
          }
        ],
        "result_schema": {
          "x-python-type": "SteinerCell"
        },
        "signature": "(self, dimension: int) -> SteinerCell"
      }
    ],
    "qualified_name": "SteinerCell.source_at",
    "releases_gil": false,
    "result_schema": {
      "x-python-type": "SteinerCell"
    },
    "signature": "(self, dimension: int) -> SteinerCell",
    "signatures": [
      "(self, dimension: int) -> SteinerCell"
    ],
    "summary": "Compute source at for this `SteinerCell`.",
    "summary_source": "derived projection vocabulary"
  },
  {
    "canonical_name": "category.SteinerCell.target",
    "determinism": "deterministic",
    "doc": "",
    "domain": "category",
    "file": "noether/category/__init__.pyi",
    "input_schema": {
      "additionalProperties": false,
      "properties": {},
      "type": "object"
    },
    "kind": "property",
    "line": 49,
    "module": "noether.category",
    "name": "target",
    "overloads": [
      {
        "input_schema": {
          "additionalProperties": false,
          "properties": {},
          "type": "object"
        },
        "parameters": [],
        "result_schema": {
          "x-python-type": "SteinerCell"
        },
        "signature": "(self) -> SteinerCell"
      }
    ],
    "qualified_name": "SteinerCell.target",
    "releases_gil": false,
    "result_schema": {
      "x-python-type": "SteinerCell"
    },
    "signature": "(self) -> SteinerCell",
    "signatures": [
      "(self) -> SteinerCell"
    ],
    "summary": "Return the target recorded by this `SteinerCell`.",
    "summary_source": "derived projection vocabulary"
  },
  {
    "canonical_name": "category.SteinerCell.target_at",
    "determinism": "deterministic",
    "doc": "",
    "domain": "category",
    "file": "noether/category/__init__.pyi",
    "input_schema": {
      "additionalProperties": false,
      "properties": {
        "dimension": {
          "type": "integer"
        }
      },
      "required": [
        "dimension"
      ],
      "type": "object"
    },
    "kind": "method",
    "line": 52,
    "module": "noether.category",
    "name": "target_at",
    "overloads": [
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    "domain": "category",
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    "signature": "(self) -> list[int]",
    "signatures": [
      "(self) -> list[int]"
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    "summary": "Return the top atoms recorded by this `SteinerCell`.",
    "summary_source": "derived projection vocabulary"
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    "canonical_name": "category.StreetNerveSkeleton",
    "determinism": "deterministic",
    "doc": "Certified finite Street-nerve skeleton.",
    "domain": "category",
    "file": "noether/category/__init__.pyi",
    "input_schema": {},
    "kind": "class",
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    "module": "noether.category",
    "name": "StreetNerveSkeleton",
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    "qualified_name": "StreetNerveSkeleton",
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    "signatures": [],
    "summary": "Certified finite Street-nerve skeleton.",
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    "doc": "",
    "domain": "category",
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          },
          {
            "kind": "positional_or_keyword",
            "name": "simplex",
            "required": true,
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    "signature": "(self, degree: int, simplex: int) -> list[int] | None",
    "signatures": [
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    "summary": "Compute degeneracies for this `StreetNerveSkeleton`.",
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    "canonical_name": "category.StreetNerveSkeleton.faces",
    "determinism": "deterministic",
    "doc": "",
    "domain": "category",
    "file": "noether/category/__init__.pyi",
    "input_schema": {
      "additionalProperties": false,
      "properties": {
        "degree": {
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        "simplex": {
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      },
      "required": [
        "degree",
        "simplex"
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      "type": "object"
    },
    "kind": "method",
    "line": 125,
    "module": "noether.category",
    "name": "faces",
    "overloads": [
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        "input_schema": {
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          "properties": {
            "degree": {
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          },
          "required": [
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            "simplex"
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          "type": "object"
        },
        "parameters": [
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            "kind": "positional_or_keyword",
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          },
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            "kind": "positional_or_keyword",
            "name": "simplex",
            "required": true,
            "schema": {
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              "type": "array",
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          ],
          "x-python-type": "list[int] | None"
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    ],
    "qualified_name": "StreetNerveSkeleton.faces",
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    "result_schema": {
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    "signature": "(self, degree: int, simplex: int) -> list[int] | None",
    "signatures": [
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    "summary": "Compute faces for this `StreetNerveSkeleton`.",
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    "canonical_name": "category.StreetNerveSkeleton.max_simplicial_degree",
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    "doc": "",
    "domain": "category",
    "file": "noether/category/__init__.pyi",
    "input_schema": {
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      "type": "object"
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    "qualified_name": "StreetNerveSkeleton.max_simplicial_degree",
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    "signature": "(self) -> int",
    "signatures": [
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    "summary": "Return the max simplicial degree recorded by this `StreetNerveSkeleton`.",
    "summary_source": "derived projection vocabulary"
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    "canonical_name": "category.StreetNerveSkeleton.simplex_counts",
    "determinism": "deterministic",
    "doc": "",
    "domain": "category",
    "file": "noether/category/__init__.pyi",
    "input_schema": {
      "additionalProperties": false,
      "properties": {},
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    "kind": "property",
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    "qualified_name": "StreetNerveSkeleton.simplex_counts",
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    "signature": "(self) -> list[int]",
    "signatures": [
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    "summary": "Return the simplex counts recorded by this `StreetNerveSkeleton`.",
    "summary_source": "derived projection vocabulary"
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    "canonical_name": "category.StreetNerveSkeleton.simplices",
    "determinism": "deterministic",
    "doc": "",
    "domain": "category",
    "file": "noether/category/__init__.pyi",
    "input_schema": {
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      "properties": {
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      "required": [
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    "module": "noether.category",
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        "result_schema": {
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          ],
          "x-python-type": "list[StreetSimplex] | None"
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    "qualified_name": "StreetNerveSkeleton.simplices",
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    "result_schema": {
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    "signature": "(self, degree: int) -> list[StreetSimplex] | None",
    "signatures": [
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    "summary": "Compute simplices for this `StreetNerveSkeleton`.",
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    "canonical_name": "category.StreetNerveSkeleton.source",
    "determinism": "deterministic",
    "doc": "",
    "domain": "category",
    "file": "noether/category/__init__.pyi",
    "input_schema": {
      "additionalProperties": false,
      "properties": {},
      "type": "object"
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    "kind": "property",
    "line": 117,
    "module": "noether.category",
    "name": "source",
    "overloads": [
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    "qualified_name": "StreetNerveSkeleton.source",
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    "result_schema": {
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    "signature": "(self) -> SteinerCategory",
    "signatures": [
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    "summary": "Return the source recorded by this `StreetNerveSkeleton`.",
    "summary_source": "derived projection vocabulary"
  },
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    "canonical_name": "category.StreetNerveSkeleton.thin_simplex_counts",
    "determinism": "deterministic",
    "doc": "",
    "domain": "category",
    "file": "noether/category/__init__.pyi",
    "input_schema": {
      "additionalProperties": false,
      "properties": {},
      "type": "object"
    },
    "kind": "property",
    "line": 123,
    "module": "noether.category",
    "name": "thin_simplex_counts",
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    "qualified_name": "StreetNerveSkeleton.thin_simplex_counts",
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    "result_schema": {
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    },
    "signature": "(self) -> list[int]",
    "signatures": [
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    "summary": "Return the thin simplex counts recorded by this `StreetNerveSkeleton`.",
    "summary_source": "derived projection vocabulary"
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    "canonical_name": "category.StreetNerveSkeleton.verify",
    "determinism": "deterministic",
    "doc": "",
    "domain": "category",
    "file": "noether/category/__init__.pyi",
    "input_schema": {
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      "properties": {},
      "type": "object"
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    "kind": "method",
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    "module": "noether.category",
    "name": "verify",
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    "qualified_name": "StreetNerveSkeleton.verify",
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    "signature": "(self) -> bool",
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    "summary": "Replay and verify the mathematical evidence carried by this `StreetNerveSkeleton`.",
    "summary_source": "derived projection vocabulary"
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    "canonical_name": "category.StreetSimplex",
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    "doc": "One Street-nerve simplex.",
    "domain": "category",
    "file": "noether/category/__init__.pyi",
    "input_schema": {},
    "kind": "class",
    "line": 107,
    "module": "noether.category",
    "name": "StreetSimplex",
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    "qualified_name": "StreetSimplex",
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    "result_schema": {},
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    "signatures": [],
    "summary": "One Street-nerve simplex.",
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    "canonical_name": "category.StreetSimplex.dimension",
    "determinism": "deterministic",
    "doc": "",
    "domain": "category",
    "file": "noether/category/__init__.pyi",
    "input_schema": {
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      "properties": {},
      "type": "object"
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    "kind": "property",
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    "module": "noether.category",
    "name": "dimension",
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    "qualified_name": "StreetSimplex.dimension",
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    "signature": "(self) -> int",
    "signatures": [
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    "summary": "Return the dimension recorded by this `StreetSimplex`.",
    "summary_source": "derived projection vocabulary"
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    "canonical_name": "category.StreetSimplex.images",
    "determinism": "deterministic",
    "doc": "",
    "domain": "category",
    "file": "noether/category/__init__.pyi",
    "input_schema": {
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      "properties": {},
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    "kind": "property",
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    "module": "noether.category",
    "name": "images",
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    "qualified_name": "StreetSimplex.images",
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    "signature": "(self) -> list[list[list[int]]]",
    "signatures": [
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    "summary": "Return the images recorded by this `StreetSimplex`.",
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    "canonical_name": "category.StreetSimplex.is_thin",
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    "doc": "",
    "domain": "category",
    "file": "noether/category/__init__.pyi",
    "input_schema": {
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      "properties": {},
      "type": "object"
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    "module": "noether.category",
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    "summary_source": "derived projection vocabulary"
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    "canonical_name": "combin.BeatPoint",
    "determinism": "deterministic",
    "doc": "One checked up- or down-beat-point certificate.",
    "domain": "combin",
    "file": "noether/combin/__init__.pyi",
    "input_schema": {},
    "kind": "class",
    "line": 264,
    "module": "noether.combin",
    "name": "BeatPoint",
    "overloads": [],
    "qualified_name": "BeatPoint",
    "releases_gil": false,
    "result_schema": {},
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    "summary": "One checked up- or down-beat-point certificate.",
    "summary_source": "public docstring"
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    "canonical_name": "combin.BeatPoint.element",
    "determinism": "deterministic",
    "doc": "Element which can be deleted.",
    "domain": "combin",
    "file": "noether/combin/__init__.pyi",
    "input_schema": {
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      "properties": {},
      "type": "object"
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    "kind": "property",
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    "module": "noether.combin",
    "name": "element",
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          "x-python-type": "VertexId"
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        "signature": "(self) -> VertexId"
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    "qualified_name": "BeatPoint.element",
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  {
    "canonical_name": "combin.BeatPoint.kind",
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    "doc": "`\"down\"` for a greatest strict predecessor, otherwise `\"up\"`.",
    "domain": "combin",
    "file": "noether/combin/__init__.pyi",
    "input_schema": {
      "additionalProperties": false,
      "properties": {},
      "type": "object"
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    "kind": "property",
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    "module": "noether.combin",
    "name": "kind",
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        "parameters": [],
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        "signature": "(self) -> BeatPointKind"
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    ],
    "qualified_name": "BeatPoint.kind",
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    "result_schema": {
      "x-python-type": "BeatPointKind"
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    "signature": "(self) -> BeatPointKind",
    "signatures": [
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    "summary": "`\"down\"` for a greatest strict predecessor, otherwise `\"up\"`.",
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    "canonical_name": "combin.BeatPoint.witness",
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    "doc": "Neighbor certifying the beat-point condition.",
    "domain": "combin",
    "file": "noether/combin/__init__.pyi",
    "input_schema": {
      "additionalProperties": false,
      "properties": {},
      "type": "object"
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    "kind": "property",
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    "domain": "combin",
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  },
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    "doc": "The degree of each vertex, sorted descending (undirected; a self-loop\nwould contribute 2, but simple graphs have none).",
    "domain": "combin",
    "file": "noether/combin/__init__.pyi",
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    "summary_source": "public docstring",
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    "domain": "combin",
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    "summary": "The number of vertices.",
    "summary_source": "public docstring",
    "type_id": "noether/combin/Graph@1"
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    "canonical_name": "combin.Graph.reduced_laplacian",
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          "type": "object"
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        ],
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    "signatures": [
      "(self, delete_vertex: VertexId=0) -> list[list[int]]"
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    "summary": "The reduced Laplacian: the Laplacian with the row and column of `delete_vertex` removed.",
    "summary_source": "public docstring",
    "type_id": "noether/combin/Graph@1"
  },
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    "canonical_name": "combin.Graph.shortest_path",
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    "domain": "combin",
    "file": "noether/combin/__init__.pyi",
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      "properties": {
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          "x-python-type": "VertexId"
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    "module": "noether.combin",
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    "file": "noether/combin/__init__.pyi",
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          "required": [
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    "domain": "combin",
    "file": "noether/combin/__init__.pyi",
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    "summary_source": "public docstring",
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    "canonical_name": "combin.GraphLabeling",
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    "file": "noether/combin/__init__.pyi",
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        "graph": {
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          "items": {},
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    "line": 209,
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    "doc": "The label of edge `e` (by edge index), or raises `IndexError`.",
    "domain": "combin",
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    "qualified_name": "GraphLabeling.edge_label",
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    "domain": "combin",
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      "additionalProperties": false,
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    "signatures": [
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    "signatures": [
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    "summary": "The labeling carried across [`Graph.subgraph`].",
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    "canonical_name": "combin.GraphLabeling.transpose",
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    "domain": "combin",
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    "canonical_name": "combin.Poset",
    "determinism": "deterministic",
    "doc": "The thin Python handle around Noether's validated Rust [`Poset`].\n\nConstruction accepts strict generating relations on `0..n_elements`.\nRedundant transitive relations and duplicates are removed, so `covers` is\nalways the directed Hasse graph. Python stores no parallel order relation or\ncaller-supplied claims about lattice or graded structure.\n\n```python\nfrom noether.combin import Poset\n\ndiamond = Poset(4, [(0, 1), (0, 2), (1, 3), (2, 3)])\nassert diamond.leq(0, 3)\nassert diamond.covers.edge_count == 4\n```",
    "domain": "combin",
    "example": "from noether.combin import Poset\ndiamond = Poset(4, [(0, 1), (0, 2), (1, 3), (2, 3)])\nassert diamond.leq(0, 3)\nassert diamond.covers.edge_count == 4",
    "file": "noether/combin/__init__.pyi",
    "input_schema": {
      "additionalProperties": false,
      "properties": {
        "n_elements": {
          "type": "integer"
        },
        "relations": {
          "items": {
            "items": {
              "type": "integer"
            },
            "type": "array",
            "x-python-type": "Sequence[int]"
          },
          "type": "array",
          "x-python-type": "Sequence[Sequence[int]]"
        }
      },
      "required": [
        "n_elements",
        "relations"
      ],
      "type": "object"
    },
    "kind": "class",
    "line": 297,
    "module": "noether.combin",
    "name": "Poset",
    "overloads": [
      {
        "input_schema": {
          "additionalProperties": false,
          "properties": {
            "n_elements": {
              "type": "integer"
            },
            "relations": {
              "items": {
                "items": {
                  "type": "integer"
                },
                "type": "array",
                "x-python-type": "Sequence[int]"
              },
              "type": "array",
              "x-python-type": "Sequence[Sequence[int]]"
            }
          },
          "required": [
            "n_elements",
            "relations"
          ],
          "type": "object"
        },
        "parameters": [
          {
            "kind": "positional_or_keyword",
            "name": "n_elements",
            "required": true,
            "schema": {
              "type": "integer"
            }
          },
          {
            "kind": "positional_or_keyword",
            "name": "relations",
            "required": true,
            "schema": {
              "items": {
                "items": {
                  "type": "integer"
                },
                "type": "array",
                "x-python-type": "Sequence[int]"
              },
              "type": "array",
              "x-python-type": "Sequence[Sequence[int]]"
            }
          }
        ],
        "result_schema": {
          "type": "null"
        },
        "signature": "(self, n_elements: int, relations: Sequence[Sequence[int]]) -> None"
      }
    ],
    "qualified_name": "Poset",
    "releases_gil": false,
    "result_schema": {
      "type": "null"
    },
    "signature": "(self, n_elements: int, relations: Sequence[Sequence[int]]) -> None",
    "signatures": [
      "(self, n_elements: int, relations: Sequence[Sequence[int]]) -> None"
    ],
    "summary": "The thin Python handle around Noether's validated Rust [`Poset`].",
    "summary_source": "public docstring",
    "type_id": "noether/combin/Poset@1"
  },
  {
    "canonical_name": "combin.Poset.__eq__",
    "determinism": "deterministic",
    "doc": "",
    "domain": "combin",
    "file": "noether/combin/__init__.pyi",
    "input_schema": {
      "additionalProperties": false,
      "properties": {
        "other": {}
      },
      "required": [
        "other"
      ],
      "type": "object"
    },
    "kind": "method",
    "line": 331,
    "module": "noether.combin",
    "name": "__eq__",
    "overloads": [
      {
        "input_schema": {
          "additionalProperties": false,
          "properties": {
            "other": {}
          },
          "required": [
            "other"
          ],
          "type": "object"
        },
        "parameters": [
          {
            "kind": "positional_or_keyword",
            "name": "other",
            "required": true,
            "schema": {}
          }
        ],
        "result_schema": {
          "type": "boolean"
        },
        "signature": "(self, other: object) -> bool"
      }
    ],
    "qualified_name": "Poset.__eq__",
    "releases_gil": false,
    "result_schema": {
      "type": "boolean"
    },
    "signature": "(self, other: object) -> bool",
    "signatures": [
      "(self, other: object) -> bool"
    ],
    "summary": "Test mathematical equality with another value.",
    "summary_source": "derived projection vocabulary",
    "type_id": "noether/combin/Poset@1"
  },
  {
    "canonical_name": "combin.Poset.automorphism_group",
    "determinism": "deterministic",
    "doc": "Compute the exact automorphism group on the element ids.",
    "domain": "combin",
    "file": "noether/combin/__init__.pyi",
    "input_schema": {
      "additionalProperties": false,
      "properties": {},
      "type": "object"
    },
    "kind": "method",
    "line": 314,
    "module": "noether.combin",
    "name": "automorphism_group",
    "overloads": [
      {
        "input_schema": {
          "additionalProperties": false,
          "properties": {},
          "type": "object"
        },
        "parameters": [],
        "result_schema": {
          "x-python-type": "PermutationGroup"
        },
        "signature": "(self) -> PermutationGroup"
      }
    ],
    "qualified_name": "Poset.automorphism_group",
    "releases_gil": true,
    "result_schema": {
      "x-python-type": "PermutationGroup"
    },
    "signature": "(self) -> PermutationGroup",
    "signatures": [
      "(self) -> PermutationGroup"
    ],
    "summary": "Compute the exact automorphism group on the element ids.",
    "summary_source": "public docstring",
    "type_id": "noether/combin/Poset@1"
  },
  {
    "canonical_name": "combin.Poset.beat_points",
    "determinism": "deterministic",
    "doc": "Current up- and down-beat-point certificates.",
    "domain": "combin",
    "file": "noether/combin/__init__.pyi",
    "input_schema": {
      "additionalProperties": false,
      "properties": {},
      "type": "object"
    },
    "kind": "method",
    "line": 326,
    "module": "noether.combin",
    "name": "beat_points",
    "overloads": [
      {
        "input_schema": {
          "additionalProperties": false,
          "properties": {},
          "type": "object"
        },
        "parameters": [],
        "result_schema": {
          "items": {
            "x-python-type": "BeatPoint"
          },
          "type": "array",
          "x-python-type": "list[BeatPoint]"
        },
        "signature": "(self) -> list[BeatPoint]"
      }
    ],
    "qualified_name": "Poset.beat_points",
    "releases_gil": false,
    "result_schema": {
      "items": {
        "x-python-type": "BeatPoint"
      },
      "type": "array",
      "x-python-type": "list[BeatPoint]"
    },
    "signature": "(self) -> list[BeatPoint]",
    "signatures": [
      "(self) -> list[BeatPoint]"
    ],
    "summary": "Current up- and down-beat-point certificates.",
    "summary_source": "public docstring",
    "type_id": "noether/combin/Poset@1"
  },
  {
    "algorithms": [
      "principal_downset_canonicalization"
    ],
    "canonical_name": "combin.Poset.canonical_form",
    "determinism": "deterministic",
    "doc": "Return the representative minimizing the principal-downset tuple.",
    "domain": "combin",
    "example": "from noether.combin import Poset\nposet = Poset(3, [(2, 1), (1, 0)])\nresult = poset.canonical_form()\nassert result.to_json() == '{\"order\":3,\"covers\":[[0,1],[1,2]]}'",
    "file": "noether/combin/__init__.pyi",
    "input_schema": {
      "additionalProperties": false,
      "properties": {},
      "type": "object"
    },
    "kind": "method",
    "line": 313,
    "module": "noether.combin",
    "name": "canonical_form",
    "overloads": [
      {
        "input_schema": {
          "additionalProperties": false,
          "properties": {},
          "type": "object"
        },
        "parameters": [],
        "result_schema": {
          "x-python-type": "Poset"
        },
        "signature": "(self) -> Poset"
      }
    ],
    "qualified_name": "Poset.canonical_form",
    "releases_gil": true,
    "result_schema": {
      "x-python-type": "Poset"
    },
    "signature": "(self) -> Poset",
    "signatures": [
      "(self) -> Poset"
    ],
    "summary": "Return the representative minimizing the principal-downset tuple.",
    "summary_source": "public docstring",
    "type_id": "noether/combin/Poset@1"
  },
  {
    "canonical_name": "combin.Poset.canonical_label",
    "determinism": "deterministic",
    "doc": "Compute the exact labeling minimizing the principal-downset tuple.",
    "domain": "combin",
    "file": "noether/combin/__init__.pyi",
    "input_schema": {
      "additionalProperties": false,
      "properties": {},
      "type": "object"
    },
    "kind": "method",
    "line": 312,
    "module": "noether.combin",
    "name": "canonical_label",
    "overloads": [
      {
        "input_schema": {
          "additionalProperties": false,
          "properties": {},
          "type": "object"
        },
        "parameters": [],
        "result_schema": {
          "x-python-type": "PosetCanonicalLabel"
        },
        "signature": "(self) -> PosetCanonicalLabel"
      }
    ],
    "qualified_name": "Poset.canonical_label",
    "releases_gil": true,
    "result_schema": {
      "x-python-type": "PosetCanonicalLabel"
    },
    "signature": "(self) -> PosetCanonicalLabel",
    "signatures": [
      "(self) -> PosetCanonicalLabel"
    ],
    "summary": "Compute the exact labeling minimizing the principal-downset tuple.",
    "summary_source": "public docstring",
    "type_id": "noether/combin/Poset@1"
  },
  {
    "canonical_name": "combin.Poset.comparable_pair_count",
    "determinism": "deterministic",
    "doc": "Number of strict comparable pairs.",
    "domain": "combin",
    "file": "noether/combin/__init__.pyi",
    "input_schema": {
      "additionalProperties": false,
      "properties": {},
      "type": "object"
    },
    "kind": "method",
    "line": 319,
    "module": "noether.combin",
    "name": "comparable_pair_count",
    "overloads": [
      {
        "input_schema": {
          "additionalProperties": false,
          "properties": {},
          "type": "object"
        },
        "parameters": [],
        "result_schema": {
          "type": "integer"
        },
        "signature": "(self) -> int"
      }
    ],
    "qualified_name": "Poset.comparable_pair_count",
    "releases_gil": false,
    "result_schema": {
      "type": "integer"
    },
    "signature": "(self) -> int",
    "signatures": [
      "(self) -> int"
    ],
    "summary": "Number of strict comparable pairs.",
    "summary_source": "public docstring",
    "type_id": "noether/combin/Poset@1"
  },
  {
    "canonical_name": "combin.Poset.connected_components",
    "determinism": "deterministic",
    "doc": "Weakly connected components of the Hasse diagram.",
    "domain": "combin",
    "file": "noether/combin/__init__.pyi",
    "input_schema": {
      "additionalProperties": false,
      "properties": {},
      "type": "object"
    },
    "kind": "method",
    "line": 320,
    "module": "noether.combin",
    "name": "connected_components",
    "overloads": [
      {
        "input_schema": {
          "additionalProperties": false,
          "properties": {},
          "type": "object"
        },
        "parameters": [],
        "result_schema": {
          "items": {
            "items": {
              "x-python-type": "VertexId"
            },
            "type": "array",
            "x-python-type": "list[VertexId]"
          },
          "type": "array",
          "x-python-type": "list[list[VertexId]]"
        },
        "signature": "(self) -> list[list[VertexId]]"
      }
    ],
    "qualified_name": "Poset.connected_components",
    "releases_gil": false,
    "result_schema": {
      "items": {
        "items": {
          "x-python-type": "VertexId"
        },
        "type": "array",
        "x-python-type": "list[VertexId]"
      },
      "type": "array",
      "x-python-type": "list[list[VertexId]]"
    },
    "signature": "(self) -> list[list[VertexId]]",
    "signatures": [
      "(self) -> list[list[VertexId]]"
    ],
    "summary": "Weakly connected components of the Hasse diagram.",
    "summary_source": "public docstring",
    "type_id": "noether/combin/Poset@1"
  },
  {
    "canonical_name": "combin.Poset.core",
    "determinism": "deterministic",
    "doc": "Delete beat points and return replayable reduction evidence.",
    "domain": "combin",
    "file": "noether/combin/__init__.pyi",
    "input_schema": {
      "additionalProperties": false,
      "properties": {},
      "type": "object"
    },
    "kind": "method",
    "line": 328,
    "module": "noether.combin",
    "name": "core",
    "overloads": [
      {
        "input_schema": {
          "additionalProperties": false,
          "properties": {},
          "type": "object"
        },
        "parameters": [],
        "result_schema": {
          "x-python-type": "CoreReduction"
        },
        "signature": "(self) -> CoreReduction"
      }
    ],
    "qualified_name": "Poset.core",
    "releases_gil": true,
    "result_schema": {
      "x-python-type": "CoreReduction"
    },
    "signature": "(self) -> CoreReduction",
    "signatures": [
      "(self) -> CoreReduction"
    ],
    "summary": "Delete beat points and return replayable reduction evidence.",
    "summary_source": "public docstring",
    "type_id": "noether/combin/Poset@1"
  },
  {
    "canonical_name": "combin.Poset.covers",
    "determinism": "deterministic",
    "doc": "The directed Hasse diagram as the public graph type.",
    "domain": "combin",
    "file": "noether/combin/__init__.pyi",
    "input_schema": {
      "additionalProperties": false,
      "properties": {},
      "type": "object"
    },
    "kind": "property",
    "line": 310,
    "module": "noether.combin",
    "name": "covers",
    "overloads": [
      {
        "input_schema": {
          "additionalProperties": false,
          "properties": {},
          "type": "object"
        },
        "parameters": [],
        "result_schema": {
          "x-python-type": "Graph"
        },
        "signature": "(self) -> Graph"
      }
    ],
    "qualified_name": "Poset.covers",
    "releases_gil": false,
    "result_schema": {
      "x-python-type": "Graph"
    },
    "signature": "(self) -> Graph",
    "signatures": [
      "(self) -> Graph"
    ],
    "summary": "The directed Hasse diagram as the public graph type.",
    "summary_source": "public docstring",
    "type_id": "noether/combin/Poset@1"
  },
  {
    "canonical_name": "combin.Poset.from_json",
    "determinism": "deterministic",
    "doc": "Deserialize and revalidate a finite poset.",
    "domain": "combin",
    "file": "noether/combin/__init__.pyi",
    "input_schema": {
      "additionalProperties": false,
      "properties": {
        "value": {
          "type": "string"
        }
      },
      "required": [
        "value"
      ],
      "type": "object"
    },
    "kind": "staticmethod",
    "line": 305,
    "module": "noether.combin",
    "name": "from_json",
    "overloads": [
      {
        "input_schema": {
          "additionalProperties": false,
          "properties": {
            "value": {
              "type": "string"
            }
          },
          "required": [
            "value"
          ],
          "type": "object"
        },
        "parameters": [
          {
            "kind": "positional_or_keyword",
            "name": "value",
            "required": true,
            "schema": {
              "type": "string"
            }
          }
        ],
        "result_schema": {
          "x-python-type": "Poset"
        },
        "signature": "(value: str) -> Poset"
      }
    ],
    "qualified_name": "Poset.from_json",
    "releases_gil": false,
    "result_schema": {
      "x-python-type": "Poset"
    },
    "signature": "(value: str) -> Poset",
    "signatures": [
      "(value: str) -> Poset"
    ],
    "summary": "Deserialize and revalidate a finite poset.",
    "summary_source": "public docstring",
    "type_id": "noether/combin/Poset@1"
  },
  {
    "canonical_name": "combin.Poset.graded_homology_invariants",
    "determinism": "deterministic",
    "doc": "Compute lean integral homology after beat-point reduction.",
    "domain": "combin",
    "file": "noether/combin/__init__.pyi",
    "input_schema": {
      "additionalProperties": false,
      "properties": {},
      "type": "object"
    },
    "kind": "method",
    "line": 329,
    "module": "noether.combin",
    "name": "graded_homology_invariants",
    "overloads": [
      {
        "input_schema": {
          "additionalProperties": false,
          "properties": {},
          "type": "object"
        },
        "parameters": [],
        "result_schema": {
          "items": {
            "x-python-type": "FinitelyGeneratedAbelianGroup"
          },
          "type": "array",
          "x-python-type": "list[FinitelyGeneratedAbelianGroup]"
        },
        "signature": "(self) -> list[FinitelyGeneratedAbelianGroup]"
      }
    ],
    "qualified_name": "Poset.graded_homology_invariants",
    "releases_gil": true,
    "result_schema": {
      "items": {
        "x-python-type": "FinitelyGeneratedAbelianGroup"
      },
      "type": "array",
      "x-python-type": "list[FinitelyGeneratedAbelianGroup]"
    },
    "signature": "(self) -> list[FinitelyGeneratedAbelianGroup]",
    "signatures": [
      "(self) -> list[FinitelyGeneratedAbelianGroup]"
    ],
    "summary": "Compute lean integral homology after beat-point reduction.",
    "summary_source": "public docstring",
    "type_id": "noether/combin/Poset@1"
  },
  {
    "canonical_name": "combin.Poset.height",
    "determinism": "deterministic",
    "doc": "Number of elements in a longest chain.",
    "domain": "combin",
    "file": "noether/combin/__init__.pyi",
    "input_schema": {
      "additionalProperties": false,
      "properties": {},
      "type": "object"
    },
    "kind": "method",
    "line": 321,
    "module": "noether.combin",
    "name": "height",
    "overloads": [
      {
        "input_schema": {
          "additionalProperties": false,
          "properties": {},
          "type": "object"
        },
        "parameters": [],
        "result_schema": {
          "type": "integer"
        },
        "signature": "(self) -> int"
      }
    ],
    "qualified_name": "Poset.height",
    "releases_gil": false,
    "result_schema": {
      "type": "integer"
    },
    "signature": "(self) -> int",
    "signatures": [
      "(self) -> int"
    ],
    "summary": "Number of elements in a longest chain.",
    "summary_source": "public docstring",
    "type_id": "noether/combin/Poset@1"
  },
  {
    "canonical_name": "combin.Poset.is_core",
    "determinism": "deterministic",
    "doc": "Whether this poset has no beat points.",
    "domain": "combin",
    "file": "noether/combin/__init__.pyi",
    "input_schema": {
      "additionalProperties": false,
      "properties": {},
      "type": "object"
    },
    "kind": "method",
    "line": 327,
    "module": "noether.combin",
    "name": "is_core",
    "overloads": [
      {
        "input_schema": {
          "additionalProperties": false,
          "properties": {},
          "type": "object"
        },
        "parameters": [],
        "result_schema": {
          "type": "boolean"
        },
        "signature": "(self) -> bool"
      }
    ],
    "qualified_name": "Poset.is_core",
    "releases_gil": false,
    "result_schema": {
      "type": "boolean"
    },
    "signature": "(self) -> bool",
    "signatures": [
      "(self) -> bool"
    ],
    "summary": "Whether this poset has no beat points.",
    "summary_source": "public docstring",
    "type_id": "noether/combin/Poset@1"
  },
  {
    "canonical_name": "combin.Poset.is_lattice",
    "determinism": "deterministic",
    "doc": "Whether every pair has a meet and a join.",
    "domain": "combin",
    "file": "noether/combin/__init__.pyi",
    "input_schema": {
      "additionalProperties": false,
      "properties": {},
      "type": "object"
    },
    "kind": "method",
    "line": 324,
    "module": "noether.combin",
    "name": "is_lattice",
    "overloads": [
      {
        "input_schema": {
          "additionalProperties": false,
          "properties": {},
          "type": "object"
        },
        "parameters": [],
        "result_schema": {
          "type": "boolean"
        },
        "signature": "(self) -> bool"
      }
    ],
    "qualified_name": "Poset.is_lattice",
    "releases_gil": true,
    "result_schema": {
      "type": "boolean"
    },
    "signature": "(self) -> bool",
    "signatures": [
      "(self) -> bool"
    ],
    "summary": "Whether every pair has a meet and a join.",
    "summary_source": "public docstring",
    "type_id": "noether/combin/Poset@1"
  },
  {
    "canonical_name": "combin.Poset.is_self_dual",
    "determinism": "deterministic",
    "doc": "Whether this poset is isomorphic to its opposite.",
    "domain": "combin",
    "file": "noether/combin/__init__.pyi",
    "input_schema": {
      "additionalProperties": false,
      "properties": {},
      "type": "object"
    },
    "kind": "method",
    "line": 325,
    "module": "noether.combin",
    "name": "is_self_dual",
    "overloads": [
      {
        "input_schema": {
          "additionalProperties": false,
          "properties": {},
          "type": "object"
        },
        "parameters": [],
        "result_schema": {
          "type": "boolean"
        },
        "signature": "(self) -> bool"
      }
    ],
    "qualified_name": "Poset.is_self_dual",
    "releases_gil": true,
    "result_schema": {
      "type": "boolean"
    },
    "signature": "(self) -> bool",
    "signatures": [
      "(self) -> bool"
    ],
    "summary": "Whether this poset is isomorphic to its opposite.",
    "summary_source": "public docstring",
    "type_id": "noether/combin/Poset@1"
  },
  {
    "canonical_name": "combin.Poset.leq",
    "determinism": "deterministic",
    "doc": "Return whether `lower <= upper`.",
    "domain": "combin",
    "file": "noether/combin/__init__.pyi",
    "input_schema": {
      "additionalProperties": false,
      "properties": {
        "lower": {
          "x-python-type": "VertexId"
        },
        "upper": {
          "x-python-type": "VertexId"
        }
      },
      "required": [
        "lower",
        "upper"
      ],
      "type": "object"
    },
    "kind": "method",
    "line": 311,
    "module": "noether.combin",
    "name": "leq",
    "overloads": [
      {
        "input_schema": {
          "additionalProperties": false,
          "properties": {
            "lower": {
              "x-python-type": "VertexId"
            },
            "upper": {
              "x-python-type": "VertexId"
            }
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    "qualified_name": "Tree.centroid_decomposition",
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    "signatures": [
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    "summary": "The recursive **centroid decomposition** as a `dict`: `{\"root\": int, \"parent\": list[int | None]}`, where `parent[v]` is `v`'s parent in the centroid tree (`None` for the decomposition root).",
    "summary_source": "public docstring",
    "type_id": "noether/combin/Tree@1"
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    "canonical_name": "combin.Tree.depth",
    "determinism": "deterministic",
    "doc": "`v`'s depth — edges from the root to `v` (the root has depth `0`).\nRaises `IndexError` if `v >= order`.",
    "domain": "combin",
    "file": "noether/combin/__init__.pyi",
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    "summary": "`v`'s depth — edges from the root to `v` (the root has depth `0`).",
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    "type_id": "noether/combin/Tree@1"
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    "canonical_name": "combin.Tree.diameter",
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    "domain": "combin",
    "file": "noether/combin/__init__.pyi",
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    "summary_source": "public docstring",
    "type_id": "noether/combin/Tree@1"
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    "canonical_name": "combin.Tree.distance",
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    "domain": "combin",
    "file": "noether/combin/__init__.pyi",
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    "name": "distance",
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    "summary": "The tree distance between `u` and `v` — `depth(u) + depth(v) − 2·depth(lca(u, v))`.",
    "summary_source": "public docstring",
    "type_id": "noether/combin/Tree@1"
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    "canonical_name": "combin.Tree.edges",
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    "domain": "combin",
    "file": "noether/combin/__init__.pyi",
    "input_schema": {
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    "domain": "combin",
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    "type_id": "noether/combin/Tree@1"
  },
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    "doc": "The **heavy-light decomposition** structure as a `dict`:\n`{\"head\": list[int], \"position\": list[int], \"chain\": list[int]}`,\nindexed by vertex id (`head[v]` the chain head, `position[v]` the\nlinearised HLD position, `chain[v]` the chain id).",
    "domain": "combin",
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    "input_schema": {
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      "properties": {},
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    "summary": "The **heavy-light decomposition** structure as a `dict`: `{\"head\": list[int], \"position\": list[int], \"chain\": list[int]}`, indexed by vertex id (`head[v]` the chain head, `position[v]` the linearised HLD position, `chain[v]` the chain id).",
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    "canonical_name": "combin.Tree.hld_path_intervals",
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    "file": "noether/combin/__init__.pyi",
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    "line": 253,
    "module": "noether.combin",
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    "summary": "`True` if `a` is an ancestor of `d` (a vertex is its own ancestor).",
    "summary_source": "public docstring",
    "type_id": "noether/combin/Tree@1"
  },
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    "canonical_name": "combin.Tree.kth_ancestor",
    "determinism": "deterministic",
    "doc": "`v`'s `k`-th ancestor (parent for `k == 1`, `v` itself for `k == 0`),\nor `None` if `k` exceeds `v`'s depth. `O(log n)` (binary lifting).\nRaises `IndexError` if `v >= order`.",
    "domain": "combin",
    "file": "noether/combin/__init__.pyi",
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    "module": "noether.combin",
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    "summary": "`v`'s `k`-th ancestor (parent for `k == 1`, `v` itself for `k == 0`), or `None` if `k` exceeds `v`'s depth.",
    "summary_source": "public docstring",
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    "canonical_name": "combin.Tree.lca",
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    "doc": "The lowest common ancestor of `u` and `v` — `O(1)` (Euler-tour\nsparse-table RMQ). Raises `IndexError` if `u` or `v >= order`.",
    "domain": "combin",
    "file": "noether/combin/__init__.pyi",
    "input_schema": {
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    "line": 250,
    "module": "noether.combin",
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    "summary": "The lowest common ancestor of `u` and `v` — `O(1)` (Euler-tour sparse-table RMQ).",
    "summary_source": "public docstring",
    "type_id": "noether/combin/Tree@1"
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    "canonical_name": "combin.Tree.order",
    "determinism": "deterministic",
    "doc": "The number of vertices. Valid vertex ids are `0..order`.",
    "domain": "combin",
    "file": "noether/combin/__init__.pyi",
    "input_schema": {
      "additionalProperties": false,
      "properties": {},
      "type": "object"
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    "kind": "property",
    "line": 243,
    "module": "noether.combin",
    "name": "order",
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    "summary": "The number of vertices.",
    "summary_source": "public docstring",
    "type_id": "noether/combin/Tree@1"
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    "canonical_name": "combin.Tree.root",
    "determinism": "deterministic",
    "doc": "The root vertex.",
    "domain": "combin",
    "file": "noether/combin/__init__.pyi",
    "input_schema": {
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      "properties": {},
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    "kind": "property",
    "line": 245,
    "module": "noether.combin",
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    "summary": "The root vertex.",
    "summary_source": "public docstring",
    "type_id": "noether/combin/Tree@1"
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    "canonical_name": "combin.Tree.subtree",
    "determinism": "deterministic",
    "doc": "`v`'s subtree as a preorder `list[int]` (it begins with `v`). `O(1)` to\nobtain; length `subtree_size(v)`. Raises `IndexError` if `v >= order`.",
    "domain": "combin",
    "file": "noether/combin/__init__.pyi",
    "input_schema": {
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      "properties": {
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    "module": "noether.combin",
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    "qualified_name": "Tree.subtree",
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    "signatures": [
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    "summary": "`v`'s subtree as a preorder `list[int]` (it begins with `v`).",
    "summary_source": "public docstring",
    "type_id": "noether/combin/Tree@1"
  },
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    "canonical_name": "combin.Tree.subtree_size",
    "determinism": "deterministic",
    "doc": "The number of vertices in `v`'s subtree (including `v`). `O(1)`.\nRaises `IndexError` if `v >= order`.",
    "domain": "combin",
    "file": "noether/combin/__init__.pyi",
    "input_schema": {
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      "properties": {
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    "line": 254,
    "module": "noether.combin",
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    "result_schema": {
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    "signatures": [
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    "summary": "The number of vertices in `v`'s subtree (including `v`).",
    "summary_source": "public docstring",
    "type_id": "noether/combin/Tree@1"
  },
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    "canonical_name": "combin.Tree.to_json",
    "determinism": "deterministic",
    "doc": "Serialize the tree's validated defining content.",
    "domain": "combin",
    "file": "noether/combin/__init__.pyi",
    "input_schema": {
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      "properties": {},
      "type": "object"
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    "kind": "method",
    "line": 241,
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    "qualified_name": "Tree.to_json",
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    "result_schema": {
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    "signature": "(self) -> str",
    "signatures": [
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    "summary": "Serialize the tree's validated defining content.",
    "summary_source": "public docstring",
    "type_id": "noether/combin/Tree@1"
  },
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    "canonical_name": "combin.Tree.tree_dp",
    "determinism": "deterministic",
    "doc": "The canonical **tree DP**: the post-order subtree fold instantiated as\nthe subtree-size dynamic program — `result[v] == subtree_size(v)` for\nevery vertex, computed by the deep-safe (explicit-stack, no recursion)\nreverse-preorder fold. Returns a `list[int]` indexed by vertex id.\n\nThis surfaces the kernel's generic `Tree::fold` through the canonical\nDP; the closure (`leaf = 1`, `merge = +`) is fixed\nbecause Python closures cannot cross the kernel's `Fn` boundary safely.",
    "domain": "combin",
    "file": "noether/combin/__init__.pyi",
    "input_schema": {
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      "properties": {},
      "type": "object"
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    "kind": "method",
    "line": 261,
    "module": "noether.combin",
    "name": "tree_dp",
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          "properties": {},
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    "qualified_name": "Tree.tree_dp",
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      "x-python-type": "list[int]"
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    "signature": "(self) -> list[int]",
    "signatures": [
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    "summary": "The canonical **tree DP**: the post-order subtree fold instantiated as the subtree-size dynamic program — `result[v] == subtree_size(v)` for every vertex, computed by the deep-safe (explicit-stack, no recursion) reverse-preorder fold.",
    "summary_source": "public docstring",
    "type_id": "noether/combin/Tree@1"
  },
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    "canonical_name": "commalg.AffineNormalization",
    "determinism": "deterministic",
    "doc": "A verified finite birational normalization and its conductor.",
    "domain": "commalg",
    "file": "noether/commalg/__init__.pyi",
    "input_schema": {},
    "kind": "class",
    "line": 1034,
    "module": "noether.commalg",
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    "qualified_name": "AffineNormalization",
    "releases_gil": false,
    "result_schema": {},
    "signature": "",
    "signatures": [],
    "summary": "A verified finite birational normalization and its conductor.",
    "summary_source": "public docstring",
    "type_id": "noether/commalg/AffineNormalization@1"
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    "canonical_name": "commalg.AffineNormalization.conductor_theorem",
    "determinism": "deterministic",
    "doc": "",
    "domain": "commalg",
    "file": "noether/commalg/__init__.pyi",
    "input_schema": {
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      "properties": {},
      "type": "object"
    },
    "kind": "property",
    "line": 1073,
    "module": "noether.commalg",
    "name": "conductor_theorem",
    "overloads": [
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          "properties": {},
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        "signature": "(self) -> str"
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    "qualified_name": "AffineNormalization.conductor_theorem",
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    "result_schema": {
      "type": "string"
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    "signature": "(self) -> str",
    "signatures": [
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    "summary": "Return the conductor theorem recorded by this `AffineNormalization`.",
    "summary_source": "derived projection vocabulary",
    "type_id": "noether/commalg/AffineNormalization@1"
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    "canonical_name": "commalg.AffineNormalization.coordinate_images",
    "determinism": "deterministic",
    "doc": "",
    "domain": "commalg",
    "file": "noether/commalg/__init__.pyi",
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      "properties": {},
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    "line": 1053,
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          "type": "object"
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        "signature": "(self) -> list[Polynomial[C]]"
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    "qualified_name": "AffineNormalization.coordinate_images",
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    "result_schema": {
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    },
    "signature": "(self) -> list[Polynomial[C]]",
    "signatures": [
      "(self) -> list[Polynomial[C]]"
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    "summary": "Return the coordinate images recorded by this `AffineNormalization`.",
    "summary_source": "derived projection vocabulary",
    "type_id": "noether/commalg/AffineNormalization@1"
  },
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    "canonical_name": "commalg.AffineNormalization.fraction_denominators",
    "determinism": "deterministic",
    "doc": "",
    "domain": "commalg",
    "file": "noether/commalg/__init__.pyi",
    "input_schema": {
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      "properties": {},
      "type": "object"
    },
    "kind": "property",
    "line": 1059,
    "module": "noether.commalg",
    "name": "fraction_denominators",
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    ],
    "qualified_name": "AffineNormalization.fraction_denominators",
    "releases_gil": false,
    "result_schema": {
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    },
    "signature": "(self) -> list[Polynomial[C]]",
    "signatures": [
      "(self) -> list[Polynomial[C]]"
    ],
    "summary": "Return the fraction denominators recorded by this `AffineNormalization`.",
    "summary_source": "derived projection vocabulary",
    "type_id": "noether/commalg/AffineNormalization@1"
  },
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    "canonical_name": "commalg.AffineNormalization.fraction_numerators",
    "determinism": "deterministic",
    "doc": "",
    "domain": "commalg",
    "file": "noether/commalg/__init__.pyi",
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      "type": "object"
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    "kind": "property",
    "line": 1057,
    "module": "noether.commalg",
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    "qualified_name": "AffineNormalization.fraction_numerators",
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    },
    "signature": "(self) -> list[Polynomial[C]]",
    "signatures": [
      "(self) -> list[Polynomial[C]]"
    ],
    "summary": "Return the fraction numerators recorded by this `AffineNormalization`.",
    "summary_source": "derived projection vocabulary",
    "type_id": "noether/commalg/AffineNormalization@1"
  },
  {
    "canonical_name": "commalg.AffineNormalization.from_json",
    "determinism": "deterministic",
    "doc": "",
    "domain": "commalg",
    "file": "noether/commalg/__init__.pyi",
    "input_schema": {
      "additionalProperties": false,
      "properties": {
        "value": {
          "type": "string"
        }
      },
      "required": [
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      "type": "object"
    },
    "kind": "classmethod",
    "line": 1077,
    "module": "noether.commalg",
    "name": "from_json",
    "overloads": [
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        "input_schema": {
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          "properties": {
            "value": {
              "type": "string"
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          "type": "object"
        },
        "parameters": [
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            "kind": "positional_or_keyword",
            "name": "value",
            "required": true,
            "schema": {
              "type": "string"
            }
          }
        ],
        "result_schema": {
          "x-python-type": "AffineNormalization[RationalField | FiniteField]"
        },
        "signature": "(cls, value: str) -> AffineNormalization[RationalField | FiniteField]"
      }
    ],
    "qualified_name": "AffineNormalization.from_json",
    "releases_gil": false,
    "result_schema": {
      "x-python-type": "AffineNormalization[RationalField | FiniteField]"
    },
    "signature": "(cls, value: str) -> AffineNormalization[RationalField | FiniteField]",
    "signatures": [
      "(cls, value: str) -> AffineNormalization[RationalField | FiniteField]"
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    "summary": "Return the multipliers recorded by this `ArithmeticNullstellensatzCertificate`.",
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    "summary": "Return the minimal primes recorded by this `AssociatedPrimeExtStage`.",
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    "domain": "commalg",
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    "summary": "Serialize this `AssociatedPrimeExtStage` to its canonical typed JSON representation.",
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    "doc": "",
    "domain": "commalg",
    "file": "noether/commalg/__init__.pyi",
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    "doc": "Complete verified associated primes of a polynomial ideal.",
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    "module": "noether.commalg",
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    "summary": "Complete verified associated primes of a polynomial ideal.",
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    "domain": "commalg",
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    "canonical_name": "commalg.AssociatedPrimeSet.prime_certificates",
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    "doc": "",
    "domain": "commalg",
    "file": "noether/commalg/__init__.pyi",
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    "signature": "(self) -> list[PolynomialPrimeCertificate[C]]",
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      "(self) -> list[PolynomialPrimeCertificate[C]]"
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    "summary": "Return the prime certificates recorded by this `AssociatedPrimeSet`.",
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    "type_id": "noether/commalg/AssociatedPrimeSet@1"
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    "canonical_name": "commalg.AssociatedPrimeSet.primes",
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    "doc": "",
    "domain": "commalg",
    "file": "noether/commalg/__init__.pyi",
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    "summary": "Return the primes recorded by this `AssociatedPrimeSet`.",
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    "canonical_name": "commalg.AssociatedPrimeSet.resolution",
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    "domain": "commalg",
    "file": "noether/commalg/__init__.pyi",
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      "additionalProperties": false,
      "properties": {},
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    "module": "noether.commalg",
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        "parameters": [],
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    "qualified_name": "AssociatedPrimeSet.resolution",
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    "summary": "Return the resolution recorded by this `AssociatedPrimeSet`.",
    "summary_source": "derived projection vocabulary",
    "type_id": "noether/commalg/AssociatedPrimeSet@1"
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  {
    "canonical_name": "commalg.AssociatedPrimeSet.source",
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    "doc": "",
    "domain": "commalg",
    "file": "noether/commalg/__init__.pyi",
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      "properties": {},
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    "module": "noether.commalg",
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        "parameters": [],
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    "qualified_name": "AssociatedPrimeSet.source",
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    "signature": "(self) -> Ideal[PolynomialRing[C]]",
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    "summary": "Return the source recorded by this `AssociatedPrimeSet`.",
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    "type_id": "noether/commalg/AssociatedPrimeSet@1"
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    "canonical_name": "commalg.AssociatedPrimeSet.stages",
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    "doc": "",
    "domain": "commalg",
    "file": "noether/commalg/__init__.pyi",
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      "properties": {},
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    "module": "noether.commalg",
    "name": "stages",
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        "parameters": [],
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            "x-python-type": "AssociatedPrimeExtStage[C]"
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    "summary": "Return the stages recorded by this `AssociatedPrimeSet`.",
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    "canonical_name": "commalg.AssociatedPrimeSet.theorem",
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    "domain": "commalg",
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    "module": "noether.commalg",
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    "qualified_name": "AssociatedPrimeSet.theorem",
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      "(self) -> str"
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    "summary": "Return the theorem recorded by this `AssociatedPrimeSet`.",
    "summary_source": "derived projection vocabulary",
    "type_id": "noether/commalg/AssociatedPrimeSet@1"
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    "canonical_name": "commalg.AssociatedPrimeSet.to_json",
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    "doc": "",
    "domain": "commalg",
    "file": "noether/commalg/__init__.pyi",
    "input_schema": {
      "additionalProperties": false,
      "properties": {},
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    "kind": "method",
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    "module": "noether.commalg",
    "name": "to_json",
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    "qualified_name": "AssociatedPrimeSet.to_json",
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    "summary": "Serialize this `AssociatedPrimeSet` to its canonical typed JSON representation.",
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    "canonical_name": "commalg.AssociatedPrimeSet.verify",
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    "domain": "commalg",
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    "input_schema": {
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    "kind": "method",
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    "module": "noether.commalg",
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    "qualified_name": "AssociatedPrimeSet.verify",
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    "summary": "Replay and verify the mathematical evidence carried by this `AssociatedPrimeSet`.",
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    "canonical_name": "commalg.BlowUp",
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    "doc": "Typed point blow-up with all canonical exceptional-neighbourhood charts.",
    "domain": "commalg",
    "file": "noether/commalg/__init__.pyi",
    "input_schema": {},
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    "module": "noether.commalg",
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    "qualified_name": "BlowUp",
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    "summary": "Typed point blow-up with all canonical exceptional-neighbourhood charts.",
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    "canonical_name": "commalg.BlowUp.center",
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    "doc": "",
    "domain": "commalg",
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    "module": "noether.commalg",
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    "summary": "Return the center recorded by this `BlowUp`.",
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    "canonical_name": "commalg.BlowUp.charts",
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    "domain": "commalg",
    "file": "noether/commalg/__init__.pyi",
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      "properties": {},
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    "kind": "property",
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    "module": "noether.commalg",
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        "parameters": [],
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    "qualified_name": "BlowUp.charts",
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    "canonical_name": "commalg.BlowUp.from_json",
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    "domain": "commalg",
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      "type": "object"
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    "module": "noether.commalg",
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            "kind": "positional_or_keyword",
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        "result_schema": {
          "x-python-type": "BlowUp[FiniteField]"
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    "qualified_name": "BlowUp.from_json",
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    "signature": "(cls, value: str) -> BlowUp[FiniteField]",
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      "(cls, value: str) -> BlowUp[FiniteField]"
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    "summary": "Deserialize and revalidate a `BlowUp` from typed JSON.",
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    "canonical_name": "commalg.BlowUp.local_source",
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    "domain": "commalg",
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    "kind": "property",
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    "module": "noether.commalg",
    "name": "local_source",
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    "qualified_name": "BlowUp.local_source",
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      "x-python-type": "Scheme[C]"
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    "signatures": [
      "(self) -> Scheme[C]"
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    "summary": "Return the local source recorded by this `BlowUp`.",
    "summary_source": "derived projection vocabulary",
    "type_id": "noether/commalg/BlowUp@1"
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  {
    "canonical_name": "commalg.BlowUp.source",
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    "doc": "",
    "domain": "commalg",
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    "input_schema": {
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      "properties": {},
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    "kind": "property",
    "line": 2722,
    "module": "noether.commalg",
    "name": "source",
    "overloads": [
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        "parameters": [],
        "result_schema": {
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    "qualified_name": "BlowUp.source",
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    "result_schema": {
      "x-python-type": "Scheme[C]"
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    "signature": "(self) -> Scheme[C]",
    "signatures": [
      "(self) -> Scheme[C]"
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    "summary": "Return the source recorded by this `BlowUp`.",
    "summary_source": "derived projection vocabulary",
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    "domain": "commalg",
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    "domain": "commalg",
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    "domain": "commalg",
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    "domain": "commalg",
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    "domain": "commalg",
    "file": "noether/commalg/__init__.pyi",
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    "domain": "commalg",
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    "domain": "commalg",
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    "module": "noether.commalg",
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    "domain": "commalg",
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    "summary": "Return the additive inverse of this value.",
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    "domain": "commalg",
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    "summary": "Subtract another compatible value from this value.",
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    "doc": "",
    "domain": "commalg",
    "file": "noether/commalg/__init__.pyi",
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    "summary": "Return the codimension recorded by this `ChowClass`.",
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    "summary": "Deserialize and revalidate a `ChowClass` from typed JSON.",
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    "domain": "commalg",
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    "canonical_name": "commalg.ChowClass.over_rationals",
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    "domain": "commalg",
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    "domain": "commalg",
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    "summary": "Replay and verify the mathematical evidence carried by this `ChowClass`.",
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    "doc": "The exact integral Chow ring of a product of projective spaces.",
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    "qualified_name": "CrtDecomposition.verify",
    "releases_gil": false,
    "result_schema": {
      "type": "null"
    },
    "signature": "(self) -> None",
    "signatures": [
      "(self) -> None"
    ],
    "summary": "Replay and verify the mathematical evidence carried by this `CrtDecomposition`.",
    "summary_source": "derived projection vocabulary",
    "type_id": "noether/commalg/CrtDecomposition@1"
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  {
    "canonical_name": "commalg.Curve",
    "determinism": "deterministic",
    "doc": "A checked smooth projective plane or Weierstrass curve.",
    "domain": "commalg",
    "file": "noether/commalg/__init__.pyi",
    "input_schema": {},
    "kind": "class",
    "line": 2833,
    "module": "noether.commalg",
    "name": "Curve",
    "overloads": [],
    "qualified_name": "Curve",
    "releases_gil": false,
    "result_schema": {},
    "signature": "",
    "signatures": [],
    "summary": "A checked smooth projective plane or Weierstrass curve.",
    "summary_source": "public docstring",
    "type_id": "noether/commalg/Curve@1"
  },
  {
    "canonical_name": "commalg.Curve.base",
    "determinism": "deterministic",
    "doc": "",
    "domain": "commalg",
    "file": "noether/commalg/__init__.pyi",
    "input_schema": {
      "additionalProperties": false,
      "properties": {},
      "type": "object"
    },
    "kind": "property",
    "line": 2840,
    "module": "noether.commalg",
    "name": "base",
    "overloads": [
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        "input_schema": {
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          "properties": {},
          "type": "object"
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        "parameters": [],
        "result_schema": {
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        "signature": "(self) -> C"
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    ],
    "qualified_name": "Curve.base",
    "releases_gil": false,
    "result_schema": {
      "x-python-type": "C"
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    "signature": "(self) -> C",
    "signatures": [
      "(self) -> C"
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    "summary": "Return the base recorded by this `Curve`.",
    "summary_source": "derived projection vocabulary",
    "type_id": "noether/commalg/Curve@1"
  },
  {
    "canonical_name": "commalg.Curve.equation",
    "determinism": "deterministic",
    "doc": "",
    "domain": "commalg",
    "file": "noether/commalg/__init__.pyi",
    "input_schema": {
      "additionalProperties": false,
      "properties": {},
      "type": "object"
    },
    "kind": "property",
    "line": 2842,
    "module": "noether.commalg",
    "name": "equation",
    "overloads": [
      {
        "input_schema": {
          "additionalProperties": false,
          "properties": {},
          "type": "object"
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        "parameters": [],
        "result_schema": {
          "x-python-type": "Polynomial[C]"
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        "signature": "(self) -> Polynomial[C]"
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    ],
    "qualified_name": "Curve.equation",
    "releases_gil": false,
    "result_schema": {
      "x-python-type": "Polynomial[C]"
    },
    "signature": "(self) -> Polynomial[C]",
    "signatures": [
      "(self) -> Polynomial[C]"
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    "summary": "Return the equation recorded by this `Curve`.",
    "summary_source": "derived projection vocabulary",
    "type_id": "noether/commalg/Curve@1"
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  {
    "canonical_name": "commalg.Curve.from_json",
    "determinism": "deterministic",
    "doc": "",
    "domain": "commalg",
    "file": "noether/commalg/__init__.pyi",
    "input_schema": {
      "additionalProperties": false,
      "properties": {
        "value": {
          "type": "string"
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      },
      "required": [
        "value"
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      "type": "object"
    },
    "kind": "classmethod",
    "line": 2858,
    "module": "noether.commalg",
    "name": "from_json",
    "overloads": [
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        "input_schema": {
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          "properties": {
            "value": {
              "type": "string"
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          },
          "required": [
            "value"
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          "type": "object"
        },
        "parameters": [
          {
            "kind": "positional_or_keyword",
            "name": "value",
            "required": true,
            "schema": {
              "type": "string"
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          }
        ],
        "result_schema": {
          "x-python-type": "Curve[CurveBase]"
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        "signature": "(cls, value: str) -> Curve[CurveBase]"
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    ],
    "qualified_name": "Curve.from_json",
    "releases_gil": false,
    "result_schema": {
      "x-python-type": "Curve[CurveBase]"
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    "signature": "(cls, value: str) -> Curve[CurveBase]",
    "signatures": [
      "(cls, value: str) -> Curve[CurveBase]"
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    "summary": "Deserialize and revalidate a `Curve` from typed JSON.",
    "summary_source": "derived projection vocabulary",
    "type_id": "noether/commalg/Curve@1"
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  {
    "canonical_name": "commalg.Curve.function_field",
    "determinism": "deterministic",
    "doc": "",
    "domain": "commalg",
    "file": "noether/commalg/__init__.pyi",
    "input_schema": {
      "additionalProperties": false,
      "properties": {},
      "type": "object"
    },
    "kind": "method",
    "line": 2853,
    "module": "noether.commalg",
    "name": "function_field",
    "overloads": [
      {
        "input_schema": {
          "additionalProperties": false,
          "properties": {},
          "type": "object"
        },
        "parameters": [],
        "result_schema": {
          "x-python-type": "CurveFunctionField[C]"
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        "signature": "(self) -> CurveFunctionField[C]"
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    ],
    "qualified_name": "Curve.function_field",
    "releases_gil": false,
    "result_schema": {
      "x-python-type": "CurveFunctionField[C]"
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    "signature": "(self) -> CurveFunctionField[C]",
    "signatures": [
      "(self) -> CurveFunctionField[C]"
    ],
    "summary": "Compute function field for this `Curve`.",
    "summary_source": "derived projection vocabulary",
    "type_id": "noether/commalg/Curve@1"
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  {
    "canonical_name": "commalg.Curve.point",
    "determinism": "deterministic",
    "doc": "",
    "domain": "commalg",
    "file": "noether/commalg/__init__.pyi",
    "input_schema": {
      "additionalProperties": false,
      "properties": {
        "coordinates": {
          "items": {
            "x-python-type": "CurveCoefficient"
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          "type": "array",
          "x-python-type": "Iterable[CurveCoefficient]"
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      },
      "required": [
        "coordinates"
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      "type": "object"
    },
    "kind": "method",
    "line": 2854,
    "module": "noether.commalg",
    "name": "point",
    "overloads": [
      {
        "input_schema": {
          "additionalProperties": false,
          "properties": {
            "coordinates": {
              "items": {
                "x-python-type": "CurveCoefficient"
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              "type": "array",
              "x-python-type": "Iterable[CurveCoefficient]"
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          },
          "required": [
            "coordinates"
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          "type": "object"
        },
        "parameters": [
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            "kind": "positional_or_keyword",
            "name": "coordinates",
            "required": true,
            "schema": {
              "items": {
                "x-python-type": "CurveCoefficient"
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              "type": "array",
              "x-python-type": "Iterable[CurveCoefficient]"
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          }
        ],
        "result_schema": {
          "x-python-type": "CurvePoint[C]"
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        "signature": "(self, coordinates: Iterable[CurveCoefficient]) -> CurvePoint[C]"
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    ],
    "qualified_name": "Curve.point",
    "releases_gil": false,
    "result_schema": {
      "x-python-type": "CurvePoint[C]"
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    "signature": "(self, coordinates: Iterable[CurveCoefficient]) -> CurvePoint[C]",
    "signatures": [
      "(self, coordinates: Iterable[CurveCoefficient]) -> CurvePoint[C]"
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    "summary": "Compute point for this `Curve`.",
    "summary_source": "derived projection vocabulary",
    "type_id": "noether/commalg/Curve@1"
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  {
    "canonical_name": "commalg.Curve.presentation_kind",
    "determinism": "deterministic",
    "doc": "",
    "domain": "commalg",
    "file": "noether/commalg/__init__.pyi",
    "input_schema": {
      "additionalProperties": false,
      "properties": {},
      "type": "object"
    },
    "kind": "property",
    "line": 2844,
    "module": "noether.commalg",
    "name": "presentation_kind",
    "overloads": [
      {
        "input_schema": {
          "additionalProperties": false,
          "properties": {},
          "type": "object"
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        "parameters": [],
        "result_schema": {
          "enum": [
            "smooth_projective_plane",
            "smooth_projective_weierstrass"
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          "x-python-type": "Literal['smooth_projective_plane', 'smooth_projective_weierstrass']"
        },
        "signature": "(self) -> Literal['smooth_projective_plane', 'smooth_projective_weierstrass']"
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    ],
    "qualified_name": "Curve.presentation_kind",
    "releases_gil": false,
    "result_schema": {
      "enum": [
        "smooth_projective_plane",
        "smooth_projective_weierstrass"
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      "x-python-type": "Literal['smooth_projective_plane', 'smooth_projective_weierstrass']"
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    "signature": "(self) -> Literal['smooth_projective_plane', 'smooth_projective_weierstrass']",
    "signatures": [
      "(self) -> Literal['smooth_projective_plane', 'smooth_projective_weierstrass']"
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    "summary": "Return the presentation kind recorded by this `Curve`.",
    "summary_source": "derived projection vocabulary",
    "type_id": "noether/commalg/Curve@1"
  },
  {
    "canonical_name": "commalg.Curve.quotient_dimension",
    "determinism": "deterministic",
    "doc": "",
    "domain": "commalg",
    "file": "noether/commalg/__init__.pyi",
    "input_schema": {
      "additionalProperties": false,
      "properties": {},
      "type": "object"
    },
    "kind": "property",
    "line": 2848,
    "module": "noether.commalg",
    "name": "quotient_dimension",
    "overloads": [
      {
        "input_schema": {
          "additionalProperties": false,
          "properties": {},
          "type": "object"
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        "parameters": [],
        "result_schema": {
          "type": "integer"
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        "signature": "(self) -> int"
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    ],
    "qualified_name": "Curve.quotient_dimension",
    "releases_gil": false,
    "result_schema": {
      "type": "integer"
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    "signature": "(self) -> int",
    "signatures": [
      "(self) -> int"
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    "summary": "Return the quotient dimension recorded by this `Curve`.",
    "summary_source": "derived projection vocabulary",
    "type_id": "noether/commalg/Curve@1"
  },
  {
    "canonical_name": "commalg.Curve.singular_cone_dimension",
    "determinism": "deterministic",
    "doc": "",
    "domain": "commalg",
    "file": "noether/commalg/__init__.pyi",
    "input_schema": {
      "additionalProperties": false,
      "properties": {},
      "type": "object"
    },
    "kind": "property",
    "line": 2850,
    "module": "noether.commalg",
    "name": "singular_cone_dimension",
    "overloads": [
      {
        "input_schema": {
          "additionalProperties": false,
          "properties": {},
          "type": "object"
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        "parameters": [],
        "result_schema": {
          "type": "integer"
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        "signature": "(self) -> int"
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    ],
    "qualified_name": "Curve.singular_cone_dimension",
    "releases_gil": false,
    "result_schema": {
      "type": "integer"
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    "signature": "(self) -> int",
    "signatures": [
      "(self) -> int"
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    "summary": "Return the singular cone dimension recorded by this `Curve`.",
    "summary_source": "derived projection vocabulary",
    "type_id": "noether/commalg/Curve@1"
  },
  {
    "canonical_name": "commalg.Curve.smooth_projective_plane",
    "determinism": "deterministic",
    "doc": "",
    "domain": "commalg",
    "file": "noether/commalg/__init__.pyi",
    "input_schema": {
      "additionalProperties": false,
      "properties": {
        "equation": {
          "x-python-type": "Polynomial[C]"
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      },
      "required": [
        "equation"
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      "type": "object"
    },
    "kind": "staticmethod",
    "line": 2836,
    "module": "noether.commalg",
    "name": "smooth_projective_plane",
    "overloads": [
      {
        "input_schema": {
          "additionalProperties": false,
          "properties": {
            "equation": {
              "x-python-type": "Polynomial[C]"
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          },
          "required": [
            "equation"
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          "type": "object"
        },
        "parameters": [
          {
            "kind": "positional_or_keyword",
            "name": "equation",
            "required": true,
            "schema": {
              "x-python-type": "Polynomial[C]"
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          }
        ],
        "result_schema": {
          "x-python-type": "Curve[C]"
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        "signature": "(equation: Polynomial[C]) -> Curve[C]"
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    ],
    "qualified_name": "Curve.smooth_projective_plane",
    "releases_gil": false,
    "result_schema": {
      "x-python-type": "Curve[C]"
    },
    "signature": "(equation: Polynomial[C]) -> Curve[C]",
    "signatures": [
      "(equation: Polynomial[C]) -> Curve[C]"
    ],
    "summary": "Construct a `Curve` using smooth projective plane.",
    "summary_source": "derived projection vocabulary",
    "type_id": "noether/commalg/Curve@1"
  },
  {
    "canonical_name": "commalg.Curve.to_json",
    "determinism": "deterministic",
    "doc": "",
    "domain": "commalg",
    "file": "noether/commalg/__init__.pyi",
    "input_schema": {
      "additionalProperties": false,
      "properties": {},
      "type": "object"
    },
    "kind": "method",
    "line": 2856,
    "module": "noether.commalg",
    "name": "to_json",
    "overloads": [
      {
        "input_schema": {
          "additionalProperties": false,
          "properties": {},
          "type": "object"
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        "parameters": [],
        "result_schema": {
          "type": "string"
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        "signature": "(self) -> str"
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    "qualified_name": "Curve.to_json",
    "releases_gil": false,
    "result_schema": {
      "type": "string"
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    "signature": "(self) -> str",
    "signatures": [
      "(self) -> str"
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    "summary": "Serialize this `Curve` to its canonical typed JSON representation.",
    "summary_source": "derived projection vocabulary",
    "type_id": "noether/commalg/Curve@1"
  },
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    "canonical_name": "commalg.Curve.verify",
    "determinism": "deterministic",
    "doc": "",
    "domain": "commalg",
    "file": "noether/commalg/__init__.pyi",
    "input_schema": {
      "additionalProperties": false,
      "properties": {},
      "type": "object"
    },
    "kind": "method",
    "line": 2855,
    "module": "noether.commalg",
    "name": "verify",
    "overloads": [
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        "input_schema": {
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          "properties": {},
          "type": "object"
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        "parameters": [],
        "result_schema": {
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        "signature": "(self) -> None"
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    ],
    "qualified_name": "Curve.verify",
    "releases_gil": false,
    "result_schema": {
      "type": "null"
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    "signature": "(self) -> None",
    "signatures": [
      "(self) -> None"
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    "summary": "Replay and verify the mathematical evidence carried by this `Curve`.",
    "summary_source": "derived projection vocabulary",
    "type_id": "noether/commalg/Curve@1"
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    "canonical_name": "commalg.Curve.weierstrass",
    "determinism": "deterministic",
    "doc": "",
    "domain": "commalg",
    "file": "noether/commalg/__init__.pyi",
    "input_schema": {
      "additionalProperties": false,
      "properties": {
        "base": {
          "x-python-type": "C"
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        "coefficients": {
          "items": {
            "x-python-type": "CurveCoefficient"
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          "type": "array",
          "x-python-type": "Iterable[CurveCoefficient]"
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      },
      "required": [
        "base",
        "coefficients"
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      "type": "object"
    },
    "kind": "staticmethod",
    "line": 2838,
    "module": "noether.commalg",
    "name": "weierstrass",
    "overloads": [
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          "properties": {
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              "x-python-type": "C"
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            "coefficients": {
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                "x-python-type": "CurveCoefficient"
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              "type": "array",
              "x-python-type": "Iterable[CurveCoefficient]"
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          },
          "required": [
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          "type": "object"
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        "parameters": [
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            "kind": "positional_or_keyword",
            "name": "base",
            "required": true,
            "schema": {
              "x-python-type": "C"
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          },
          {
            "kind": "positional_or_keyword",
            "name": "coefficients",
            "required": true,
            "schema": {
              "items": {
                "x-python-type": "CurveCoefficient"
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              "type": "array",
              "x-python-type": "Iterable[CurveCoefficient]"
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          }
        ],
        "result_schema": {
          "x-python-type": "Curve[C]"
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        "signature": "(base: C, coefficients: Iterable[CurveCoefficient]) -> Curve[C]"
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    ],
    "qualified_name": "Curve.weierstrass",
    "releases_gil": false,
    "result_schema": {
      "x-python-type": "Curve[C]"
    },
    "signature": "(base: C, coefficients: Iterable[CurveCoefficient]) -> Curve[C]",
    "signatures": [
      "(base: C, coefficients: Iterable[CurveCoefficient]) -> Curve[C]"
    ],
    "summary": "Construct a `Curve` using weierstrass.",
    "summary_source": "derived projection vocabulary",
    "type_id": "noether/commalg/Curve@1"
  },
  {
    "canonical_name": "commalg.Curve.weierstrass_coefficients",
    "determinism": "deterministic",
    "doc": "",
    "domain": "commalg",
    "file": "noether/commalg/__init__.pyi",
    "input_schema": {
      "additionalProperties": false,
      "properties": {},
      "type": "object"
    },
    "kind": "property",
    "line": 2852,
    "module": "noether.commalg",
    "name": "weierstrass_coefficients",
    "overloads": [
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          "properties": {},
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        "result_schema": {
          "anyOf": [
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              "items": {
                "x-python-type": "CurveCoefficient"
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              "type": "array",
              "x-python-type": "list[CurveCoefficient]"
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              "type": "null"
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          ],
          "x-python-type": "list[CurveCoefficient] | None"
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        "signature": "(self) -> list[CurveCoefficient] | None"
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    "qualified_name": "Curve.weierstrass_coefficients",
    "releases_gil": false,
    "result_schema": {
      "anyOf": [
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          "type": "array",
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        {
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      ],
      "x-python-type": "list[CurveCoefficient] | None"
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    "signature": "(self) -> list[CurveCoefficient] | None",
    "signatures": [
      "(self) -> list[CurveCoefficient] | None"
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    "summary": "Return the weierstrass coefficients recorded by this `Curve`.",
    "summary_source": "derived projection vocabulary",
    "type_id": "noether/commalg/Curve@1"
  },
  {
    "canonical_name": "commalg.CurveClosedPoint",
    "determinism": "deterministic",
    "doc": "A closed point (place) on one curve.",
    "domain": "commalg",
    "file": "noether/commalg/__init__.pyi",
    "input_schema": {},
    "kind": "class",
    "line": 2872,
    "module": "noether.commalg",
    "name": "CurveClosedPoint",
    "overloads": [],
    "qualified_name": "CurveClosedPoint",
    "releases_gil": false,
    "result_schema": {},
    "signature": "",
    "signatures": [],
    "summary": "A closed point (place) on one curve.",
    "summary_source": "public docstring",
    "type_id": "noether/commalg/CurveClosedPoint@1"
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  {
    "canonical_name": "commalg.CurveClosedPoint.curve",
    "determinism": "deterministic",
    "doc": "",
    "domain": "commalg",
    "file": "noether/commalg/__init__.pyi",
    "input_schema": {
      "additionalProperties": false,
      "properties": {},
      "type": "object"
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    "kind": "property",
    "line": 2875,
    "module": "noether.commalg",
    "name": "curve",
    "overloads": [
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        "input_schema": {
          "additionalProperties": false,
          "properties": {},
          "type": "object"
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        "parameters": [],
        "result_schema": {
          "x-python-type": "Curve[C]"
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        "signature": "(self) -> Curve[C]"
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    ],
    "qualified_name": "CurveClosedPoint.curve",
    "releases_gil": false,
    "result_schema": {
      "x-python-type": "Curve[C]"
    },
    "signature": "(self) -> Curve[C]",
    "signatures": [
      "(self) -> Curve[C]"
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    "summary": "Return the curve recorded by this `CurveClosedPoint`.",
    "summary_source": "derived projection vocabulary",
    "type_id": "noether/commalg/CurveClosedPoint@1"
  },
  {
    "canonical_name": "commalg.CurveClosedPoint.defining_ideal",
    "determinism": "deterministic",
    "doc": "",
    "domain": "commalg",
    "file": "noether/commalg/__init__.pyi",
    "input_schema": {
      "additionalProperties": false,
      "properties": {},
      "type": "object"
    },
    "kind": "property",
    "line": 2879,
    "module": "noether.commalg",
    "name": "defining_ideal",
    "overloads": [
      {
        "input_schema": {
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          "properties": {},
          "type": "object"
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        "parameters": [],
        "result_schema": {
          "x-python-type": "Ideal[PolynomialRing[C]]"
        },
        "signature": "(self) -> Ideal[PolynomialRing[C]]"
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    ],
    "qualified_name": "CurveClosedPoint.defining_ideal",
    "releases_gil": false,
    "result_schema": {
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    "domain": "commalg",
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    "domain": "commalg",
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    "domain": "commalg",
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    "domain": "commalg",
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      "properties": {},
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    "canonical_name": "commalg.CurveFunction.inverse",
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    "domain": "commalg",
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      "additionalProperties": false,
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    "domain": "commalg",
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      "additionalProperties": false,
      "properties": {},
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    "module": "noether.commalg",
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      "properties": {},
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    "module": "noether.commalg",
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    "canonical_name": "commalg.CurveFunction.order_at",
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    "doc": "",
    "domain": "commalg",
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    "module": "noether.commalg",
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    "canonical_name": "commalg.CurveFunction.principal_divisor",
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    "domain": "commalg",
    "file": "noether/commalg/__init__.pyi",
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      "additionalProperties": false,
      "properties": {},
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    "module": "noether.commalg",
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    "summary": "Compute principal divisor for this `CurveFunction`.",
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    "type_id": "noether/commalg/CurveFunction@1"
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    "canonical_name": "commalg.CurveFunction.to_json",
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    "domain": "commalg",
    "file": "noether/commalg/__init__.pyi",
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      "additionalProperties": false,
      "properties": {},
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    "module": "noether.commalg",
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    "canonical_name": "commalg.CurveFunction.verify",
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    "doc": "",
    "domain": "commalg",
    "file": "noether/commalg/__init__.pyi",
    "input_schema": {
      "additionalProperties": false,
      "properties": {},
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    "kind": "method",
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    "module": "noether.commalg",
    "name": "verify",
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    "qualified_name": "CurveFunction.verify",
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    "signatures": [
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    "summary": "Replay and verify the mathematical evidence carried by this `CurveFunction`.",
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    "canonical_name": "commalg.CurveFunction.verify_principal_divisor",
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    "domain": "commalg",
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    "module": "noether.commalg",
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    "qualified_name": "CurveFunction.verify_principal_divisor",
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    "signature": "(self, divisor: WeilDivisor[C]) -> None",
    "signatures": [
      "(self, divisor: WeilDivisor[C]) -> None"
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    "summary": "Compute verify principal divisor for this `CurveFunction`.",
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    "canonical_name": "commalg.CurveFunctionEvaluation",
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    "doc": "Typed regular value, zero, or pole of a curve function.",
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    "canonical_name": "commalg.CurveFunctionEvaluation.kind",
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    "doc": "",
    "domain": "commalg",
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    "domain": "commalg",
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    "summary": "Return the order recorded by this `CurveFunctionEvaluation`.",
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    "domain": "commalg",
    "file": "noether/commalg/__init__.pyi",
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    "domain": "commalg",
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    "domain": "commalg",
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    "domain": "commalg",
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    "summary": "Return the curve recorded by this `CurveFunctionField`.",
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    "summary": "Return the extension degree recorded by this `CurveFunctionField`.",
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    "canonical_name": "commalg.CurveFunctionField.from_homogeneous_polynomial",
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    "domain": "commalg",
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    "domain": "commalg",
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    "domain": "commalg",
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    "summary": "Return the transcendental coordinate recorded by this `CurveFunctionField`.",
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    "domain": "commalg",
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    "summary": "Replay and verify the mathematical evidence carried by this `CurveFunctionField`.",
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    "domain": "commalg",
    "file": "noether/commalg/__init__.pyi",
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    "signatures": [
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    "summary": "Compute x for this `CurveFunctionField`.",
    "summary_source": "derived projection vocabulary",
    "type_id": "noether/commalg/CurveFunctionField@1"
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  {
    "canonical_name": "commalg.CurveFunctionField.y",
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    "doc": "",
    "domain": "commalg",
    "file": "noether/commalg/__init__.pyi",
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      "properties": {},
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    "kind": "method",
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    "module": "noether.commalg",
    "name": "y",
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    "summary": "Compute y for this `CurveFunctionField`.",
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    "canonical_name": "commalg.CurveFunctionField.zero",
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    "doc": "",
    "domain": "commalg",
    "file": "noether/commalg/__init__.pyi",
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    "kind": "method",
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    "module": "noether.commalg",
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    "summary": "Compute zero for this `CurveFunctionField`.",
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    "doc": "Exact finite or infinite order of a curve function at a point.",
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    "file": "noether/commalg/__init__.pyi",
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    "summary": "Exact finite or infinite order of a curve function at a point.",
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    "canonical_name": "commalg.CurveFunctionOrder.is_infinite",
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    "doc": "",
    "domain": "commalg",
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    "summary": "Test whether this `CurveFunctionOrder` is infinite.",
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    "canonical_name": "commalg.CurveFunctionOrder.value",
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    "domain": "commalg",
    "file": "noether/commalg/__init__.pyi",
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    "module": "noether.commalg",
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    "summary": "Return the value recorded by this `CurveFunctionOrder`.",
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    "canonical_name": "commalg.CurveMorphism",
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    "doc": "A checked projective morphism between curves.",
    "domain": "commalg",
    "file": "noether/commalg/__init__.pyi",
    "input_schema": {
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      "properties": {
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        "source": {
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      "type": "object"
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    "kind": "class",
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    "module": "noether.commalg",
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            "source": {
              "x-python-type": "Curve[C]"
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          },
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    "signatures": [
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    "canonical_name": "commalg.CurveMorphism.coordinate_degree",
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    "domain": "commalg",
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      "properties": {},
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    "kind": "property",
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    "module": "noether.commalg",
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    "qualified_name": "CurveMorphism.coordinate_degree",
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    "summary": "Return the coordinate degree recorded by this `CurveMorphism`.",
    "summary_source": "derived projection vocabulary"
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    "canonical_name": "commalg.CurveMorphism.coordinate_images",
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    "doc": "",
    "domain": "commalg",
    "file": "noether/commalg/__init__.pyi",
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      "properties": {},
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    "kind": "property",
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    "module": "noether.commalg",
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    "signature": "(self) -> list[Polynomial[C]]",
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    "summary": "Return the coordinate images recorded by this `CurveMorphism`.",
    "summary_source": "derived projection vocabulary"
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    "canonical_name": "commalg.CurveMorphism.from_json",
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    "doc": "",
    "domain": "commalg",
    "file": "noether/commalg/__init__.pyi",
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    "module": "noether.commalg",
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    "summary": "Deserialize and revalidate a `CurveMorphism` from typed JSON.",
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    "canonical_name": "commalg.CurveMorphism.identity",
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    "doc": "",
    "domain": "commalg",
    "file": "noether/commalg/__init__.pyi",
    "input_schema": {
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      "properties": {
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      "type": "object"
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    "module": "noether.commalg",
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        "parameters": [
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    "signatures": [
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    "summary": "Construct a `CurveMorphism` using identity.",
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    "canonical_name": "commalg.CurveMorphism.source",
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    "doc": "",
    "domain": "commalg",
    "file": "noether/commalg/__init__.pyi",
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      "properties": {},
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    "module": "noether.commalg",
    "name": "source",
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    "summary": "Return the source recorded by this `CurveMorphism`.",
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    "canonical_name": "commalg.CurveMorphism.target",
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    "doc": "",
    "domain": "commalg",
    "file": "noether/commalg/__init__.pyi",
    "input_schema": {
      "additionalProperties": false,
      "properties": {},
      "type": "object"
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    "kind": "property",
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    "module": "noether.commalg",
    "name": "target",
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    "qualified_name": "CurveMorphism.target",
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    "summary": "Return the target recorded by this `CurveMorphism`.",
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    "canonical_name": "commalg.CurveMorphism.to_json",
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    "doc": "",
    "domain": "commalg",
    "file": "noether/commalg/__init__.pyi",
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      "properties": {},
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    "module": "noether.commalg",
    "name": "to_json",
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    "canonical_name": "commalg.CurveMorphism.verify",
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    "doc": "",
    "domain": "commalg",
    "file": "noether/commalg/__init__.pyi",
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    "module": "noether.commalg",
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    "qualified_name": "CurveMorphism.verify",
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    "summary": "Replay and verify the mathematical evidence carried by this `CurveMorphism`.",
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    "canonical_name": "commalg.CurvePoint",
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    "doc": "A checked rational point on one curve.",
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    "file": "noether/commalg/__init__.pyi",
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    "module": "noether.commalg",
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    "signatures": [],
    "summary": "A checked rational point on one curve.",
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    "canonical_name": "commalg.CurvePoint.as_closed_point",
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    "doc": "",
    "domain": "commalg",
    "file": "noether/commalg/__init__.pyi",
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    "kind": "method",
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    "module": "noether.commalg",
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    "qualified_name": "CurvePoint.as_closed_point",
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      "x-python-type": "CurveClosedPoint[C]"
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    "signatures": [
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    "summary": "View this `CurvePoint` as closed point.",
    "summary_source": "derived projection vocabulary",
    "type_id": "noether/commalg/CurvePoint@1"
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    "canonical_name": "commalg.CurvePoint.coordinates",
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    "doc": "",
    "domain": "commalg",
    "file": "noether/commalg/__init__.pyi",
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      "properties": {},
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    "kind": "property",
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    "module": "noether.commalg",
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          "type": "array",
          "x-python-type": "list[CurveCoefficient]"
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        "signature": "(self) -> list[CurveCoefficient]"
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    ],
    "qualified_name": "CurvePoint.coordinates",
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    "result_schema": {
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      "type": "array",
      "x-python-type": "list[CurveCoefficient]"
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    "signature": "(self) -> list[CurveCoefficient]",
    "signatures": [
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    "summary": "Return the coordinates recorded by this `CurvePoint`.",
    "summary_source": "derived projection vocabulary",
    "type_id": "noether/commalg/CurvePoint@1"
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    "canonical_name": "commalg.CurvePoint.curve",
    "determinism": "deterministic",
    "doc": "",
    "domain": "commalg",
    "file": "noether/commalg/__init__.pyi",
    "input_schema": {
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      "properties": {},
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    "kind": "property",
    "line": 2863,
    "module": "noether.commalg",
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    "signature": "(self) -> Curve[C]",
    "signatures": [
      "(self) -> Curve[C]"
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    "summary": "Return the curve recorded by this `CurvePoint`.",
    "summary_source": "derived projection vocabulary",
    "type_id": "noether/commalg/CurvePoint@1"
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    "canonical_name": "commalg.CurvePoint.from_json",
    "determinism": "deterministic",
    "doc": "",
    "domain": "commalg",
    "file": "noether/commalg/__init__.pyi",
    "input_schema": {
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      "properties": {
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    "module": "noether.commalg",
    "name": "from_json",
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          "type": "object"
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            "required": true,
            "schema": {
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        "result_schema": {
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    "qualified_name": "CurvePoint.from_json",
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    "result_schema": {
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    "signature": "(cls, value: str) -> CurvePoint[CurveBase]",
    "signatures": [
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    "summary": "Deserialize and revalidate a `CurvePoint` from typed JSON.",
    "summary_source": "derived projection vocabulary",
    "type_id": "noether/commalg/CurvePoint@1"
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    "canonical_name": "commalg.CurvePoint.to_json",
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    "doc": "",
    "domain": "commalg",
    "file": "noether/commalg/__init__.pyi",
    "input_schema": {
      "additionalProperties": false,
      "properties": {},
      "type": "object"
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    "kind": "method",
    "line": 2868,
    "module": "noether.commalg",
    "name": "to_json",
    "overloads": [
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    "summary": "Serialize this `CurvePoint` to its canonical typed JSON representation.",
    "summary_source": "derived projection vocabulary",
    "type_id": "noether/commalg/CurvePoint@1"
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    "canonical_name": "commalg.CurvePoint.verify",
    "determinism": "deterministic",
    "doc": "",
    "domain": "commalg",
    "file": "noether/commalg/__init__.pyi",
    "input_schema": {
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      "properties": {},
      "type": "object"
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    "kind": "method",
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    "module": "noether.commalg",
    "name": "verify",
    "overloads": [
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    "signatures": [
      "(self) -> None"
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    "summary": "Replay and verify the mathematical evidence carried by this `CurvePoint`.",
    "summary_source": "derived projection vocabulary",
    "type_id": "noether/commalg/CurvePoint@1"
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    "canonical_name": "commalg.CyclicCoverSmoothnessEvidence",
    "determinism": "deterministic",
    "doc": "Regime-aware smoothness evidence for a cyclic cover.",
    "domain": "commalg",
    "file": "noether/commalg/__init__.pyi",
    "input_schema": {},
    "kind": "class",
    "line": 2431,
    "module": "noether.commalg",
    "name": "CyclicCoverSmoothnessEvidence",
    "overloads": [],
    "qualified_name": "CyclicCoverSmoothnessEvidence",
    "releases_gil": false,
    "result_schema": {},
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    "signatures": [],
    "summary": "Regime-aware smoothness evidence for a cyclic cover.",
    "summary_source": "public docstring"
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    "canonical_name": "commalg.CyclicCoverSmoothnessEvidence.branch_is_smooth",
    "determinism": "deterministic",
    "doc": "",
    "domain": "commalg",
    "file": "noether/commalg/__init__.pyi",
    "input_schema": {
      "additionalProperties": false,
      "properties": {},
      "type": "object"
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    "kind": "property",
    "line": 2435,
    "module": "noether.commalg",
    "name": "branch_is_smooth",
    "overloads": [
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    "signature": "(self) -> bool | None",
    "signatures": [
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    "summary": "Return the branch is smooth recorded by this `CyclicCoverSmoothnessEvidence`.",
    "summary_source": "derived projection vocabulary"
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  {
    "canonical_name": "commalg.CyclicCoverSmoothnessEvidence.cover",
    "determinism": "deterministic",
    "doc": "",
    "domain": "commalg",
    "file": "noether/commalg/__init__.pyi",
    "input_schema": {
      "additionalProperties": false,
      "properties": {},
      "type": "object"
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    "kind": "property",
    "line": 2433,
    "module": "noether.commalg",
    "name": "cover",
    "overloads": [
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        "input_schema": {
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          "properties": {},
          "type": "object"
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        "parameters": [],
        "result_schema": {
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        "signature": "(self) -> CyclicHypersurfaceCover[C]"
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    "qualified_name": "CyclicCoverSmoothnessEvidence.cover",
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    "result_schema": {
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    "signature": "(self) -> CyclicHypersurfaceCover[C]",
    "signatures": [
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    "summary": "Return the cover recorded by this `CyclicCoverSmoothnessEvidence`.",
    "summary_source": "derived projection vocabulary"
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  {
    "canonical_name": "commalg.CyclicCoverSmoothnessEvidence.is_smooth",
    "determinism": "deterministic",
    "doc": "",
    "domain": "commalg",
    "file": "noether/commalg/__init__.pyi",
    "input_schema": {
      "additionalProperties": false,
      "properties": {},
      "type": "object"
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    "kind": "method",
    "line": 2438,
    "module": "noether.commalg",
    "name": "is_smooth",
    "overloads": [
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    "qualified_name": "CyclicCoverSmoothnessEvidence.is_smooth",
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    "signature": "(self) -> bool",
    "signatures": [
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    "summary": "Test whether this `CyclicCoverSmoothnessEvidence` is smooth.",
    "summary_source": "derived projection vocabulary"
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  {
    "canonical_name": "commalg.CyclicCoverSmoothnessEvidence.weighted_jacobian",
    "determinism": "deterministic",
    "doc": "",
    "domain": "commalg",
    "file": "noether/commalg/__init__.pyi",
    "input_schema": {
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      "properties": {},
      "type": "object"
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    "kind": "property",
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    "module": "noether.commalg",
    "name": "weighted_jacobian",
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        "parameters": [],
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        "signature": "(self) -> WeightedHypersurfaceJacobianEvidence[C]"
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    "qualified_name": "CyclicCoverSmoothnessEvidence.weighted_jacobian",
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    "signature": "(self) -> WeightedHypersurfaceJacobianEvidence[C]",
    "signatures": [
      "(self) -> WeightedHypersurfaceJacobianEvidence[C]"
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    "summary": "Return the weighted jacobian recorded by this `CyclicCoverSmoothnessEvidence`.",
    "summary_source": "derived projection vocabulary"
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    "canonical_name": "commalg.CyclicHypersurfaceCover",
    "determinism": "deterministic",
    "doc": "A general cyclic cover `w^m=f` of projective space.",
    "domain": "commalg",
    "file": "noether/commalg/__init__.pyi",
    "input_schema": {
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      "properties": {
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        "degree": {
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      "required": [
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      "type": "object"
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    "kind": "class",
    "line": 2440,
    "module": "noether.commalg",
    "name": "CyclicHypersurfaceCover",
    "overloads": [
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            "degree": {
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          },
          "required": [
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          "type": "object"
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        "parameters": [
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          },
          {
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            "schema": {
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    "summary": "A general cyclic cover `w^m=f` of projective space.",
    "summary_source": "public docstring",
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    "canonical_name": "commalg.CyclicHypersurfaceCover.base",
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    "doc": "",
    "domain": "commalg",
    "file": "noether/commalg/__init__.pyi",
    "input_schema": {
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      "properties": {},
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    "kind": "property",
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    "module": "noether.commalg",
    "name": "base",
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    "qualified_name": "CyclicHypersurfaceCover.base",
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    "signature": "(self) -> ProjectiveSpace[C]",
    "signatures": [
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    "summary": "Return the base recorded by this `CyclicHypersurfaceCover`.",
    "summary_source": "derived projection vocabulary",
    "type_id": "noether/commalg/CyclicHypersurfaceCover@1"
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    "canonical_name": "commalg.CyclicHypersurfaceCover.branch",
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    "doc": "",
    "domain": "commalg",
    "file": "noether/commalg/__init__.pyi",
    "input_schema": {
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      "properties": {},
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    "kind": "property",
    "line": 2450,
    "module": "noether.commalg",
    "name": "branch",
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        "parameters": [],
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    "qualified_name": "CyclicHypersurfaceCover.branch",
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    "signature": "(self) -> Scheme[C]",
    "signatures": [
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    "summary": "Return the branch recorded by this `CyclicHypersurfaceCover`.",
    "summary_source": "derived projection vocabulary",
    "type_id": "noether/commalg/CyclicHypersurfaceCover@1"
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  {
    "canonical_name": "commalg.CyclicHypersurfaceCover.branch_equation",
    "determinism": "deterministic",
    "doc": "",
    "domain": "commalg",
    "file": "noether/commalg/__init__.pyi",
    "input_schema": {
      "additionalProperties": false,
      "properties": {},
      "type": "object"
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    "kind": "property",
    "line": 2448,
    "module": "noether.commalg",
    "name": "branch_equation",
    "overloads": [
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        "input_schema": {
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          "properties": {},
          "type": "object"
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        "parameters": [],
        "result_schema": {
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    "qualified_name": "CyclicHypersurfaceCover.branch_equation",
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    "result_schema": {
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    "signature": "(self) -> Polynomial[C]",
    "signatures": [
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    "summary": "Return the branch equation recorded by this `CyclicHypersurfaceCover`.",
    "summary_source": "derived projection vocabulary",
    "type_id": "noether/commalg/CyclicHypersurfaceCover@1"
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  {
    "canonical_name": "commalg.CyclicHypersurfaceCover.cover_degree",
    "determinism": "deterministic",
    "doc": "",
    "domain": "commalg",
    "file": "noether/commalg/__init__.pyi",
    "input_schema": {
      "additionalProperties": false,
      "properties": {},
      "type": "object"
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    "kind": "property",
    "line": 2446,
    "module": "noether.commalg",
    "name": "cover_degree",
    "overloads": [
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        "signature": "(self) -> int"
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    "qualified_name": "CyclicHypersurfaceCover.cover_degree",
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    "result_schema": {
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    "signature": "(self) -> int",
    "signatures": [
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    "summary": "Return the cover degree recorded by this `CyclicHypersurfaceCover`.",
    "summary_source": "derived projection vocabulary",
    "type_id": "noether/commalg/CyclicHypersurfaceCover@1"
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  {
    "canonical_name": "commalg.CyclicHypersurfaceCover.from_json",
    "determinism": "deterministic",
    "doc": "",
    "domain": "commalg",
    "file": "noether/commalg/__init__.pyi",
    "input_schema": {
      "additionalProperties": false,
      "properties": {
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      },
      "required": [
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      "type": "object"
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    "kind": "classmethod",
    "line": 2464,
    "module": "noether.commalg",
    "name": "from_json",
    "overloads": [
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        "parameters": [
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            "name": "value",
            "required": true,
            "schema": {
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        "result_schema": {
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        "signature": "(cls, value: str) -> CyclicHypersurfaceCover[SchemeBase]"
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    ],
    "qualified_name": "CyclicHypersurfaceCover.from_json",
    "releases_gil": false,
    "result_schema": {
      "x-python-type": "CyclicHypersurfaceCover[SchemeBase]"
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    "signature": "(cls, value: str) -> CyclicHypersurfaceCover[SchemeBase]",
    "signatures": [
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    "summary": "Deserialize and revalidate a `CyclicHypersurfaceCover` from typed JSON.",
    "summary_source": "derived projection vocabulary",
    "type_id": "noether/commalg/CyclicHypersurfaceCover@1"
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    "doc": "",
    "domain": "commalg",
    "file": "noether/commalg/__init__.pyi",
    "input_schema": {
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      "properties": {
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          "x-python-type": "Sequence[int]"
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      },
      "required": [
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      "type": "object"
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    "module": "noether.commalg",
    "name": "point_counts",
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        ],
        "result_schema": {
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        "signature": "(self: CyclicHypersurfaceCover[FiniteField], degrees: Sequence[int]) -> PointCountSequence"
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    "qualified_name": "CyclicHypersurfaceCover.point_counts",
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    "summary": "Compute point counts for this `CyclicHypersurfaceCover`.",
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    "doc": "",
    "domain": "commalg",
    "file": "noether/commalg/__init__.pyi",
    "input_schema": {
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      "properties": {},
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    "kind": "property",
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    "module": "noether.commalg",
    "name": "ramification",
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        "parameters": [],
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          "x-python-type": "Literal['tame', 'wild']"
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        "signature": "(self) -> Literal['tame', 'wild']"
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    "qualified_name": "CyclicHypersurfaceCover.ramification",
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    "result_schema": {
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      "x-python-type": "Literal['tame', 'wild']"
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    "signature": "(self) -> Literal['tame', 'wild']",
    "signatures": [
      "(self) -> Literal['tame', 'wild']"
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    "summary": "Return the ramification recorded by this `CyclicHypersurfaceCover`.",
    "summary_source": "derived projection vocabulary",
    "type_id": "noether/commalg/CyclicHypersurfaceCover@1"
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  {
    "canonical_name": "commalg.CyclicHypersurfaceCover.smoothness_evidence",
    "determinism": "deterministic",
    "doc": "",
    "domain": "commalg",
    "file": "noether/commalg/__init__.pyi",
    "input_schema": {
      "additionalProperties": false,
      "properties": {},
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    "kind": "method",
    "line": 2455,
    "module": "noether.commalg",
    "name": "smoothness_evidence",
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          "properties": {},
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    "qualified_name": "CyclicHypersurfaceCover.smoothness_evidence",
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    "signature": "(self) -> CyclicCoverSmoothnessEvidence[C]",
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    "summary": "Compute smoothness evidence for this `CyclicHypersurfaceCover`.",
    "summary_source": "derived projection vocabulary",
    "type_id": "noether/commalg/CyclicHypersurfaceCover@1"
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  {
    "canonical_name": "commalg.CyclicHypersurfaceCover.surface_cohomology",
    "determinism": "deterministic",
    "doc": "",
    "domain": "commalg",
    "file": "noether/commalg/__init__.pyi",
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      "additionalProperties": false,
      "properties": {},
      "type": "object"
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    "kind": "method",
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    "module": "noether.commalg",
    "name": "surface_cohomology",
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    "summary": "Compute surface cohomology for this `CyclicHypersurfaceCover`.",
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    "canonical_name": "commalg.CyclicHypersurfaceCover.to_json",
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    "doc": "",
    "domain": "commalg",
    "file": "noether/commalg/__init__.pyi",
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      "properties": {},
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    "module": "noether.commalg",
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    "summary": "Serialize this `CyclicHypersurfaceCover` to its canonical typed JSON representation.",
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    "type_id": "noether/commalg/CyclicHypersurfaceCover@1"
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    "canonical_name": "commalg.CyclicHypersurfaceCover.weighted_realization",
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    "doc": "",
    "domain": "commalg",
    "file": "noether/commalg/__init__.pyi",
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    "kind": "property",
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    "module": "noether.commalg",
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    "summary": "Return the weighted realization recorded by this `CyclicHypersurfaceCover`.",
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  {
    "canonical_name": "commalg.Degree",
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    "doc": "One arbitrary-precision degree in a free abelian grading group.",
    "domain": "commalg",
    "file": "noether/commalg/__init__.pyi",
    "input_schema": {
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    "module": "noether.commalg",
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    "summary": "One arbitrary-precision degree in a free abelian grading group.",
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    "canonical_name": "commalg.Degree.__add__",
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    "doc": "",
    "domain": "commalg",
    "file": "noether/commalg/__init__.pyi",
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      "type": "object"
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    "module": "noether.commalg",
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        ],
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    "signatures": [
      "(self, other: Degree) -> Degree"
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    "summary": "Add another compatible value to this value.",
    "summary_source": "derived projection vocabulary",
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  {
    "canonical_name": "commalg.Degree.__neg__",
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    "doc": "",
    "domain": "commalg",
    "file": "noether/commalg/__init__.pyi",
    "input_schema": {
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      "properties": {},
      "type": "object"
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    "kind": "method",
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    "module": "noether.commalg",
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        "parameters": [],
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        "signature": "(self) -> Degree"
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    ],
    "qualified_name": "Degree.__neg__",
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    "result_schema": {
      "x-python-type": "Degree"
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    "signature": "(self) -> Degree",
    "signatures": [
      "(self) -> Degree"
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    "summary": "Return the additive inverse of this value.",
    "summary_source": "derived projection vocabulary",
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  {
    "canonical_name": "commalg.Degree.__sub__",
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    "doc": "",
    "domain": "commalg",
    "file": "noether/commalg/__init__.pyi",
    "input_schema": {
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      "properties": {
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      },
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      "type": "object"
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    "module": "noether.commalg",
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    "overloads": [
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            "schema": {
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          }
        ],
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    "qualified_name": "Degree.__sub__",
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    "signature": "(self, other: Degree) -> Degree",
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    "summary": "Subtract another compatible value from this value.",
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  {
    "canonical_name": "commalg.Degree.coordinates",
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    "doc": "",
    "domain": "commalg",
    "file": "noether/commalg/__init__.pyi",
    "input_schema": {
      "additionalProperties": false,
      "properties": {},
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    "kind": "property",
    "line": 1497,
    "module": "noether.commalg",
    "name": "coordinates",
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    "signature": "(self) -> list[int]",
    "signatures": [
      "(self) -> list[int]"
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    "summary": "Return the coordinates recorded by this `Degree`.",
    "summary_source": "derived projection vocabulary",
    "type_id": "noether/commalg/Degree@1"
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    "canonical_name": "commalg.Degree.from_json",
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    "doc": "",
    "domain": "commalg",
    "file": "noether/commalg/__init__.pyi",
    "input_schema": {
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      "properties": {
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      "type": "object"
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    "kind": "classmethod",
    "line": 1503,
    "module": "noether.commalg",
    "name": "from_json",
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          }
        ],
        "result_schema": {
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        "signature": "(cls, value: str) -> Degree"
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    "qualified_name": "Degree.from_json",
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    "signature": "(cls, value: str) -> Degree",
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    "summary": "Deserialize and revalidate a `Degree` from typed JSON.",
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    "type_id": "noether/commalg/Degree@1"
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    "canonical_name": "commalg.Degree.rank",
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    "doc": "",
    "domain": "commalg",
    "file": "noether/commalg/__init__.pyi",
    "input_schema": {
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      "properties": {},
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    "kind": "property",
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    "module": "noether.commalg",
    "name": "rank",
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    "qualified_name": "Degree.rank",
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      "(self) -> int"
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    "summary": "Return the rank recorded by this `Degree`.",
    "summary_source": "derived projection vocabulary",
    "type_id": "noether/commalg/Degree@1"
  },
  {
    "canonical_name": "commalg.Degree.to_json",
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    "doc": "",
    "domain": "commalg",
    "file": "noether/commalg/__init__.pyi",
    "input_schema": {
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      "properties": {},
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    "kind": "method",
    "line": 1501,
    "module": "noether.commalg",
    "name": "to_json",
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          "properties": {},
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        "signature": "(self) -> str"
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    ],
    "qualified_name": "Degree.to_json",
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    "result_schema": {
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    "signature": "(self) -> str",
    "signatures": [
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    "summary": "Serialize this `Degree` to its canonical typed JSON representation.",
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    "canonical_name": "commalg.Degree.zero",
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    "doc": "",
    "domain": "commalg",
    "file": "noether/commalg/__init__.pyi",
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      "additionalProperties": false,
      "properties": {
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      },
      "required": [
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    "module": "noether.commalg",
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        "input_schema": {
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    "summary": "Construct a `Degree` using zero.",
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    "canonical_name": "commalg.DivisorRankLowerBound",
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    "module": "noether.commalg",
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    "summary": "Lower divisor-rank bound from an exact component intersection pairing.",
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    "canonical_name": "commalg.DivisorRankLowerBound.from_json",
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    "domain": "commalg",
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    "kind": "classmethod",
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    "module": "noether.commalg",
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        ],
        "result_schema": {
          "x-python-type": "DivisorRankLowerBound"
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        "signature": "(cls, value: str) -> DivisorRankLowerBound"
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    "signature": "(cls, value: str) -> DivisorRankLowerBound",
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    "summary": "Deserialize and revalidate a `DivisorRankLowerBound` from typed JSON.",
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    "type_id": "noether/commalg/DivisorRankLowerBound@1"
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    "canonical_name": "commalg.DivisorRankLowerBound.intersection_matrix",
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    "doc": "",
    "domain": "commalg",
    "file": "noether/commalg/__init__.pyi",
    "input_schema": {
      "additionalProperties": false,
      "properties": {},
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    "line": 2614,
    "module": "noether.commalg",
    "name": "intersection_matrix",
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        "input_schema": {
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      "x-python-type": "Matrix[IntegerRing]"
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    "signature": "(self) -> Matrix[IntegerRing]",
    "signatures": [
      "(self) -> Matrix[IntegerRing]"
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    "summary": "Return the intersection matrix recorded by this `DivisorRankLowerBound`.",
    "summary_source": "derived projection vocabulary",
    "type_id": "noether/commalg/DivisorRankLowerBound@1"
  },
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    "canonical_name": "commalg.DivisorRankLowerBound.lower_bound",
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    "doc": "",
    "domain": "commalg",
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      "properties": {},
      "type": "object"
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    "kind": "property",
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    "module": "noether.commalg",
    "name": "lower_bound",
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    "qualified_name": "DivisorRankLowerBound.lower_bound",
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    "result_schema": {
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      "(self) -> int"
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    "summary": "Return the lower bound recorded by this `DivisorRankLowerBound`.",
    "summary_source": "derived projection vocabulary",
    "type_id": "noether/commalg/DivisorRankLowerBound@1"
  },
  {
    "canonical_name": "commalg.DivisorRankLowerBound.split_pullback",
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            "required": true,
            "schema": {
              "x-python-type": "Degree"
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          }
        ],
        "result_schema": {
          "x-python-type": "PositiveGrading"
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        "signature": "(self, functional: Degree) -> PositiveGrading"
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    ],
    "qualified_name": "FreeAbelianGrading.certify_positive",
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    "result_schema": {
      "x-python-type": "PositiveGrading"
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    "signature": "(self, functional: Degree) -> PositiveGrading",
    "signatures": [
      "(self, functional: Degree) -> PositiveGrading"
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    "summary": "Compute certify positive for this `FreeAbelianGrading`.",
    "summary_source": "derived projection vocabulary",
    "type_id": "noether/commalg/FreeAbelianGrading@1"
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    "canonical_name": "commalg.FreeAbelianGrading.degree",
    "determinism": "deterministic",
    "doc": "",
    "domain": "commalg",
    "file": "noether/commalg/__init__.pyi",
    "input_schema": {
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      "properties": {
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          "type": "array",
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      },
      "required": [
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    "kind": "method",
    "line": 1518,
    "module": "noether.commalg",
    "name": "degree",
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          "properties": {
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          "required": [
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          "type": "object"
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        "parameters": [
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            "kind": "positional_or_keyword",
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            "schema": {
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              "type": "array",
              "x-python-type": "Sequence[int]"
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          }
        ],
        "result_schema": {
          "x-python-type": "Degree"
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        "signature": "(self, exponents: Sequence[int]) -> Degree"
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    ],
    "qualified_name": "FreeAbelianGrading.degree",
    "releases_gil": false,
    "result_schema": {
      "x-python-type": "Degree"
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    "signature": "(self, exponents: Sequence[int]) -> Degree",
    "signatures": [
      "(self, exponents: Sequence[int]) -> Degree"
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    "summary": "Compute degree for this `FreeAbelianGrading`.",
    "summary_source": "derived projection vocabulary",
    "type_id": "noether/commalg/FreeAbelianGrading@1"
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  {
    "canonical_name": "commalg.FreeAbelianGrading.from_json",
    "determinism": "deterministic",
    "doc": "",
    "domain": "commalg",
    "file": "noether/commalg/__init__.pyi",
    "input_schema": {
      "additionalProperties": false,
      "properties": {
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      },
      "required": [
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      "type": "object"
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    "kind": "classmethod",
    "line": 1523,
    "module": "noether.commalg",
    "name": "from_json",
    "overloads": [
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          "properties": {
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          },
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          "type": "object"
        },
        "parameters": [
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            "kind": "positional_or_keyword",
            "name": "value",
            "required": true,
            "schema": {
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          }
        ],
        "result_schema": {
          "x-python-type": "FreeAbelianGrading"
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        "signature": "(cls, value: str) -> FreeAbelianGrading"
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    ],
    "qualified_name": "FreeAbelianGrading.from_json",
    "releases_gil": false,
    "result_schema": {
      "x-python-type": "FreeAbelianGrading"
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    "signature": "(cls, value: str) -> FreeAbelianGrading",
    "signatures": [
      "(cls, value: str) -> FreeAbelianGrading"
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    "summary": "Deserialize and revalidate a `FreeAbelianGrading` from typed JSON.",
    "summary_source": "derived projection vocabulary",
    "type_id": "noether/commalg/FreeAbelianGrading@1"
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  {
    "canonical_name": "commalg.FreeAbelianGrading.rank",
    "determinism": "deterministic",
    "doc": "",
    "domain": "commalg",
    "file": "noether/commalg/__init__.pyi",
    "input_schema": {
      "additionalProperties": false,
      "properties": {},
      "type": "object"
    },
    "kind": "property",
    "line": 1513,
    "module": "noether.commalg",
    "name": "rank",
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        "parameters": [],
        "result_schema": {
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        "signature": "(self) -> int"
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    ],
    "qualified_name": "FreeAbelianGrading.rank",
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    "result_schema": {
      "type": "integer"
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    "signature": "(self) -> int",
    "signatures": [
      "(self) -> int"
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    "summary": "Return the rank recorded by this `FreeAbelianGrading`.",
    "summary_source": "derived projection vocabulary",
    "type_id": "noether/commalg/FreeAbelianGrading@1"
  },
  {
    "canonical_name": "commalg.FreeAbelianGrading.standard",
    "determinism": "deterministic",
    "doc": "",
    "domain": "commalg",
    "file": "noether/commalg/__init__.pyi",
    "input_schema": {
      "additionalProperties": false,
      "properties": {
        "arity": {
          "type": "integer"
        }
      },
      "required": [
        "arity"
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      "type": "object"
    },
    "kind": "classmethod",
    "line": 1509,
    "module": "noether.commalg",
    "name": "standard",
    "overloads": [
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        "input_schema": {
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          "properties": {
            "arity": {
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          },
          "required": [
            "arity"
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          "type": "object"
        },
        "parameters": [
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            "kind": "positional_or_keyword",
            "name": "arity",
            "required": true,
            "schema": {
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          }
        ],
        "result_schema": {
          "x-python-type": "FreeAbelianGrading"
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        "signature": "(cls, arity: int) -> FreeAbelianGrading"
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    ],
    "qualified_name": "FreeAbelianGrading.standard",
    "releases_gil": false,
    "result_schema": {
      "x-python-type": "FreeAbelianGrading"
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    "signature": "(cls, arity: int) -> FreeAbelianGrading",
    "signatures": [
      "(cls, arity: int) -> FreeAbelianGrading"
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    "summary": "Construct a `FreeAbelianGrading` using standard.",
    "summary_source": "derived projection vocabulary",
    "type_id": "noether/commalg/FreeAbelianGrading@1"
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  {
    "canonical_name": "commalg.FreeAbelianGrading.to_json",
    "determinism": "deterministic",
    "doc": "",
    "domain": "commalg",
    "file": "noether/commalg/__init__.pyi",
    "input_schema": {
      "additionalProperties": false,
      "properties": {},
      "type": "object"
    },
    "kind": "method",
    "line": 1521,
    "module": "noether.commalg",
    "name": "to_json",
    "overloads": [
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        "input_schema": {
          "additionalProperties": false,
          "properties": {},
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        "parameters": [],
        "result_schema": {
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        "signature": "(self) -> str"
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    "qualified_name": "FreeAbelianGrading.to_json",
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    "result_schema": {
      "type": "string"
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    "signature": "(self) -> str",
    "signatures": [
      "(self) -> str"
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    "summary": "Serialize this `FreeAbelianGrading` to its canonical typed JSON representation.",
    "summary_source": "derived projection vocabulary",
    "type_id": "noether/commalg/FreeAbelianGrading@1"
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  {
    "canonical_name": "commalg.FreeAbelianGrading.trivial",
    "determinism": "deterministic",
    "doc": "",
    "domain": "commalg",
    "file": "noether/commalg/__init__.pyi",
    "input_schema": {
      "additionalProperties": false,
      "properties": {
        "arity": {
          "type": "integer"
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      },
      "required": [
        "arity"
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      "type": "object"
    },
    "kind": "classmethod",
    "line": 1511,
    "module": "noether.commalg",
    "name": "trivial",
    "overloads": [
      {
        "input_schema": {
          "additionalProperties": false,
          "properties": {
            "arity": {
              "type": "integer"
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          },
          "required": [
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          "type": "object"
        },
        "parameters": [
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            "kind": "positional_or_keyword",
            "name": "arity",
            "required": true,
            "schema": {
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          }
        ],
        "result_schema": {
          "x-python-type": "FreeAbelianGrading"
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        "signature": "(cls, arity: int) -> FreeAbelianGrading"
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    ],
    "qualified_name": "FreeAbelianGrading.trivial",
    "releases_gil": false,
    "result_schema": {
      "x-python-type": "FreeAbelianGrading"
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    "signature": "(cls, arity: int) -> FreeAbelianGrading",
    "signatures": [
      "(cls, arity: int) -> FreeAbelianGrading"
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    "summary": "Construct a `FreeAbelianGrading` using trivial.",
    "summary_source": "derived projection vocabulary",
    "type_id": "noether/commalg/FreeAbelianGrading@1"
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  {
    "canonical_name": "commalg.FreeAbelianGrading.variable_degrees",
    "determinism": "deterministic",
    "doc": "",
    "domain": "commalg",
    "file": "noether/commalg/__init__.pyi",
    "input_schema": {
      "additionalProperties": false,
      "properties": {},
      "type": "object"
    },
    "kind": "property",
    "line": 1517,
    "module": "noether.commalg",
    "name": "variable_degrees",
    "overloads": [
      {
        "input_schema": {
          "additionalProperties": false,
          "properties": {},
          "type": "object"
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        "parameters": [],
        "result_schema": {
          "items": {
            "x-python-type": "Degree"
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          "type": "array",
          "x-python-type": "list[Degree]"
        },
        "signature": "(self) -> list[Degree]"
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    ],
    "qualified_name": "FreeAbelianGrading.variable_degrees",
    "releases_gil": false,
    "result_schema": {
      "items": {
        "x-python-type": "Degree"
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      "type": "array",
      "x-python-type": "list[Degree]"
    },
    "signature": "(self) -> list[Degree]",
    "signatures": [
      "(self) -> list[Degree]"
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    "summary": "Return the variable degrees recorded by this `FreeAbelianGrading`.",
    "summary_source": "derived projection vocabulary",
    "type_id": "noether/commalg/FreeAbelianGrading@1"
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  {
    "canonical_name": "commalg.FreeAbelianGrading.weighted_view",
    "determinism": "deterministic",
    "doc": "",
    "domain": "commalg",
    "file": "noether/commalg/__init__.pyi",
    "input_schema": {
      "additionalProperties": false,
      "properties": {},
      "type": "object"
    },
    "kind": "method",
    "line": 1519,
    "module": "noether.commalg",
    "name": "weighted_view",
    "overloads": [
      {
        "input_schema": {
          "additionalProperties": false,
          "properties": {},
          "type": "object"
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        "parameters": [],
        "result_schema": {
          "anyOf": [
            {
              "x-python-type": "WeightedGrading"
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            {
              "type": "null"
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          ],
          "x-python-type": "WeightedGrading | None"
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        "signature": "(self) -> WeightedGrading | None"
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    "qualified_name": "FreeAbelianGrading.weighted_view",
    "releases_gil": false,
    "result_schema": {
      "anyOf": [
        {
          "x-python-type": "WeightedGrading"
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        {
          "type": "null"
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      "x-python-type": "WeightedGrading | None"
    },
    "signature": "(self) -> WeightedGrading | None",
    "signatures": [
      "(self) -> WeightedGrading | None"
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    "summary": "Compute weighted view for this `FreeAbelianGrading`.",
    "summary_source": "derived projection vocabulary",
    "type_id": "noether/commalg/FreeAbelianGrading@1"
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    "canonical_name": "commalg.FrobeniusPolynomial",
    "determinism": "deterministic",
    "doc": "Exact characteristic polynomial of Frobenius on surface `H^2`.",
    "domain": "commalg",
    "file": "noether/commalg/__init__.pyi",
    "input_schema": {},
    "kind": "class",
    "line": 2540,
    "module": "noether.commalg",
    "name": "FrobeniusPolynomial",
    "overloads": [],
    "qualified_name": "FrobeniusPolynomial",
    "releases_gil": false,
    "result_schema": {},
    "signature": "",
    "signatures": [],
    "summary": "Exact characteristic polynomial of Frobenius on surface `H^2`.",
    "summary_source": "public docstring",
    "type_id": "noether/commalg/FrobeniusPolynomial@1"
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  {
    "canonical_name": "commalg.FrobeniusPolynomial.cohomological_degree",
    "determinism": "deterministic",
    "doc": "",
    "domain": "commalg",
    "file": "noether/commalg/__init__.pyi",
    "input_schema": {
      "additionalProperties": false,
      "properties": {},
      "type": "object"
    },
    "kind": "property",
    "line": 2547,
    "module": "noether.commalg",
    "name": "cohomological_degree",
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    "qualified_name": "FrobeniusPolynomial.cohomological_degree",
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    "result_schema": {
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    "signature": "(self) -> int",
    "signatures": [
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    "summary": "Return the cohomological degree recorded by this `FrobeniusPolynomial`.",
    "summary_source": "derived projection vocabulary",
    "type_id": "noether/commalg/FrobeniusPolynomial@1"
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  {
    "canonical_name": "commalg.FrobeniusPolynomial.cohomological_weight",
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    "doc": "",
    "domain": "commalg",
    "file": "noether/commalg/__init__.pyi",
    "input_schema": {
      "additionalProperties": false,
      "properties": {},
      "type": "object"
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    "kind": "property",
    "line": 2549,
    "module": "noether.commalg",
    "name": "cohomological_weight",
    "overloads": [
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        "input_schema": {
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          "properties": {},
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        "result_schema": {
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        "signature": "(self) -> int"
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    ],
    "qualified_name": "FrobeniusPolynomial.cohomological_weight",
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    "result_schema": {
      "type": "integer"
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    "signature": "(self) -> int",
    "signatures": [
      "(self) -> int"
    ],
    "summary": "Return the cohomological weight recorded by this `FrobeniusPolynomial`.",
    "summary_source": "derived projection vocabulary",
    "type_id": "noether/commalg/FrobeniusPolynomial@1"
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  {
    "canonical_name": "commalg.FrobeniusPolynomial.cohomology",
    "determinism": "deterministic",
    "doc": "",
    "domain": "commalg",
    "file": "noether/commalg/__init__.pyi",
    "input_schema": {
      "additionalProperties": false,
      "properties": {},
      "type": "object"
    },
    "kind": "property",
    "line": 2551,
    "module": "noether.commalg",
    "name": "cohomology",
    "overloads": [
      {
        "input_schema": {
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          "properties": {},
          "type": "object"
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        "parameters": [],
        "result_schema": {
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        "signature": "(self) -> SmoothProjectiveSurfaceCohomology"
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    ],
    "qualified_name": "FrobeniusPolynomial.cohomology",
    "releases_gil": false,
    "result_schema": {
      "x-python-type": "SmoothProjectiveSurfaceCohomology"
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    "signature": "(self) -> SmoothProjectiveSurfaceCohomology",
    "signatures": [
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    "summary": "Return the cohomology recorded by this `FrobeniusPolynomial`.",
    "summary_source": "derived projection vocabulary",
    "type_id": "noether/commalg/FrobeniusPolynomial@1"
  },
  {
    "canonical_name": "commalg.FrobeniusPolynomial.degree",
    "determinism": "deterministic",
    "doc": "",
    "domain": "commalg",
    "file": "noether/commalg/__init__.pyi",
    "input_schema": {
      "additionalProperties": false,
      "properties": {},
      "type": "object"
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    "kind": "property",
    "line": 2555,
    "module": "noether.commalg",
    "name": "degree",
    "overloads": [
      {
        "input_schema": {
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          "properties": {},
          "type": "object"
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        "parameters": [],
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        "signature": "(self) -> int"
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    ],
    "qualified_name": "FrobeniusPolynomial.degree",
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    "result_schema": {
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    "signature": "(self) -> int",
    "signatures": [
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    "summary": "Return the degree recorded by this `FrobeniusPolynomial`.",
    "summary_source": "derived projection vocabulary",
    "type_id": "noether/commalg/FrobeniusPolynomial@1"
  },
  {
    "canonical_name": "commalg.FrobeniusPolynomial.field_order",
    "determinism": "deterministic",
    "doc": "",
    "domain": "commalg",
    "file": "noether/commalg/__init__.pyi",
    "input_schema": {
      "additionalProperties": false,
      "properties": {},
      "type": "object"
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    "kind": "property",
    "line": 2545,
    "module": "noether.commalg",
    "name": "field_order",
    "overloads": [
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        "parameters": [],
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    "qualified_name": "FrobeniusPolynomial.field_order",
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    "result_schema": {
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    "signature": "(self) -> int",
    "signatures": [
      "(self) -> int"
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    "summary": "Return the field order recorded by this `FrobeniusPolynomial`.",
    "summary_source": "derived projection vocabulary",
    "type_id": "noether/commalg/FrobeniusPolynomial@1"
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  {
    "canonical_name": "commalg.FrobeniusPolynomial.from_json",
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    "doc": "",
    "domain": "commalg",
    "file": "noether/commalg/__init__.pyi",
    "input_schema": {
      "additionalProperties": false,
      "properties": {
        "value": {
          "type": "string"
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      },
      "required": [
        "value"
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      "type": "object"
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    "kind": "classmethod",
    "line": 2562,
    "module": "noether.commalg",
    "name": "from_json",
    "overloads": [
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        "input_schema": {
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          "properties": {
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          },
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          "type": "object"
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        "parameters": [
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            "kind": "positional_or_keyword",
            "name": "value",
            "required": true,
            "schema": {
              "type": "string"
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          }
        ],
        "result_schema": {
          "x-python-type": "FrobeniusPolynomial"
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        "signature": "(cls, value: str) -> FrobeniusPolynomial"
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    ],
    "qualified_name": "FrobeniusPolynomial.from_json",
    "releases_gil": false,
    "result_schema": {
      "x-python-type": "FrobeniusPolynomial"
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    "signature": "(cls, value: str) -> FrobeniusPolynomial",
    "signatures": [
      "(cls, value: str) -> FrobeniusPolynomial"
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    "summary": "Deserialize and revalidate a `FrobeniusPolynomial` from typed JSON.",
    "summary_source": "derived projection vocabulary",
    "type_id": "noether/commalg/FrobeniusPolynomial@1"
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    "canonical_name": "commalg.FrobeniusPolynomial.geometric_divisor_rank_upper_bound",
    "determinism": "deterministic",
    "doc": "",
    "domain": "commalg",
    "file": "noether/commalg/__init__.pyi",
    "input_schema": {
      "additionalProperties": false,
      "properties": {},
      "type": "object"
    },
    "kind": "method",
    "line": 2559,
    "module": "noether.commalg",
    "name": "geometric_divisor_rank_upper_bound",
    "overloads": [
      {
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          "properties": {},
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        "parameters": [],
        "result_schema": {
          "x-python-type": "GeometricDivisorRankUpperBound"
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        "signature": "(self) -> GeometricDivisorRankUpperBound"
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    ],
    "qualified_name": "FrobeniusPolynomial.geometric_divisor_rank_upper_bound",
    "releases_gil": false,
    "result_schema": {
      "x-python-type": "GeometricDivisorRankUpperBound"
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    "signature": "(self) -> GeometricDivisorRankUpperBound",
    "signatures": [
      "(self) -> GeometricDivisorRankUpperBound"
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    "summary": "Compute geometric divisor rank upper bound for this `FrobeniusPolynomial`.",
    "summary_source": "derived projection vocabulary",
    "type_id": "noether/commalg/FrobeniusPolynomial@1"
  },
  {
    "canonical_name": "commalg.FrobeniusPolynomial.point_counts",
    "determinism": "deterministic",
    "doc": "",
    "domain": "commalg",
    "file": "noether/commalg/__init__.pyi",
    "input_schema": {
      "additionalProperties": false,
      "properties": {},
      "type": "object"
    },
    "kind": "property",
    "line": 2553,
    "module": "noether.commalg",
    "name": "point_counts",
    "overloads": [
      {
        "input_schema": {
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    "domain": "commalg",
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        ],
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    "signature": "(self, degree: int) -> list[GradedMonomial]",
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    "summary": "Compute basis for this `GradedCommutativePresentation`.",
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      },
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    "summary": "Compute dimension for this `GradedCommutativePresentation`.",
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        "signature": "(cls, value: str) -> GradedCommutativePresentation"
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    "summary": "Deserialize and revalidate a `GradedCommutativePresentation` from typed JSON.",
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    "canonical_name": "commalg.GradedCommutativePresentation.hilbert_function",
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    "doc": "",
    "domain": "commalg",
    "file": "noether/commalg/__init__.pyi",
    "input_schema": {
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      "properties": {
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      },
      "required": [
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      "type": "object"
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    "module": "noether.commalg",
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    "summary": "Compute hilbert function for this `GradedCommutativePresentation`.",
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    "type_id": "noether/commalg/GradedCommutativePresentation@1"
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    "canonical_name": "commalg.GradedCommutativePresentation.normal_form",
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    "doc": "",
    "domain": "commalg",
    "file": "noether/commalg/__init__.pyi",
    "input_schema": {
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        "value": {
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      },
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    "module": "noether.commalg",
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        ],
        "result_schema": {
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    "signatures": [
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    "summary": "Compute normal form for this `GradedCommutativePresentation`.",
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    "canonical_name": "commalg.GradedCommutativePresentation.relations",
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    "doc": "",
    "domain": "commalg",
    "file": "noether/commalg/__init__.pyi",
    "input_schema": {
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          "type": "array",
          "x-python-type": "list[GradedCommutativePolynomial]"
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        "signature": "(self) -> list[GradedCommutativePolynomial]"
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    ],
    "qualified_name": "GradedCommutativePresentation.relations",
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    "signature": "(self) -> list[GradedCommutativePolynomial]",
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    "summary": "Return the relations recorded by this `GradedCommutativePresentation`.",
    "summary_source": "derived projection vocabulary",
    "type_id": "noether/commalg/GradedCommutativePresentation@1"
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  {
    "canonical_name": "commalg.GradedCommutativePresentation.ring",
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    "doc": "",
    "domain": "commalg",
    "file": "noether/commalg/__init__.pyi",
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    "kind": "property",
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        "parameters": [],
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    ],
    "qualified_name": "GradedCommutativePresentation.ring",
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    "result_schema": {
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    "signature": "(self) -> GradedCommutativePolynomialRing",
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    "summary": "Return the ring recorded by this `GradedCommutativePresentation`.",
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    "type_id": "noether/commalg/GradedCommutativePresentation@1"
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    "canonical_name": "commalg.GradedCommutativePresentation.to_json",
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    "domain": "commalg",
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      "properties": {},
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    "module": "noether.commalg",
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    "qualified_name": "GradedCommutativePresentation.to_json",
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    "summary": "Serialize this `GradedCommutativePresentation` to its canonical typed JSON representation.",
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    "canonical_name": "commalg.GradedCommutativePresentation.verify",
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    "domain": "commalg",
    "file": "noether/commalg/__init__.pyi",
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      "properties": {},
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    "module": "noether.commalg",
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    "qualified_name": "GradedCommutativePresentation.verify",
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    "summary": "Replay and verify the mathematical evidence carried by this `GradedCommutativePresentation`.",
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    "domain": "commalg",
    "file": "noether/commalg/__init__.pyi",
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      "required": [
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      "type": "object"
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    "kind": "class",
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    "module": "noether.commalg",
    "name": "GradedGenerator",
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          },
          {
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    "qualified_name": "GradedGenerator",
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    "summary": "Return the degree recorded by this `GradedGenerator`.",
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    "module": "noether.commalg",
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          "required": [
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    "file": "noether/commalg/__init__.pyi",
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    "module": "noether.commalg",
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    "domain": "commalg",
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          "x-python-type": "list[Polynomial[C]]"
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    "summary": "Compute restrict for this `LocalCompletionMap`.",
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    "type_id": "noether/commalg/LocalCompletionMap@1"
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    "domain": "commalg",
    "file": "noether/commalg/__init__.pyi",
    "input_schema": {
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      "properties": {
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        "value": {
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    "domain": "commalg",
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    "summary": "Return the source recorded by this `LocalCompletionMap`.",
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    "canonical_name": "commalg.LocalCompletionMap.target",
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    "domain": "commalg",
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      "properties": {},
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    "summary": "Return the target recorded by this `LocalCompletionMap`.",
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    "type_id": "noether/commalg/LocalCompletionMap@1"
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    "canonical_name": "commalg.LocalCompletionMap.to_json",
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    "domain": "commalg",
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      "properties": {},
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    "module": "noether.commalg",
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    "domain": "commalg",
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    "doc": "A canonical ideal in one polynomial localization.",
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    "file": "noether/commalg/__init__.pyi",
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    "canonical_name": "commalg.LocalIdeal.associated_graded",
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    "domain": "commalg",
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    "domain": "commalg",
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    "domain": "commalg",
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      "properties": {},
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    "canonical_name": "commalg.LocalIdeal.multiplicity",
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    "summary": "Return the multiplicity recorded by this `LocalIdeal`.",
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    "type_id": "noether/commalg/LocalIdeal@1"
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    "doc": "",
    "domain": "commalg",
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    "module": "noether.commalg",
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    "domain": "commalg",
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    "summary": "Compute ideal for this `LocalPolynomialRing`.",
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    "doc": "",
    "domain": "commalg",
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    "domain": "commalg",
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    "module": "noether.commalg",
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    "summary": "Return the order recorded by this `LocalPolynomialRing`.",
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    "canonical_name": "commalg.LocalPolynomialRing.point",
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    "domain": "commalg",
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    "module": "noether.commalg",
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    "summary": "Return the point recorded by this `LocalPolynomialRing`.",
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    "domain": "commalg",
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    "module": "noether.commalg",
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    "canonical_name": "commalg.LocalPolynomialRing.to_json",
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      "properties": {},
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    "domain": "commalg",
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    "module": "noether.commalg",
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    "input_schema": {},
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    "module": "noether.commalg",
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    "canonical_name": "commalg.LocalReduction.basis",
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    "domain": "commalg",
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      "properties": {},
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    "kind": "property",
    "line": 1805,
    "module": "noether.commalg",
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        "parameters": [],
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        "signature": "(self) -> LocalStandardBasis[_ProjectedIdealFieldContext]"
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    ],
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    "domain": "commalg",
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    "qualified_name": "MonomialIdeal.colon",
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    "summary": "Compute colon for this `MonomialIdeal`.",
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    "canonical_name": "commalg.MonomialIdeal.colon_monomial",
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    "signature": "(self, exponent: Sequence[int]) -> MonomialIdeal[C]",
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    "summary": "Compute colon monomial for this `MonomialIdeal`.",
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    "canonical_name": "commalg.MonomialIdeal.contains_monomial",
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    "domain": "commalg",
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    "summary": "Compute contains monomial for this `MonomialIdeal`.",
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    "canonical_name": "commalg.MonomialIdeal.degree",
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    "doc": "",
    "domain": "commalg",
    "file": "noether/commalg/__init__.pyi",
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    "summary": "Compute degree for this `MonomialIdeal`.",
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    "canonical_name": "commalg.MonomialIdeal.dimension",
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    "domain": "commalg",
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    "qualified_name": "MonomialIdeal.dimension",
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    "summary": "Compute dimension for this `MonomialIdeal`.",
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    "canonical_name": "commalg.MonomialIdeal.finite_standard_monomials",
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    "doc": "",
    "domain": "commalg",
    "file": "noether/commalg/__init__.pyi",
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    "summary": "Compute finite standard monomials for this `MonomialIdeal`.",
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            "kind": "positional_or_keyword",
            "name": "value",
            "required": true,
            "schema": {
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        "signature": "(cls, value: str) -> MonomialIdeal[CoefficientRing]"
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    "qualified_name": "MonomialIdeal.from_json",
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    "signature": "(cls, value: str) -> MonomialIdeal[CoefficientRing]",
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    "canonical_name": "commalg.MonomialIdeal.hilbert_numerator",
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    "domain": "commalg",
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      "additionalProperties": false,
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    "summary": "Compute hilbert polynomial for this `MonomialIdeal`.",
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    "domain": "commalg",
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      "additionalProperties": false,
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    "summary": "Compute hilbert series for this `MonomialIdeal`.",
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    "canonical_name": "commalg.MonomialIdeal.intersection",
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    "domain": "commalg",
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        "signature": "(self, other: MonomialIdeal[C]) -> MonomialIdeal[C]"
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    "qualified_name": "MonomialIdeal.intersection",
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    "signature": "(self, other: MonomialIdeal[C]) -> MonomialIdeal[C]",
    "signatures": [
      "(self, other: MonomialIdeal[C]) -> MonomialIdeal[C]"
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      "properties": {},
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    "canonical_name": "commalg.MonomialIdeal.is_zero",
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      "properties": {},
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    "summary": "Test whether this `MonomialIdeal` is zero.",
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    "input_schema": {
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      "properties": {},
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    "kind": "property",
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    "module": "noether.commalg",
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            "type": "array",
            "x-python-type": "list[int]"
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          "type": "array",
          "x-python-type": "list[list[int]]"
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        "signature": "(self) -> list[list[int]]"
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    ],
    "qualified_name": "MonomialIdeal.minimal_generators",
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    "result_schema": {
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        "type": "array",
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    "doc": "",
    "domain": "commalg",
    "file": "noether/commalg/__init__.pyi",
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    "domain": "commalg",
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    "domain": "commalg",
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    "domain": "commalg",
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    "summary": "Compute count for this `PointCountSequence`.",
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    "doc": "",
    "domain": "commalg",
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    "domain": "commalg",
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    "summary": "Return the target recorded by this `PointCountSequence`.",
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    "canonical_name": "commalg.PointCountSequence.to_json",
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    "domain": "commalg",
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    "domain": "commalg",
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    "domain": "commalg",
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    "domain": "commalg",
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    "kind": "method",
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    "domain": "commalg",
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    "domain": "commalg",
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    "canonical_name": "commalg.PolynomialPrimeLocalMembership.ideal",
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    "doc": "",
    "domain": "commalg",
    "file": "noether/commalg/__init__.pyi",
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      "properties": {},
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    "doc": "",
    "domain": "commalg",
    "file": "noether/commalg/__init__.pyi",
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    "module": "noether.commalg",
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    "doc": "",
    "domain": "commalg",
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    "module": "noether.commalg",
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    "summary": "Return the multiplier recorded by this `PolynomialPrimeLocalMembership`.",
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    "canonical_name": "commalg.PolynomialPrimeLocalMembership.quotients",
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    "doc": "",
    "domain": "commalg",
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    "module": "noether.commalg",
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    "summary": "Return the quotients recorded by this `PolynomialPrimeLocalMembership`.",
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    "canonical_name": "commalg.PolynomialPrimeLocalMembership.to_json",
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    "doc": "",
    "domain": "commalg",
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    "module": "noether.commalg",
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    "summary": "Serialize this `PolynomialPrimeLocalMembership` to its canonical typed JSON representation.",
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    "domain": "commalg",
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    "module": "noether.commalg",
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    "summary": "Return the value recorded by this `PolynomialPrimeLocalMembership`.",
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    "canonical_name": "commalg.PolynomialPrimeLocalMembership.verify",
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    "doc": "",
    "domain": "commalg",
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    "module": "noether.commalg",
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    "summary": "Replay and verify the mathematical evidence carried by this `PolynomialPrimeLocalMembership`.",
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    "canonical_name": "commalg.PolynomialPrimeLocalization",
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    "file": "noether/commalg/__init__.pyi",
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    "module": "noether.commalg",
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    "domain": "commalg",
    "file": "noether/commalg/__init__.pyi",
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    "module": "noether.commalg",
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    "canonical_name": "commalg.PolynomialPrimeLocalization.from_json",
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    "domain": "commalg",
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    "module": "noether.commalg",
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        "signature": "(cls, value: str) -> PolynomialPrimeLocalization[CoefficientRing]"
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    "domain": "commalg",
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    "module": "noether.commalg",
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    "summary": "Construct a `PolynomialPrimeLocalization` using polynomial.",
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    "domain": "commalg",
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    "summary": "Compute generators for this `PolynomialPrimeLocalization`.",
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    "canonical_name": "commalg.PolynomialPrimeLocalization.ideal",
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    "domain": "commalg",
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      "properties": {
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    "canonical_name": "commalg.PolynomialPrimeLocalization.one",
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    "domain": "commalg",
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      "properties": {},
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    "module": "noether.commalg",
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    "canonical_name": "commalg.PolynomialRing.ideal_with_hilbert_series",
    "determinism": "deterministic",
    "doc": "Construct a field-polynomial ideal with an exact Hilbert-series target.",
    "domain": "commalg",
    "file": "noether/commalg/__init__.pyi",
    "input_schema": {
      "additionalProperties": false,
      "properties": {
        "generators": {
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          },
          "type": "array",
          "x-python-type": "Sequence[Polynomial[C]]"
        },
        "target": {
          "x-python-type": "HilbertSeries"
        }
      },
      "required": [
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        "target"
      ],
      "type": "object"
    },
    "kind": "method",
    "line": 1958,
    "module": "noether.commalg",
    "name": "ideal_with_hilbert_series",
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        "input_schema": {
          "additionalProperties": false,
          "properties": {
            "generators": {
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              },
              "type": "array",
              "x-python-type": "Sequence[Polynomial[C]]"
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            "target": {
              "x-python-type": "HilbertSeries"
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          },
          "required": [
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            "target"
          ],
          "type": "object"
        },
        "parameters": [
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            "kind": "positional_or_keyword",
            "name": "generators",
            "required": true,
            "schema": {
              "items": {
                "x-python-type": "Polynomial[C]"
              },
              "type": "array",
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          },
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            "kind": "positional_or_keyword",
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            "schema": {
              "x-python-type": "HilbertSeries"
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          }
        ],
        "result_schema": {
          "x-python-type": "Ideal[PolynomialRing[C]]"
        },
        "signature": "(self, generators: Sequence[Polynomial[C]], target: HilbertSeries) -> Ideal[PolynomialRing[C]]"
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    "qualified_name": "PolynomialRing.ideal_with_hilbert_series",
    "releases_gil": false,
    "result_schema": {
      "x-python-type": "Ideal[PolynomialRing[C]]"
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    "signature": "(self, generators: Sequence[Polynomial[C]], target: HilbertSeries) -> Ideal[PolynomialRing[C]]",
    "signatures": [
      "(self, generators: Sequence[Polynomial[C]], target: HilbertSeries) -> Ideal[PolynomialRing[C]]"
    ],
    "summary": "Construct a field-polynomial ideal with an exact Hilbert-series target.",
    "summary_source": "public docstring",
    "type_id": "noether/commalg/PolynomialRing@1"
  },
  {
    "canonical_name": "commalg.PolynomialRing.kahler_differentials",
    "determinism": "deterministic",
    "doc": "Exact Kähler differentials over the coefficient field.",
    "domain": "commalg",
    "file": "noether/commalg/__init__.pyi",
    "input_schema": {
      "additionalProperties": false,
      "properties": {},
      "type": "object"
    },
    "kind": "method",
    "line": 1895,
    "module": "noether.commalg",
    "name": "kahler_differentials",
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        "input_schema": {
          "additionalProperties": false,
          "properties": {},
          "type": "object"
        },
        "parameters": [],
        "result_schema": {
          "x-python-type": "KahlerDifferentials[RationalField]"
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        "signature": "(self: PolynomialRing[RationalField]) -> KahlerDifferentials[RationalField]"
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        "input_schema": {
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          "properties": {},
          "type": "object"
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        "parameters": [],
        "result_schema": {
          "x-python-type": "KahlerDifferentials[FiniteField]"
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        "signature": "(self: PolynomialRing[FiniteField]) -> KahlerDifferentials[FiniteField]"
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    "qualified_name": "PolynomialRing.kahler_differentials",
    "releases_gil": true,
    "result_schema": {
      "x-python-type": "KahlerDifferentials[RationalField]"
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    "signature": "(self: PolynomialRing[RationalField]) -> KahlerDifferentials[RationalField]",
    "signatures": [
      "(self: PolynomialRing[RationalField]) -> KahlerDifferentials[RationalField]",
      "(self: PolynomialRing[FiniteField]) -> KahlerDifferentials[FiniteField]"
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    "summary": "Exact Kähler differentials over the coefficient field.",
    "summary_source": "public docstring",
    "type_id": "noether/commalg/PolynomialRing@1"
  },
  {
    "canonical_name": "commalg.PolynomialRing.monomial_ideal",
    "determinism": "deterministic",
    "doc": "Construct a canonical monomial ideal from exponent vectors.",
    "domain": "commalg",
    "file": "noether/commalg/__init__.pyi",
    "input_schema": {
      "additionalProperties": false,
      "properties": {
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              "type": "integer"
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            "type": "array",
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          "x-python-type": "Iterable[Sequence[int]]"
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      },
      "required": [
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      "type": "object"
    },
    "kind": "method",
    "line": 1963,
    "module": "noether.commalg",
    "name": "monomial_ideal",
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        "input_schema": {
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          "properties": {
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              "type": "array",
              "x-python-type": "Iterable[Sequence[int]]"
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          },
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          "type": "object"
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        "parameters": [
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            "kind": "positional_or_keyword",
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              "type": "array",
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        ],
        "result_schema": {
          "x-python-type": "MonomialIdeal[C]"
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        "signature": "(self, generators: Iterable[Sequence[int]]) -> MonomialIdeal[C]"
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    "qualified_name": "PolynomialRing.monomial_ideal",
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    "result_schema": {
      "x-python-type": "MonomialIdeal[C]"
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    "signature": "(self, generators: Iterable[Sequence[int]]) -> MonomialIdeal[C]",
    "signatures": [
      "(self, generators: Iterable[Sequence[int]]) -> MonomialIdeal[C]"
    ],
    "summary": "Construct a canonical monomial ideal from exponent vectors.",
    "summary_source": "public docstring",
    "type_id": "noether/commalg/PolynomialRing@1"
  },
  {
    "canonical_name": "commalg.PolynomialRing.monomials_in_region",
    "determinism": "deterministic",
    "doc": "Monomials in one explicit finite truncation region, in term order.",
    "domain": "commalg",
    "file": "noether/commalg/__init__.pyi",
    "input_schema": {
      "additionalProperties": false,
      "properties": {
        "max_count": {
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        "region": {
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      },
      "required": [
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        "max_count"
      ],
      "type": "object"
    },
    "kind": "method",
    "line": 1891,
    "module": "noether.commalg",
    "name": "monomials_in_region",
    "overloads": [
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          "properties": {
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        "parameters": [
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            "kind": "positional_or_keyword",
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          },
          {
            "kind": "keyword_only",
            "name": "max_count",
            "required": true,
            "schema": {
              "type": "integer"
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          }
        ],
        "result_schema": {
          "items": {
            "x-python-type": "Polynomial[C]"
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          "type": "array",
          "x-python-type": "list[Polynomial[C]]"
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        "signature": "(self, region: SeriesTruncation, *, max_count: int) -> list[Polynomial[C]]"
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    ],
    "qualified_name": "PolynomialRing.monomials_in_region",
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    "result_schema": {
      "items": {
        "x-python-type": "Polynomial[C]"
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      "type": "array",
      "x-python-type": "list[Polynomial[C]]"
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    "signature": "(self, region: SeriesTruncation, *, max_count: int) -> list[Polynomial[C]]",
    "signatures": [
      "(self, region: SeriesTruncation, *, max_count: int) -> list[Polynomial[C]]"
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    "summary": "Monomials in one explicit finite truncation region, in term order.",
    "summary_source": "public docstring",
    "type_id": "noether/commalg/PolynomialRing@1"
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  {
    "canonical_name": "commalg.PolynomialRing.nullstellensatz_certificate",
    "determinism": "deterministic",
    "doc": "Compute rational Nullstellensatz multipliers for integral generators\nwithout first constructing a strong basis over `Z`.",
    "domain": "commalg",
    "file": "noether/commalg/__init__.pyi",
    "input_schema": {
      "additionalProperties": false,
      "properties": {
        "generators": {
          "items": {
            "x-python-type": "Polynomial[IntegerRing]"
          },
          "type": "array",
          "x-python-type": "Sequence[Polynomial[IntegerRing]]"
        }
      },
      "required": [
        "generators"
      ],
      "type": "object"
    },
    "kind": "method",
    "line": 1964,
    "module": "noether.commalg",
    "name": "nullstellensatz_certificate",
    "overloads": [
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        "input_schema": {
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          "properties": {
            "generators": {
              "items": {
                "x-python-type": "Polynomial[IntegerRing]"
              },
              "type": "array",
              "x-python-type": "Sequence[Polynomial[IntegerRing]]"
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          },
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          ],
          "type": "object"
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        "parameters": [
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            "kind": "positional_or_keyword",
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              },
              "type": "array",
              "x-python-type": "Sequence[Polynomial[IntegerRing]]"
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        "result_schema": {
          "x-python-type": "ArithmeticNullstellensatzCertificate"
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        "signature": "(self: PolynomialRing[IntegerRing], generators: Sequence[Polynomial[IntegerRing]]) -> ArithmeticNullstellensatzCertificate"
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    ],
    "qualified_name": "PolynomialRing.nullstellensatz_certificate",
    "releases_gil": true,
    "result_schema": {
      "x-python-type": "ArithmeticNullstellensatzCertificate"
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    "signature": "(self: PolynomialRing[IntegerRing], generators: Sequence[Polynomial[IntegerRing]]) -> ArithmeticNullstellensatzCertificate",
    "signatures": [
      "(self: PolynomialRing[IntegerRing], generators: Sequence[Polynomial[IntegerRing]]) -> ArithmeticNullstellensatzCertificate"
    ],
    "summary": "Compute rational Nullstellensatz multipliers for integral generators without first constructing a strong basis over `Z`.",
    "summary_source": "public docstring",
    "type_id": "noether/commalg/PolynomialRing@1"
  },
  {
    "canonical_name": "commalg.PolynomialRing.one",
    "determinism": "deterministic",
    "doc": "",
    "domain": "commalg",
    "file": "noether/commalg/__init__.pyi",
    "input_schema": {
      "additionalProperties": false,
      "properties": {},
      "type": "object"
    },
    "kind": "method",
    "line": 1881,
    "module": "noether.commalg",
    "name": "one",
    "overloads": [
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        "input_schema": {
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          "properties": {},
          "type": "object"
        },
        "parameters": [],
        "result_schema": {
          "x-python-type": "Polynomial[C]"
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        "signature": "(self) -> Polynomial[C]"
      }
    ],
    "qualified_name": "PolynomialRing.one",
    "releases_gil": false,
    "result_schema": {
      "x-python-type": "Polynomial[C]"
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    "signature": "(self) -> Polynomial[C]",
    "signatures": [
      "(self) -> Polynomial[C]"
    ],
    "summary": "Compute one for this `PolynomialRing`.",
    "summary_source": "derived projection vocabulary",
    "type_id": "noether/commalg/PolynomialRing@1"
  },
  {
    "canonical_name": "commalg.PolynomialRing.quotient",
    "determinism": "deterministic",
    "doc": "Form the exact quotient by an ideal in this polynomial ring.",
    "domain": "commalg",
    "file": "noether/commalg/__init__.pyi",
    "input_schema": {
      "additionalProperties": false,
      "properties": {
        "ideal": {
          "x-python-type": "Ideal[PolynomialRing[_ProjectedIdealFieldContext]]"
        }
      },
      "required": [
        "ideal"
      ],
      "type": "object"
    },
    "kind": "method",
    "line": 1954,
    "module": "noether.commalg",
    "name": "quotient",
    "overloads": [
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        "input_schema": {
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          "properties": {
            "ideal": {
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          },
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        "parameters": [
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            "required": true,
            "schema": {
              "x-python-type": "Ideal[PolynomialRing[_ProjectedIdealFieldContext]]"
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          }
        ],
        "result_schema": {
          "x-python-type": "QuotientRing[_ProjectedIdealFieldContext]"
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        "signature": "(self: PolynomialRing[_ProjectedIdealFieldContext], ideal: Ideal[PolynomialRing[_ProjectedIdealFieldContext]]) -> QuotientRing[_ProjectedIdealFieldContext]"
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    ],
    "qualified_name": "PolynomialRing.quotient",
    "releases_gil": false,
    "result_schema": {
      "x-python-type": "QuotientRing[_ProjectedIdealFieldContext]"
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    "signature": "(self: PolynomialRing[_ProjectedIdealFieldContext], ideal: Ideal[PolynomialRing[_ProjectedIdealFieldContext]]) -> QuotientRing[_ProjectedIdealFieldContext]",
    "signatures": [
      "(self: PolynomialRing[_ProjectedIdealFieldContext], ideal: Ideal[PolynomialRing[_ProjectedIdealFieldContext]]) -> QuotientRing[_ProjectedIdealFieldContext]"
    ],
    "summary": "Form the exact quotient by an ideal in this polynomial ring.",
    "summary_source": "public docstring",
    "type_id": "noether/commalg/PolynomialRing@1"
  },
  {
    "canonical_name": "commalg.PolynomialRing.strong_groebner_basis",
    "determinism": "deterministic",
    "doc": "Compute a normalized strong Groebner basis over `Z`.",
    "domain": "commalg",
    "file": "noether/commalg/__init__.pyi",
    "input_schema": {
      "additionalProperties": false,
      "properties": {
        "generators": {
          "items": {
            "x-python-type": "Polynomial[IntegerRing]"
          },
          "type": "array",
          "x-python-type": "Sequence[Polynomial[IntegerRing]]"
        }
      },
      "required": [
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      ],
      "type": "object"
    },
    "kind": "method",
    "line": 1968,
    "module": "noether.commalg",
    "name": "strong_groebner_basis",
    "overloads": [
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        ],
        "result_schema": {
          "x-python-type": "StrongGroebnerBasis"
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        "signature": "(self: PolynomialRing[IntegerRing], generators: Sequence[Polynomial[IntegerRing]]) -> StrongGroebnerBasis"
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    ],
    "qualified_name": "PolynomialRing.strong_groebner_basis",
    "releases_gil": true,
    "result_schema": {
      "x-python-type": "StrongGroebnerBasis"
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    "signature": "(self: PolynomialRing[IntegerRing], generators: Sequence[Polynomial[IntegerRing]]) -> StrongGroebnerBasis",
    "signatures": [
      "(self: PolynomialRing[IntegerRing], generators: Sequence[Polynomial[IntegerRing]]) -> StrongGroebnerBasis"
    ],
    "summary": "Compute a normalized strong Groebner basis over `Z`.",
    "summary_source": "public docstring",
    "type_id": "noether/commalg/PolynomialRing@1"
  },
  {
    "canonical_name": "commalg.PolynomialRing.term_order",
    "determinism": "deterministic",
    "doc": "",
    "domain": "commalg",
    "file": "noether/commalg/__init__.pyi",
    "input_schema": {
      "additionalProperties": false,
      "properties": {},
      "type": "object"
    },
    "kind": "property",
    "line": 1875,
    "module": "noether.commalg",
    "name": "term_order",
    "overloads": [
      {
        "input_schema": {
          "additionalProperties": false,
          "properties": {},
          "type": "object"
        },
        "parameters": [],
        "result_schema": {
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    ],
    "qualified_name": "PolynomialRing.term_order",
    "releases_gil": false,
    "result_schema": {
      "x-python-type": "MonomialOrder"
    },
    "signature": "(self) -> MonomialOrder",
    "signatures": [
      "(self) -> MonomialOrder"
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    "summary": "Return the term order recorded by this `PolynomialRing`.",
    "summary_source": "derived projection vocabulary",
    "type_id": "noether/commalg/PolynomialRing@1"
  },
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    "canonical_name": "commalg.PolynomialRing.to_json",
    "determinism": "deterministic",
    "doc": "Serialize the complete coefficient, variable, order, and grading context.",
    "domain": "commalg",
    "file": "noether/commalg/__init__.pyi",
    "input_schema": {
      "additionalProperties": false,
      "properties": {},
      "type": "object"
    },
    "kind": "method",
    "line": 1971,
    "module": "noether.commalg",
    "name": "to_json",
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        "input_schema": {
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          "properties": {},
          "type": "object"
        },
        "parameters": [],
        "result_schema": {
          "type": "string"
        },
        "signature": "(self) -> str"
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    ],
    "qualified_name": "PolynomialRing.to_json",
    "releases_gil": false,
    "result_schema": {
      "type": "string"
    },
    "signature": "(self) -> str",
    "signatures": [
      "(self) -> str"
    ],
    "summary": "Serialize the complete coefficient, variable, order, and grading context.",
    "summary_source": "public docstring",
    "type_id": "noether/commalg/PolynomialRing@1"
  },
  {
    "canonical_name": "commalg.PolynomialRing.variables",
    "determinism": "deterministic",
    "doc": "",
    "domain": "commalg",
    "file": "noether/commalg/__init__.pyi",
    "input_schema": {
      "additionalProperties": false,
      "properties": {},
      "type": "object"
    },
    "kind": "property",
    "line": 1873,
    "module": "noether.commalg",
    "name": "variables",
    "overloads": [
      {
        "input_schema": {
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          "properties": {},
          "type": "object"
        },
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          "items": {
            "type": "string"
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          "x-python-type": "list[str]"
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    ],
    "qualified_name": "PolynomialRing.variables",
    "releases_gil": false,
    "result_schema": {
      "items": {
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      "type": "array",
      "x-python-type": "list[str]"
    },
    "signature": "(self) -> list[str]",
    "signatures": [
      "(self) -> list[str]"
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    "summary": "Return the variables recorded by this `PolynomialRing`.",
    "summary_source": "derived projection vocabulary",
    "type_id": "noether/commalg/PolynomialRing@1"
  },
  {
    "canonical_name": "commalg.PolynomialRing.weighted_grading",
    "determinism": "deterministic",
    "doc": "Positive rank-one view of the grading, when one exists.",
    "domain": "commalg",
    "file": "noether/commalg/__init__.pyi",
    "input_schema": {
      "additionalProperties": false,
      "properties": {},
      "type": "object"
    },
    "kind": "property",
    "line": 1879,
    "module": "noether.commalg",
    "name": "weighted_grading",
    "overloads": [
      {
        "input_schema": {
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        "result_schema": {
          "anyOf": [
            {
              "x-python-type": "WeightedGrading"
            },
            {
              "type": "null"
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          ],
          "x-python-type": "WeightedGrading | None"
        },
        "signature": "(self) -> WeightedGrading | None"
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    ],
    "qualified_name": "PolynomialRing.weighted_grading",
    "releases_gil": false,
    "result_schema": {
      "anyOf": [
        {
          "x-python-type": "WeightedGrading"
        },
        {
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      "x-python-type": "WeightedGrading | None"
    },
    "signature": "(self) -> WeightedGrading | None",
    "signatures": [
      "(self) -> WeightedGrading | None"
    ],
    "summary": "Positive rank-one view of the grading, when one exists.",
    "summary_source": "public docstring",
    "type_id": "noether/commalg/PolynomialRing@1"
  },
  {
    "canonical_name": "commalg.PolynomialRing.zero",
    "determinism": "deterministic",
    "doc": "",
    "domain": "commalg",
    "file": "noether/commalg/__init__.pyi",
    "input_schema": {
      "additionalProperties": false,
      "properties": {},
      "type": "object"
    },
    "kind": "method",
    "line": 1880,
    "module": "noether.commalg",
    "name": "zero",
    "overloads": [
      {
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    "domain": "commalg",
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          }
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    "qualified_name": "PowerSeries.monomial_substitution",
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    "signature": "(self, monomial_map: MonomialMap) -> PowerSeries[C]",
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    "summary": "Compute monomial substitution for this `PowerSeries`.",
    "summary_source": "derived projection vocabulary",
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    "domain": "commalg",
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    "canonical_name": "commalg.ProjectiveCompleteIntersection.fundamental_class",
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    "domain": "commalg",
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    "summary": "Compute fundamental class for this `ProjectiveCompleteIntersection`.",
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    "canonical_name": "commalg.ProjectiveCompleteIntersection.hypotheses",
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    "domain": "commalg",
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    "summary": "Return the hypotheses recorded by this `ProjectiveCompleteIntersection`.",
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    "canonical_name": "commalg.ProjectiveCompleteIntersection.integrate_restriction",
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    "domain": "commalg",
    "file": "noether/commalg/__init__.pyi",
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    "summary": "Compute integrate restriction for this `ProjectiveCompleteIntersection`.",
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    "domain": "commalg",
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    "input_schema": {
      "additionalProperties": false,
      "properties": {
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        "result_schema": {
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    "signature": "(self, multidegree: Sequence[int]) -> RationalChowClass",
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    "summary": "Compute line bundle chern character for this `ProjectiveCompleteIntersection`.",
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    "canonical_name": "commalg.ProjectiveCompleteIntersection.line_bundle_euler_characteristic",
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    "doc": "",
    "domain": "commalg",
    "file": "noether/commalg/__init__.pyi",
    "input_schema": {
      "additionalProperties": false,
      "properties": {
        "multidegree": {
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      "required": [
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      "type": "object"
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              "type": "array",
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        "result_schema": {
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    "summary": "Compute line bundle euler characteristic for this `ProjectiveCompleteIntersection`.",
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    "canonical_name": "commalg.ProjectiveCompleteIntersection.line_bundle_euler_characteristic_koszul",
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    "domain": "commalg",
    "file": "noether/commalg/__init__.pyi",
    "input_schema": {
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      "properties": {
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    "kind": "method",
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    "summary": "Compute line bundle euler characteristic koszul for this `ProjectiveCompleteIntersection`.",
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    "summary": "Return the multidegrees recorded by this `ProjectiveCompleteIntersection`.",
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    "canonical_name": "commalg.ProjectiveCompleteIntersection.normal_total_chern_class",
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    "doc": "",
    "domain": "commalg",
    "file": "noether/commalg/__init__.pyi",
    "input_schema": {
      "additionalProperties": false,
      "properties": {},
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    "summary": "Compute normal total chern class for this `ProjectiveCompleteIntersection`.",
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    "type_id": "noether/commalg/ProjectiveCompleteIntersection@1"
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    "canonical_name": "commalg.ProjectiveCompleteIntersection.pushforward_restriction",
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    "doc": "",
    "domain": "commalg",
    "file": "noether/commalg/__init__.pyi",
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    "module": "noether.commalg",
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    "summary": "Compute pushforward restriction for this `ProjectiveCompleteIntersection`.",
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    "canonical_name": "commalg.ProjectiveCompleteIntersection.ring",
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    "domain": "commalg",
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    "qualified_name": "ProjectiveCompleteIntersection.ring",
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    "signature": "(self) -> ChowRing",
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    "summary": "Return the ring recorded by this `ProjectiveCompleteIntersection`.",
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    "canonical_name": "commalg.ProjectiveCompleteIntersection.tangent_total_chern_class",
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    "doc": "",
    "domain": "commalg",
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    "qualified_name": "ProjectiveCompleteIntersection.tangent_total_chern_class",
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    "summary": "Compute tangent total chern class for this `ProjectiveCompleteIntersection`.",
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        "signature": "(self, other: RationalFunction[C] | Polynomial[C] | FunctionCoefficient) -> RationalFunction[C]"
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    "signature": "(self, other: RationalFunction[C] | Polynomial[C] | FunctionCoefficient) -> RationalFunction[C]",
    "signatures": [
      "(self, other: RationalFunction[C] | Polynomial[C] | FunctionCoefficient) -> RationalFunction[C]"
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    "summary": "Divide this value by another compatible value.",
    "summary_source": "derived projection vocabulary",
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    "canonical_name": "commalg.RationalFunction.denominator",
    "determinism": "deterministic",
    "doc": "",
    "domain": "commalg",
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      "properties": {},
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    "line": 3086,
    "module": "noether.commalg",
    "name": "denominator",
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    "qualified_name": "RationalFunction.denominator",
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    "signatures": [
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    "summary": "Return the denominator recorded by this `RationalFunction`.",
    "summary_source": "derived projection vocabulary",
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    "canonical_name": "commalg.RationalFunction.field",
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    "doc": "",
    "domain": "commalg",
    "file": "noether/commalg/__init__.pyi",
    "input_schema": {
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      "properties": {},
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    "kind": "property",
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    "module": "noether.commalg",
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    "summary": "Return the field recorded by this `RationalFunction`.",
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    "doc": "Deserialize and revalidate a supported canonical rational function.",
    "domain": "commalg",
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    "module": "noether.commalg",
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    "qualified_name": "RationalFunction.from_json",
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    "signature": "(cls, value: str) -> RationalFunction[FunctionCoefficientRing]",
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      "(cls, value: str) -> RationalFunction[FunctionCoefficientRing]"
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    "summary": "Deserialize and revalidate a supported canonical rational function.",
    "summary_source": "public docstring",
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    "canonical_name": "commalg.RationalFunction.inverse",
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    "doc": "",
    "domain": "commalg",
    "file": "noether/commalg/__init__.pyi",
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      "properties": {},
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    "kind": "method",
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    "module": "noether.commalg",
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    "signatures": [
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    "summary": "Compute inverse for this `RationalFunction`.",
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    "canonical_name": "commalg.RationalFunction.is_nth_power",
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    "doc": "Whether this rational function is an exact positive integral power.",
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    "file": "noether/commalg/__init__.pyi",
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      "properties": {
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      },
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    "kind": "method",
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    "module": "noether.commalg",
    "name": "is_nth_power",
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    "summary": "Whether this rational function is an exact positive integral power.",
    "summary_source": "public docstring",
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    "canonical_name": "commalg.RationalFunction.is_one",
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    "doc": "",
    "domain": "commalg",
    "file": "noether/commalg/__init__.pyi",
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      "properties": {},
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    "module": "noether.commalg",
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    "signatures": [
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    "summary_source": "derived projection vocabulary",
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    "canonical_name": "commalg.RationalFunction.is_zero",
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    "domain": "commalg",
    "file": "noether/commalg/__init__.pyi",
    "input_schema": {
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      "properties": {},
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    "kind": "method",
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    "module": "noether.commalg",
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    "domain": "commalg",
    "file": "noether/commalg/__init__.pyi",
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    "module": "noether.commalg",
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    "summary": "Extract a deterministic exact root, or return `None` when none exists.",
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    "canonical_name": "commalg.RationalFunction.numerator",
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    "doc": "",
    "domain": "commalg",
    "file": "noether/commalg/__init__.pyi",
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    "module": "noether.commalg",
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    "qualified_name": "RationalFunction.numerator",
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    "summary": "Return the numerator recorded by this `RationalFunction`.",
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    "canonical_name": "commalg.RationalFunction.pow",
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    "doc": "",
    "domain": "commalg",
    "file": "noether/commalg/__init__.pyi",
    "input_schema": {
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    "module": "noether.commalg",
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    "summary": "Compute pow for this `RationalFunction`.",
    "summary_source": "derived projection vocabulary",
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    "canonical_name": "commalg.RationalFunction.specialize",
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    "domain": "commalg",
    "file": "noether/commalg/__init__.pyi",
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    "module": "noether.commalg",
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    "summary": "Compute specialize for this `RationalFunction`.",
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    "canonical_name": "commalg.RationalFunction.to_json",
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    "doc": "Serialize the field context and canonical reduced fraction.",
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    "input_schema": {
      "additionalProperties": false,
      "properties": {},
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    "module": "noether.commalg",
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    "summary": "Serialize the field context and canonical reduced fraction.",
    "summary_source": "public docstring",
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    "doc": "A rational function field over `Q` or one concrete finite field.",
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    "file": "noether/commalg/__init__.pyi",
    "input_schema": {
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    "module": "noether.commalg",
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    "summary": "Coerce a scalar, polynomial, or rational function into this fraction field.",
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    "summary": "Compute constant for this `RationalFunctionField`.",
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    "domain": "commalg",
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    "module": "noether.commalg",
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    "summary": "Construct a `RationalFunctionField` using polynomial.",
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    "domain": "commalg",
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    "module": "noether.commalg",
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    "summary": "Compute generator for this `RationalFunctionField`.",
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    "domain": "commalg",
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    "domain": "commalg",
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    "summary": "Compute one for this `RationalFunctionField`.",
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    "canonical_name": "commalg.RationalFunctionField.polynomial_ring",
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    "domain": "commalg",
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    "module": "noether.commalg",
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    "signatures": [
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    "summary": "Return the polynomial ring recorded by this `RationalFunctionField`.",
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    "type_id": "noether/commalg/RationalFunctionField@1"
  },
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    "canonical_name": "commalg.RationalFunctionField.to_json",
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    "doc": "Serialize the complete coefficient-field and variable context.",
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    "input_schema": {
      "additionalProperties": false,
      "properties": {},
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    "module": "noether.commalg",
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    "qualified_name": "RationalFunctionField.to_json",
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      "x-python-type": "ResidueClass"
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    "signatures": [
      "(self, exponent: int, modulus: None=None) -> ResidueClass"
    ],
    "summary": "Raise this value to the supplied power.",
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    "type_id": "noether/commalg/ResidueClass@1"
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    "canonical_name": "commalg.ResidueClass.__sub__",
    "determinism": "deterministic",
    "doc": "",
    "domain": "commalg",
    "file": "noether/commalg/__init__.pyi",
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    "module": "noether.commalg",
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      "(self, other: ResidueClass) -> ResidueClass"
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    "summary": "Subtract another compatible value from this value.",
    "summary_source": "derived projection vocabulary",
    "type_id": "noether/commalg/ResidueClass@1"
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    "canonical_name": "commalg.ResidueClass.from_json",
    "determinism": "deterministic",
    "doc": "Deserialize and revalidate a residue class and its ring context.",
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    "module": "noether.commalg",
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          "x-python-type": "ResidueClass"
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    "qualified_name": "ResidueClass.from_json",
    "releases_gil": false,
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      "x-python-type": "ResidueClass"
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    "signatures": [
      "(cls, value: str) -> ResidueClass"
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    "summary": "Deserialize and revalidate a residue class and its ring context.",
    "summary_source": "public docstring",
    "type_id": "noether/commalg/ResidueClass@1"
  },
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    "canonical_name": "commalg.ResidueClass.inverse",
    "determinism": "deterministic",
    "doc": "",
    "domain": "commalg",
    "file": "noether/commalg/__init__.pyi",
    "input_schema": {
      "additionalProperties": false,
      "properties": {},
      "type": "object"
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    "kind": "method",
    "line": 1433,
    "module": "noether.commalg",
    "name": "inverse",
    "overloads": [
      {
        "input_schema": {
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          "properties": {},
          "type": "object"
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          "x-python-type": "ResidueClass"
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    "qualified_name": "ResidueClass.inverse",
    "releases_gil": false,
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      "x-python-type": "ResidueClass"
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    "signatures": [
      "(self) -> ResidueClass"
    ],
    "summary": "Compute inverse for this `ResidueClass`.",
    "summary_source": "derived projection vocabulary",
    "type_id": "noether/commalg/ResidueClass@1"
  },
  {
    "canonical_name": "commalg.ResidueClass.is_one",
    "determinism": "deterministic",
    "doc": "",
    "domain": "commalg",
    "file": "noether/commalg/__init__.pyi",
    "input_schema": {
      "additionalProperties": false,
      "properties": {},
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    "kind": "method",
    "line": 1431,
    "module": "noether.commalg",
    "name": "is_one",
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      "(self) -> bool"
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    "summary": "Test whether this `ResidueClass` is one.",
    "summary_source": "derived projection vocabulary",
    "type_id": "noether/commalg/ResidueClass@1"
  },
  {
    "canonical_name": "commalg.ResidueClass.is_unit",
    "determinism": "deterministic",
    "doc": "",
    "domain": "commalg",
    "file": "noether/commalg/__init__.pyi",
    "input_schema": {
      "additionalProperties": false,
      "properties": {},
      "type": "object"
    },
    "kind": "method",
    "line": 1432,
    "module": "noether.commalg",
    "name": "is_unit",
    "overloads": [
      {
        "input_schema": {
          "additionalProperties": false,
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    "qualified_name": "ResidueClass.is_unit",
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    "summary": "Test whether this `ResidueClass` is unit.",
    "summary_source": "derived projection vocabulary",
    "type_id": "noether/commalg/ResidueClass@1"
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  {
    "canonical_name": "commalg.ResidueClass.is_zero",
    "determinism": "deterministic",
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    "domain": "commalg",
    "file": "noether/commalg/__init__.pyi",
    "input_schema": {
      "additionalProperties": false,
      "properties": {},
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    "kind": "method",
    "line": 1430,
    "module": "noether.commalg",
    "name": "is_zero",
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    "qualified_name": "ResidueClass.is_zero",
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      "(self) -> bool"
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    "summary": "Test whether this `ResidueClass` is zero.",
    "summary_source": "derived projection vocabulary",
    "type_id": "noether/commalg/ResidueClass@1"
  },
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    "canonical_name": "commalg.ResidueClass.pow",
    "determinism": "deterministic",
    "doc": "",
    "domain": "commalg",
    "file": "noether/commalg/__init__.pyi",
    "input_schema": {
      "additionalProperties": false,
      "properties": {
        "exponent": {
          "type": "integer"
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      },
      "required": [
        "exponent"
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      "type": "object"
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    "kind": "method",
    "line": 1434,
    "module": "noether.commalg",
    "name": "pow",
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        "input_schema": {
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          "properties": {
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        "parameters": [
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            "kind": "positional_or_keyword",
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              "type": "integer"
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          }
        ],
        "result_schema": {
          "x-python-type": "ResidueClass"
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        "signature": "(self, exponent: int) -> ResidueClass"
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    ],
    "qualified_name": "ResidueClass.pow",
    "releases_gil": false,
    "result_schema": {
      "x-python-type": "ResidueClass"
    },
    "signature": "(self, exponent: int) -> ResidueClass",
    "signatures": [
      "(self, exponent: int) -> ResidueClass"
    ],
    "summary": "Compute pow for this `ResidueClass`.",
    "summary_source": "derived projection vocabulary",
    "type_id": "noether/commalg/ResidueClass@1"
  },
  {
    "canonical_name": "commalg.ResidueClass.representative",
    "determinism": "deterministic",
    "doc": "",
    "domain": "commalg",
    "file": "noether/commalg/__init__.pyi",
    "input_schema": {
      "additionalProperties": false,
      "properties": {},
      "type": "object"
    },
    "kind": "property",
    "line": 1429,
    "module": "noether.commalg",
    "name": "representative",
    "overloads": [
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          "type": "object"
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        "result_schema": {
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    ],
    "qualified_name": "ResidueClass.representative",
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    "result_schema": {
      "type": "integer"
    },
    "signature": "(self) -> int",
    "signatures": [
      "(self) -> int"
    ],
    "summary": "Return the representative recorded by this `ResidueClass`.",
    "summary_source": "derived projection vocabulary",
    "type_id": "noether/commalg/ResidueClass@1"
  },
  {
    "canonical_name": "commalg.ResidueClass.ring",
    "determinism": "deterministic",
    "doc": "",
    "domain": "commalg",
    "file": "noether/commalg/__init__.pyi",
    "input_schema": {
      "additionalProperties": false,
      "properties": {},
      "type": "object"
    },
    "kind": "property",
    "line": 1427,
    "module": "noether.commalg",
    "name": "ring",
    "overloads": [
      {
        "input_schema": {
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          "properties": {},
          "type": "object"
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        "parameters": [],
        "result_schema": {
          "x-python-type": "ResidueRing"
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        "signature": "(self) -> ResidueRing"
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    ],
    "qualified_name": "ResidueClass.ring",
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    "result_schema": {
      "x-python-type": "ResidueRing"
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    "signatures": [
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    ],
    "summary": "Return the ring recorded by this `ResidueClass`.",
    "summary_source": "derived projection vocabulary",
    "type_id": "noether/commalg/ResidueClass@1"
  },
  {
    "canonical_name": "commalg.ResidueClass.to_json",
    "determinism": "deterministic",
    "doc": "Serialize the ring context and canonical representative as typed JSON.",
    "domain": "commalg",
    "file": "noether/commalg/__init__.pyi",
    "input_schema": {
      "additionalProperties": false,
      "properties": {},
      "type": "object"
    },
    "kind": "method",
    "line": 1435,
    "module": "noether.commalg",
    "name": "to_json",
    "overloads": [
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        "input_schema": {
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          "properties": {},
          "type": "object"
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        "result_schema": {
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    "qualified_name": "ResidueClass.to_json",
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      "(self) -> str"
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    "summary": "Serialize the ring context and canonical representative as typed JSON.",
    "summary_source": "public docstring",
    "type_id": "noether/commalg/ResidueClass@1"
  },
  {
    "canonical_name": "commalg.ResidueRing",
    "determinism": "deterministic",
    "doc": "The exact finite ring of integers modulo an arbitrary-size modulus.",
    "domain": "commalg",
    "file": "noether/commalg/__init__.pyi",
    "input_schema": {
      "additionalProperties": false,
      "properties": {
        "modulus": {
          "type": "integer"
        }
      },
      "required": [
        "modulus"
      ],
      "type": "object"
    },
    "kind": "class",
    "line": 1410,
    "module": "noether.commalg",
    "name": "ResidueRing",
    "overloads": [
      {
        "input_schema": {
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          "properties": {
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          },
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        "parameters": [
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            "kind": "positional_or_keyword",
            "name": "modulus",
            "required": true,
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        "result_schema": {
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        "signature": "(self, modulus: int) -> None"
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    ],
    "qualified_name": "ResidueRing",
    "releases_gil": false,
    "result_schema": {
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    },
    "signature": "(self, modulus: int) -> None",
    "signatures": [
      "(self, modulus: int) -> None"
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    "summary": "The exact finite ring of integers modulo an arbitrary-size modulus.",
    "summary_source": "public docstring",
    "type_id": "noether/commalg/ResidueRing@1"
  },
  {
    "canonical_name": "commalg.ResidueRing.__call__",
    "determinism": "deterministic",
    "doc": "",
    "domain": "commalg",
    "example": "from noether.commalg import ResidueRing\nring = ResidueRing(15)\nresult = ring(8) * ring(2)\nassert result == ring(1)",
    "file": "noether/commalg/__init__.pyi",
    "input_schema": {
      "additionalProperties": false,
      "properties": {
        "value": {
          "type": "integer"
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      },
      "required": [
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      "type": "object"
    },
    "kind": "method",
    "line": 1422,
    "module": "noether.commalg",
    "name": "__call__",
    "overloads": [
      {
        "input_schema": {
          "additionalProperties": false,
          "properties": {
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        "parameters": [
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            "kind": "positional_or_keyword",
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            "required": true,
            "schema": {
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            }
          }
        ],
        "result_schema": {
          "x-python-type": "ResidueClass"
        },
        "signature": "(self, value: int) -> ResidueClass"
      }
    ],
    "qualified_name": "ResidueRing.__call__",
    "releases_gil": false,
    "result_schema": {
      "x-python-type": "ResidueClass"
    },
    "signature": "(self, value: int) -> ResidueClass",
    "signatures": [
      "(self, value: int) -> ResidueClass"
    ],
    "summary": "Coerce the supplied input through this context or mathematical map.",
    "summary_source": "derived projection vocabulary",
    "supported_rings": [
      "Z/nZ"
    ],
    "type_id": "noether/commalg/ResidueRing@1"
  },
  {
    "canonical_name": "commalg.ResidueRing.cardinality",
    "determinism": "deterministic",
    "doc": "",
    "domain": "commalg",
    "file": "noether/commalg/__init__.pyi",
    "input_schema": {
      "additionalProperties": false,
      "properties": {},
      "type": "object"
    },
    "kind": "method",
    "line": 1417,
    "module": "noether.commalg",
    "name": "cardinality",
    "overloads": [
      {
        "input_schema": {
          "additionalProperties": false,
          "properties": {},
          "type": "object"
        },
        "parameters": [],
        "result_schema": {
          "type": "integer"
        },
        "signature": "(self) -> int"
      }
    ],
    "qualified_name": "ResidueRing.cardinality",
    "releases_gil": false,
    "result_schema": {
      "type": "integer"
    },
    "signature": "(self) -> int",
    "signatures": [
      "(self) -> int"
    ],
    "summary": "Compute cardinality for this `ResidueRing`.",
    "summary_source": "derived projection vocabulary",
    "type_id": "noether/commalg/ResidueRing@1"
  },
  {
    "canonical_name": "commalg.ResidueRing.crt",
    "determinism": "deterministic",
    "doc": "Return the canonical prime-power Chinese-remainder decomposition.",
    "domain": "commalg",
    "file": "noether/commalg/__init__.pyi",
    "input_schema": {
      "additionalProperties": false,
      "properties": {},
      "type": "object"
    },
    "kind": "method",
    "line": 1418,
    "module": "noether.commalg",
    "name": "crt",
    "overloads": [
      {
        "input_schema": {
          "additionalProperties": false,
          "properties": {},
          "type": "object"
        },
        "parameters": [],
        "result_schema": {
          "x-python-type": "CrtDecomposition"
        },
        "signature": "(self) -> CrtDecomposition"
      }
    ],
    "qualified_name": "ResidueRing.crt",
    "releases_gil": false,
    "result_schema": {
      "x-python-type": "CrtDecomposition"
    },
    "signature": "(self) -> CrtDecomposition",
    "signatures": [
      "(self) -> CrtDecomposition"
    ],
    "summary": "Return the canonical prime-power Chinese-remainder decomposition.",
    "summary_source": "public docstring",
    "type_id": "noether/commalg/ResidueRing@1"
  },
  {
    "canonical_name": "commalg.ResidueRing.from_json",
    "determinism": "deterministic",
    "doc": "Deserialize and revalidate a residue-ring context.",
    "domain": "commalg",
    "file": "noether/commalg/__init__.pyi",
    "input_schema": {
      "additionalProperties": false,
      "properties": {
        "value": {
          "type": "string"
        }
      },
      "required": [
        "value"
      ],
      "type": "object"
    },
    "kind": "classmethod",
    "line": 1421,
    "module": "noether.commalg",
    "name": "from_json",
    "overloads": [
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        "input_schema": {
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          "properties": {
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              "type": "string"
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          },
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        "parameters": [
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            "kind": "positional_or_keyword",
            "name": "value",
            "required": true,
            "schema": {
              "type": "string"
            }
          }
        ],
        "result_schema": {
          "x-python-type": "ResidueRing"
        },
        "signature": "(cls, value: str) -> ResidueRing"
      }
    ],
    "qualified_name": "ResidueRing.from_json",
    "releases_gil": false,
    "result_schema": {
      "x-python-type": "ResidueRing"
    },
    "signature": "(cls, value: str) -> ResidueRing",
    "signatures": [
      "(cls, value: str) -> ResidueRing"
    ],
    "summary": "Deserialize and revalidate a residue-ring context.",
    "summary_source": "public docstring",
    "type_id": "noether/commalg/ResidueRing@1"
  },
  {
    "canonical_name": "commalg.ResidueRing.modulus",
    "determinism": "deterministic",
    "doc": "",
    "domain": "commalg",
    "file": "noether/commalg/__init__.pyi",
    "input_schema": {
      "additionalProperties": false,
      "properties": {},
      "type": "object"
    },
    "kind": "property",
    "line": 1414,
    "module": "noether.commalg",
    "name": "modulus",
    "overloads": [
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        "parameters": [],
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        },
        "signature": "(self) -> int"
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    ],
    "qualified_name": "ResidueRing.modulus",
    "releases_gil": false,
    "result_schema": {
      "type": "integer"
    },
    "signature": "(self) -> int",
    "signatures": [
      "(self) -> int"
    ],
    "summary": "Return the modulus recorded by this `ResidueRing`.",
    "summary_source": "derived projection vocabulary",
    "type_id": "noether/commalg/ResidueRing@1"
  },
  {
    "canonical_name": "commalg.ResidueRing.one",
    "determinism": "deterministic",
    "doc": "",
    "domain": "commalg",
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    "kind": "method",
    "line": 2692,
    "module": "noether.commalg",
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        "input_schema": {
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    "qualified_name": "Scheme.to_json",
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    "summary": "Serialize this `Scheme` to its canonical typed JSON representation.",
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    "type_id": "noether/commalg/Scheme@1"
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    "canonical_name": "commalg.SchemeCompletion",
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    "doc": "Fixed-order completed local quotient presentation.",
    "domain": "commalg",
    "file": "noether/commalg/__init__.pyi",
    "input_schema": {},
    "kind": "class",
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    "module": "noether.commalg",
    "name": "SchemeCompletion",
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    "qualified_name": "SchemeCompletion",
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    "result_schema": {},
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    "signatures": [],
    "summary": "Fixed-order completed local quotient presentation.",
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    "type_id": "noether/commalg/SchemeCompletion@1"
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    "canonical_name": "commalg.SchemeCompletion.ambient_ring",
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    "doc": "",
    "domain": "commalg",
    "file": "noether/commalg/__init__.pyi",
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      "additionalProperties": false,
      "properties": {},
      "type": "object"
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    "kind": "property",
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    "module": "noether.commalg",
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        "input_schema": {
          "additionalProperties": false,
          "properties": {},
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          "x-python-type": "PowerSeriesRing[C]"
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    "qualified_name": "SchemeCompletion.ambient_ring",
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    "signatures": [
      "(self) -> PowerSeriesRing[C]"
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    "summary": "Return the ambient ring recorded by this `SchemeCompletion`.",
    "summary_source": "derived projection vocabulary",
    "type_id": "noether/commalg/SchemeCompletion@1"
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  {
    "canonical_name": "commalg.SchemeCompletion.center",
    "determinism": "deterministic",
    "doc": "",
    "domain": "commalg",
    "file": "noether/commalg/__init__.pyi",
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      "additionalProperties": false,
      "properties": {},
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    "kind": "property",
    "line": 2741,
    "module": "noether.commalg",
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        "input_schema": {
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          "properties": {},
          "type": "object"
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    "qualified_name": "SchemeCompletion.center",
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      "x-python-type": "SchemePoint"
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    "signature": "(self) -> SchemePoint",
    "signatures": [
      "(self) -> SchemePoint"
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    "summary": "Return the center recorded by this `SchemeCompletion`.",
    "summary_source": "derived projection vocabulary",
    "type_id": "noether/commalg/SchemeCompletion@1"
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  {
    "canonical_name": "commalg.SchemeCompletion.equations",
    "determinism": "deterministic",
    "doc": "",
    "domain": "commalg",
    "file": "noether/commalg/__init__.pyi",
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      "additionalProperties": false,
      "properties": {},
      "type": "object"
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    "kind": "property",
    "line": 2745,
    "module": "noether.commalg",
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        "input_schema": {
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    "qualified_name": "SchemeCompletion.equations",
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    "signatures": [
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    "summary": "Return the equations recorded by this `SchemeCompletion`.",
    "summary_source": "derived projection vocabulary",
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    "canonical_name": "commalg.SchemeCompletion.from_json",
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    "doc": "",
    "domain": "commalg",
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      "additionalProperties": false,
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    "kind": "classmethod",
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    "module": "noether.commalg",
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            "kind": "positional_or_keyword",
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        "result_schema": {
          "x-python-type": "SchemeCompletion[FiniteField]"
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        "signature": "(cls, value: str) -> SchemeCompletion[FiniteField]"
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    "qualified_name": "SchemeCompletion.from_json",
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      "x-python-type": "SchemeCompletion[FiniteField]"
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    "signature": "(cls, value: str) -> SchemeCompletion[FiniteField]",
    "signatures": [
      "(cls, value: str) -> SchemeCompletion[FiniteField]"
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    "summary": "Deserialize and revalidate a `SchemeCompletion` from typed JSON.",
    "summary_source": "derived projection vocabulary",
    "type_id": "noether/commalg/SchemeCompletion@1"
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  {
    "canonical_name": "commalg.SchemeCompletion.source",
    "determinism": "deterministic",
    "doc": "",
    "domain": "commalg",
    "file": "noether/commalg/__init__.pyi",
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      "additionalProperties": false,
      "properties": {},
      "type": "object"
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    "kind": "property",
    "line": 2739,
    "module": "noether.commalg",
    "name": "source",
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        "input_schema": {
          "additionalProperties": false,
          "properties": {},
          "type": "object"
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        "parameters": [],
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          "x-python-type": "Scheme[C]"
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    "qualified_name": "SchemeCompletion.source",
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      "x-python-type": "Scheme[C]"
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    "signatures": [
      "(self) -> Scheme[C]"
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    "summary": "Return the source recorded by this `SchemeCompletion`.",
    "summary_source": "derived projection vocabulary",
    "type_id": "noether/commalg/SchemeCompletion@1"
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    "canonical_name": "commalg.SchemeCompletion.to_json",
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    "doc": "",
    "domain": "commalg",
    "file": "noether/commalg/__init__.pyi",
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      "additionalProperties": false,
      "properties": {},
      "type": "object"
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    "kind": "method",
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    "module": "noether.commalg",
    "name": "to_json",
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    "qualified_name": "SchemeCompletion.to_json",
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      "(self) -> str"
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    "summary": "Serialize this `SchemeCompletion` to its canonical typed JSON representation.",
    "summary_source": "derived projection vocabulary",
    "type_id": "noether/commalg/SchemeCompletion@1"
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    "canonical_name": "commalg.SchemeCompletion.truncation",
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    "doc": "",
    "domain": "commalg",
    "file": "noether/commalg/__init__.pyi",
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      "additionalProperties": false,
      "properties": {},
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    "kind": "property",
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    "module": "noether.commalg",
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          "properties": {},
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    "qualified_name": "SchemeCompletion.truncation",
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      "x-python-type": "SeriesTruncation"
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      "(self) -> SeriesTruncation"
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    "summary": "Return the truncation recorded by this `SchemeCompletion`.",
    "summary_source": "derived projection vocabulary",
    "type_id": "noether/commalg/SchemeCompletion@1"
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  {
    "canonical_name": "commalg.SchemeCompletion.verify",
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    "doc": "",
    "domain": "commalg",
    "file": "noether/commalg/__init__.pyi",
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      "additionalProperties": false,
      "properties": {},
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    "kind": "method",
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    "module": "noether.commalg",
    "name": "verify",
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    "qualified_name": "SchemeCompletion.verify",
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    "summary": "Replay and verify the mathematical evidence carried by this `SchemeCompletion`.",
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    "type_id": "noether/commalg/SchemeCompletion@1"
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    "canonical_name": "commalg.SchemePoint",
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    "doc": "A checked rational point on one finite-field scheme.",
    "domain": "commalg",
    "file": "noether/commalg/__init__.pyi",
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    "kind": "class",
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    "module": "noether.commalg",
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    "qualified_name": "SchemePoint",
    "releases_gil": false,
    "result_schema": {},
    "signature": "",
    "signatures": [],
    "summary": "A checked rational point on one finite-field scheme.",
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    "canonical_name": "commalg.SchemePoint.base_change",
    "determinism": "deterministic",
    "doc": "Extend this point and its complete scheme presentation to the target\nfinite field.",
    "domain": "commalg",
    "file": "noether/commalg/__init__.pyi",
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      "additionalProperties": false,
      "properties": {
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              "x-python-type": "FiniteFieldEmbedding"
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          "x-python-type": "PrimeSubfieldEmbedding | FiniteFieldEmbedding"
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    "module": "noether.commalg",
    "name": "base_change",
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              "x-python-type": "PrimeSubfieldEmbedding | FiniteFieldEmbedding"
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                  "x-python-type": "PrimeSubfieldEmbedding"
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              "x-python-type": "PrimeSubfieldEmbedding | FiniteFieldEmbedding"
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        "result_schema": {
          "x-python-type": "SchemePoint"
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    "qualified_name": "SchemePoint.base_change",
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      "x-python-type": "SchemePoint"
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    "summary": "Extend this point and its complete scheme presentation to the target finite field.",
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    "canonical_name": "commalg.SchemePoint.coordinates",
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    "domain": "commalg",
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    "signature": "(self) -> list[FiniteFieldElement]",
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    "canonical_name": "commalg.SchemePoint.from_json",
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          "x-python-type": "SchemePoint"
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        "signature": "(cls, value: str) -> SchemePoint"
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    "summary": "Deserialize and revalidate a `SchemePoint` from typed JSON.",
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  {
    "canonical_name": "commalg.SchemePoint.scheme",
    "determinism": "deterministic",
    "doc": "",
    "domain": "commalg",
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    "kind": "property",
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    "module": "noether.commalg",
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        "input_schema": {
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          "type": "object"
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        "parameters": [],
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          "x-python-type": "Scheme[FiniteField]"
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    "summary": "Return the scheme recorded by this `SchemePoint`.",
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    "canonical_name": "commalg.SchemePoint.to_json",
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    "domain": "commalg",
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      "properties": {},
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    "kind": "method",
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    "module": "noether.commalg",
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    "summary": "Serialize this `SchemePoint` to its canonical typed JSON representation.",
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    "canonical_name": "commalg.SchemePoint.verify",
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    "kind": "method",
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    "summary": "Replay and verify the mathematical evidence carried by this `SchemePoint`.",
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    "canonical_name": "commalg.SeriesTruncation",
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    "module": "noether.commalg",
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    "canonical_name": "commalg.SeriesTruncation.bound",
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    "doc": "",
    "domain": "commalg",
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      "properties": {},
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    "kind": "property",
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    "module": "noether.commalg",
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        "parameters": [],
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          "x-python-type": "int | None"
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    "qualified_name": "SeriesTruncation.bound",
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      "x-python-type": "int | None"
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    "summary": "Return the bound recorded by this `SeriesTruncation`.",
    "summary_source": "derived projection vocabulary",
    "type_id": "noether/commalg/SeriesTruncation@1"
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  {
    "canonical_name": "commalg.SeriesTruncation.bounds",
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    "doc": "",
    "domain": "commalg",
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    ],
    "qualified_name": "SmoothProjectiveSurfaceCohomology.branch_genus",
    "releases_gil": false,
    "result_schema": {
      "type": "integer"
    },
    "signature": "(self) -> int",
    "signatures": [
      "(self) -> int"
    ],
    "summary": "Return the branch genus recorded by this `SmoothProjectiveSurfaceCohomology`.",
    "summary_source": "derived projection vocabulary",
    "type_id": "noether/commalg/SmoothProjectiveSurfaceCohomology@1"
  },
  {
    "canonical_name": "commalg.SmoothProjectiveSurfaceCohomology.cover",
    "determinism": "deterministic",
    "doc": "",
    "domain": "commalg",
    "file": "noether/commalg/__init__.pyi",
    "input_schema": {
      "additionalProperties": false,
      "properties": {},
      "type": "object"
    },
    "kind": "property",
    "line": 2521,
    "module": "noether.commalg",
    "name": "cover",
    "overloads": [
      {
        "input_schema": {
          "additionalProperties": false,
          "properties": {},
          "type": "object"
        },
        "parameters": [],
        "result_schema": {
          "x-python-type": "CyclicHypersurfaceCover[FiniteField]"
        },
        "signature": "(self) -> CyclicHypersurfaceCover[FiniteField]"
      }
    ],
    "qualified_name": "SmoothProjectiveSurfaceCohomology.cover",
    "releases_gil": false,
    "result_schema": {
      "x-python-type": "CyclicHypersurfaceCover[FiniteField]"
    },
    "signature": "(self) -> CyclicHypersurfaceCover[FiniteField]",
    "signatures": [
      "(self) -> CyclicHypersurfaceCover[FiniteField]"
    ],
    "summary": "Return the cover recorded by this `SmoothProjectiveSurfaceCohomology`.",
    "summary_source": "derived projection vocabulary",
    "type_id": "noether/commalg/SmoothProjectiveSurfaceCohomology@1"
  },
  {
    "canonical_name": "commalg.SmoothProjectiveSurfaceCohomology.euler_characteristic",
    "determinism": "deterministic",
    "doc": "",
    "domain": "commalg",
    "file": "noether/commalg/__init__.pyi",
    "input_schema": {
      "additionalProperties": false,
      "properties": {},
      "type": "object"
    },
    "kind": "property",
    "line": 2527,
    "module": "noether.commalg",
    "name": "euler_characteristic",
    "overloads": [
      {
        "input_schema": {
          "additionalProperties": false,
          "properties": {},
          "type": "object"
        },
        "parameters": [],
        "result_schema": {
          "type": "integer"
        },
        "signature": "(self) -> int"
      }
    ],
    "qualified_name": "SmoothProjectiveSurfaceCohomology.euler_characteristic",
    "releases_gil": false,
    "result_schema": {
      "type": "integer"
    },
    "signature": "(self) -> int",
    "signatures": [
      "(self) -> int"
    ],
    "summary": "Return the euler characteristic recorded by this `SmoothProjectiveSurfaceCohomology`.",
    "summary_source": "derived projection vocabulary",
    "type_id": "noether/commalg/SmoothProjectiveSurfaceCohomology@1"
  },
  {
    "canonical_name": "commalg.SmoothProjectiveSurfaceCohomology.frobenius_polynomial",
    "determinism": "deterministic",
    "doc": "",
    "domain": "commalg",
    "file": "noether/commalg/__init__.pyi",
    "input_schema": {
      "additionalProperties": false,
      "properties": {
        "counts": {
          "x-python-type": "PointCountSequence"
        }
      },
      "required": [
        "counts"
      ],
      "type": "object"
    },
    "kind": "method",
    "line": 2534,
    "module": "noether.commalg",
    "name": "frobenius_polynomial",
    "overloads": [
      {
        "input_schema": {
          "additionalProperties": false,
          "properties": {
            "counts": {
              "x-python-type": "PointCountSequence"
            }
          },
          "required": [
            "counts"
          ],
          "type": "object"
        },
        "parameters": [
          {
            "kind": "positional_or_keyword",
            "name": "counts",
            "required": true,
            "schema": {
              "x-python-type": "PointCountSequence"
            }
          }
        ],
        "result_schema": {
          "x-python-type": "FrobeniusPolynomial"
        },
        "signature": "(self, counts: PointCountSequence) -> FrobeniusPolynomial"
      }
    ],
    "qualified_name": "SmoothProjectiveSurfaceCohomology.frobenius_polynomial",
    "releases_gil": false,
    "result_schema": {
      "x-python-type": "FrobeniusPolynomial"
    },
    "signature": "(self, counts: PointCountSequence) -> FrobeniusPolynomial",
    "signatures": [
      "(self, counts: PointCountSequence) -> FrobeniusPolynomial"
    ],
    "summary": "Compute frobenius polynomial for this `SmoothProjectiveSurfaceCohomology`.",
    "summary_source": "derived projection vocabulary",
    "type_id": "noether/commalg/SmoothProjectiveSurfaceCohomology@1"
  },
  {
    "canonical_name": "commalg.SmoothProjectiveSurfaceCohomology.from_json",
    "determinism": "deterministic",
    "doc": "",
    "domain": "commalg",
    "file": "noether/commalg/__init__.pyi",
    "input_schema": {
      "additionalProperties": false,
      "properties": {
        "value": {
          "type": "string"
        }
      },
      "required": [
        "value"
      ],
      "type": "object"
    },
    "kind": "classmethod",
    "line": 2538,
    "module": "noether.commalg",
    "name": "from_json",
    "overloads": [
      {
        "input_schema": {
          "additionalProperties": false,
          "properties": {
            "value": {
              "type": "string"
            }
          },
          "required": [
            "value"
          ],
          "type": "object"
        },
        "parameters": [
          {
            "kind": "positional_or_keyword",
            "name": "value",
            "required": true,
            "schema": {
              "type": "string"
            }
          }
        ],
        "result_schema": {
          "x-python-type": "SmoothProjectiveSurfaceCohomology"
        },
        "signature": "(cls, value: str) -> SmoothProjectiveSurfaceCohomology"
      }
    ],
    "qualified_name": "SmoothProjectiveSurfaceCohomology.from_json",
    "releases_gil": false,
    "result_schema": {
      "x-python-type": "SmoothProjectiveSurfaceCohomology"
    },
    "signature": "(cls, value: str) -> SmoothProjectiveSurfaceCohomology",
    "signatures": [
      "(cls, value: str) -> SmoothProjectiveSurfaceCohomology"
    ],
    "summary": "Deserialize and revalidate a `SmoothProjectiveSurfaceCohomology` from typed JSON.",
    "summary_source": "derived projection vocabulary",
    "type_id": "noether/commalg/SmoothProjectiveSurfaceCohomology@1"
  },
  {
    "canonical_name": "commalg.SmoothProjectiveSurfaceCohomology.hyperplane_frobenius_eigenvalue",
    "determinism": "deterministic",
    "doc": "",
    "domain": "commalg",
    "file": "noether/commalg/__init__.pyi",
    "input_schema": {
      "additionalProperties": false,
      "properties": {},
      "type": "object"
    },
    "kind": "property",
    "line": 2533,
    "module": "noether.commalg",
    "name": "hyperplane_frobenius_eigenvalue",
    "overloads": [
      {
        "input_schema": {
          "additionalProperties": false,
          "properties": {},
          "type": "object"
        },
        "parameters": [],
        "result_schema": {
          "type": "integer"
        },
        "signature": "(self) -> int"
      }
    ],
    "qualified_name": "SmoothProjectiveSurfaceCohomology.hyperplane_frobenius_eigenvalue",
    "releases_gil": false,
    "result_schema": {
      "type": "integer"
    },
    "signature": "(self) -> int",
    "signatures": [
      "(self) -> int"
    ],
    "summary": "Return the hyperplane frobenius eigenvalue recorded by this `SmoothProjectiveSurfaceCohomology`.",
    "summary_source": "derived projection vocabulary",
    "type_id": "noether/commalg/SmoothProjectiveSurfaceCohomology@1"
  },
  {
    "canonical_name": "commalg.SmoothProjectiveSurfaceCohomology.required_point_count_degree",
    "determinism": "deterministic",
    "doc": "",
    "domain": "commalg",
    "file": "noether/commalg/__init__.pyi",
    "input_schema": {
      "additionalProperties": false,
      "properties": {},
      "type": "object"
    },
    "kind": "property",
    "line": 2531,
    "module": "noether.commalg",
    "name": "required_point_count_degree",
    "overloads": [
      {
        "input_schema": {
          "additionalProperties": false,
          "properties": {},
          "type": "object"
        },
        "parameters": [],
        "result_schema": {
          "type": "integer"
        },
        "signature": "(self) -> int"
      }
    ],
    "qualified_name": "SmoothProjectiveSurfaceCohomology.required_point_count_degree",
    "releases_gil": false,
    "result_schema": {
      "type": "integer"
    },
    "signature": "(self) -> int",
    "signatures": [
      "(self) -> int"
    ],
    "summary": "Return the required point count degree recorded by this `SmoothProjectiveSurfaceCohomology`.",
    "summary_source": "derived projection vocabulary",
    "type_id": "noether/commalg/SmoothProjectiveSurfaceCohomology@1"
  },
  {
    "canonical_name": "commalg.SmoothProjectiveSurfaceCohomology.second_betti_number",
    "determinism": "deterministic",
    "doc": "",
    "domain": "commalg",
    "file": "noether/commalg/__init__.pyi",
    "input_schema": {
      "additionalProperties": false,
      "properties": {},
      "type": "object"
    },
    "kind": "property",
    "line": 2529,
    "module": "noether.commalg",
    "name": "second_betti_number",
    "overloads": [
      {
        "input_schema": {
          "additionalProperties": false,
          "properties": {},
          "type": "object"
        },
        "parameters": [],
        "result_schema": {
          "type": "integer"
        },
        "signature": "(self) -> int"
      }
    ],
    "qualified_name": "SmoothProjectiveSurfaceCohomology.second_betti_number",
    "releases_gil": false,
    "result_schema": {
      "type": "integer"
    },
    "signature": "(self) -> int",
    "signatures": [
      "(self) -> int"
    ],
    "summary": "Return the second betti number recorded by this `SmoothProjectiveSurfaceCohomology`.",
    "summary_source": "derived projection vocabulary",
    "type_id": "noether/commalg/SmoothProjectiveSurfaceCohomology@1"
  },
  {
    "canonical_name": "commalg.SmoothProjectiveSurfaceCohomology.to_json",
    "determinism": "deterministic",
    "doc": "",
    "domain": "commalg",
    "file": "noether/commalg/__init__.pyi",
    "input_schema": {
      "additionalProperties": false,
      "properties": {},
      "type": "object"
    },
    "kind": "method",
    "line": 2536,
    "module": "noether.commalg",
    "name": "to_json",
    "overloads": [
      {
        "input_schema": {
          "additionalProperties": false,
          "properties": {},
          "type": "object"
        },
        "parameters": [],
        "result_schema": {
          "type": "string"
        },
        "signature": "(self) -> str"
      }
    ],
    "qualified_name": "SmoothProjectiveSurfaceCohomology.to_json",
    "releases_gil": false,
    "result_schema": {
      "type": "string"
    },
    "signature": "(self) -> str",
    "signatures": [
      "(self) -> str"
    ],
    "summary": "Serialize this `SmoothProjectiveSurfaceCohomology` to its canonical typed JSON representation.",
    "summary_source": "derived projection vocabulary",
    "type_id": "noether/commalg/SmoothProjectiveSurfaceCohomology@1"
  },
  {
    "canonical_name": "commalg.SmoothProjectiveSurfaceCohomology.verify",
    "determinism": "deterministic",
    "doc": "",
    "domain": "commalg",
    "file": "noether/commalg/__init__.pyi",
    "input_schema": {
      "additionalProperties": false,
      "properties": {},
      "type": "object"
    },
    "kind": "method",
    "line": 2535,
    "module": "noether.commalg",
    "name": "verify",
    "overloads": [
      {
        "input_schema": {
          "additionalProperties": false,
          "properties": {},
          "type": "object"
        },
        "parameters": [],
        "result_schema": {
          "type": "null"
        },
        "signature": "(self) -> None"
      }
    ],
    "qualified_name": "SmoothProjectiveSurfaceCohomology.verify",
    "releases_gil": false,
    "result_schema": {
      "type": "null"
    },
    "signature": "(self) -> None",
    "signatures": [
      "(self) -> None"
    ],
    "summary": "Replay and verify the mathematical evidence carried by this `SmoothProjectiveSurfaceCohomology`.",
    "summary_source": "derived projection vocabulary",
    "type_id": "noether/commalg/SmoothProjectiveSurfaceCohomology@1"
  },
  {
    "canonical_name": "commalg.SplitCyclicPullback",
    "determinism": "deterministic",
    "doc": "Verified splitting of a line pullback to a cyclic surface.",
    "domain": "commalg",
    "file": "noether/commalg/__init__.pyi",
    "input_schema": {
      "additionalProperties": false,
      "properties": {
        "coordinate_images": {
          "items": {
            "x-python-type": "Polynomial[FiniteField]"
          },
          "type": "array",
          "x-python-type": "Sequence[Polynomial[FiniteField]]"
        },
        "cover": {
          "x-python-type": "CyclicHypersurfaceCover[FiniteField]"
        }
      },
      "required": [
        "cover",
        "coordinate_images"
      ],
      "type": "object"
    },
    "kind": "class",
    "line": 2585,
    "module": "noether.commalg",
    "name": "SplitCyclicPullback",
    "overloads": [
      {
        "input_schema": {
          "additionalProperties": false,
          "properties": {
            "coordinate_images": {
              "items": {
                "x-python-type": "Polynomial[FiniteField]"
              },
              "type": "array",
              "x-python-type": "Sequence[Polynomial[FiniteField]]"
            },
            "cover": {
              "x-python-type": "CyclicHypersurfaceCover[FiniteField]"
            }
          },
          "required": [
            "cover",
            "coordinate_images"
          ],
          "type": "object"
        },
        "parameters": [
          {
            "kind": "positional_or_keyword",
            "name": "cover",
            "required": true,
            "schema": {
              "x-python-type": "CyclicHypersurfaceCover[FiniteField]"
            }
          },
          {
            "kind": "positional_or_keyword",
            "name": "coordinate_images",
            "required": true,
            "schema": {
              "items": {
                "x-python-type": "Polynomial[FiniteField]"
              },
              "type": "array",
              "x-python-type": "Sequence[Polynomial[FiniteField]]"
            }
          }
        ],
        "result_schema": {
          "type": "null"
        },
        "signature": "(self, cover: CyclicHypersurfaceCover[FiniteField], coordinate_images: Sequence[Polynomial[FiniteField]]) -> None"
      }
    ],
    "qualified_name": "SplitCyclicPullback",
    "releases_gil": false,
    "result_schema": {
      "type": "null"
    },
    "signature": "(self, cover: CyclicHypersurfaceCover[FiniteField], coordinate_images: Sequence[Polynomial[FiniteField]]) -> None",
    "signatures": [
      "(self, cover: CyclicHypersurfaceCover[FiniteField], coordinate_images: Sequence[Polynomial[FiniteField]]) -> None"
    ],
    "summary": "Verified splitting of a line pullback to a cyclic surface.",
    "summary_source": "public docstring",
    "type_id": "noether/commalg/SplitCyclicPullback@1"
  },
  {
    "canonical_name": "commalg.SplitCyclicPullback.branch_root",
    "determinism": "deterministic",
    "doc": "",
    "domain": "commalg",
    "file": "noether/commalg/__init__.pyi",
    "input_schema": {
      "additionalProperties": false,
      "properties": {},
      "type": "object"
    },
    "kind": "property",
    "line": 2599,
    "module": "noether.commalg",
    "name": "branch_root",
    "overloads": [
      {
        "input_schema": {
          "additionalProperties": false,
          "properties": {},
          "type": "object"
        },
        "parameters": [],
        "result_schema": {
          "x-python-type": "Polynomial[FiniteField]"
        },
        "signature": "(self) -> Polynomial[FiniteField]"
      }
    ],
    "qualified_name": "SplitCyclicPullback.branch_root",
    "releases_gil": false,
    "result_schema": {
      "x-python-type": "Polynomial[FiniteField]"
    },
    "signature": "(self) -> Polynomial[FiniteField]",
    "signatures": [
      "(self) -> Polynomial[FiniteField]"
    ],
    "summary": "Return the branch root recorded by this `SplitCyclicPullback`.",
    "summary_source": "derived projection vocabulary",
    "type_id": "noether/commalg/SplitCyclicPullback@1"
  },
  {
    "canonical_name": "commalg.SplitCyclicPullback.component_count",
    "determinism": "deterministic",
    "doc": "",
    "domain": "commalg",
    "file": "noether/commalg/__init__.pyi",
    "input_schema": {
      "additionalProperties": false,
      "properties": {},
      "type": "object"
    },
    "kind": "property",
    "line": 2601,
    "module": "noether.commalg",
    "name": "component_count",
    "overloads": [
      {
        "input_schema": {
          "additionalProperties": false,
          "properties": {},
          "type": "object"
        },
        "parameters": [],
        "result_schema": {
          "type": "integer"
        },
        "signature": "(self) -> int"
      }
    ],
    "qualified_name": "SplitCyclicPullback.component_count",
    "releases_gil": false,
    "result_schema": {
      "type": "integer"
    },
    "signature": "(self) -> int",
    "signatures": [
      "(self) -> int"
    ],
    "summary": "Return the component count recorded by this `SplitCyclicPullback`.",
    "summary_source": "derived projection vocabulary",
    "type_id": "noether/commalg/SplitCyclicPullback@1"
  },
  {
    "canonical_name": "commalg.SplitCyclicPullback.cover",
    "determinism": "deterministic",
    "doc": "",
    "domain": "commalg",
    "file": "noether/commalg/__init__.pyi",
    "input_schema": {
      "additionalProperties": false,
      "properties": {},
      "type": "object"
    },
    "kind": "property",
    "line": 2593,
    "module": "noether.commalg",
    "name": "cover",
    "overloads": [
      {
        "input_schema": {
          "additionalProperties": false,
          "properties": {},
          "type": "object"
        },
        "parameters": [],
        "result_schema": {
          "x-python-type": "CyclicHypersurfaceCover[FiniteField]"
        },
        "signature": "(self) -> CyclicHypersurfaceCover[FiniteField]"
      }
    ],
    "qualified_name": "SplitCyclicPullback.cover",
    "releases_gil": false,
    "result_schema": {
      "x-python-type": "CyclicHypersurfaceCover[FiniteField]"
    },
    "signature": "(self) -> CyclicHypersurfaceCover[FiniteField]",
    "signatures": [
      "(self) -> CyclicHypersurfaceCover[FiniteField]"
    ],
    "summary": "Return the cover recorded by this `SplitCyclicPullback`.",
    "summary_source": "derived projection vocabulary",
    "type_id": "noether/commalg/SplitCyclicPullback@1"
  },
  {
    "canonical_name": "commalg.SplitCyclicPullback.divisor_rank_lower_bound",
    "determinism": "deterministic",
    "doc": "",
    "domain": "commalg",
    "file": "noether/commalg/__init__.pyi",
    "input_schema": {
      "additionalProperties": false,
      "properties": {},
      "type": "object"
    },
    "kind": "method",
    "line": 2603,
    "module": "noether.commalg",
    "name": "divisor_rank_lower_bound",
    "overloads": [
      {
        "input_schema": {
          "additionalProperties": false,
          "properties": {},
          "type": "object"
        },
        "parameters": [],
        "result_schema": {
          "x-python-type": "DivisorRankLowerBound"
        },
        "signature": "(self) -> DivisorRankLowerBound"
      }
    ],
    "qualified_name": "SplitCyclicPullback.divisor_rank_lower_bound",
    "releases_gil": false,
    "result_schema": {
      "x-python-type": "DivisorRankLowerBound"
    },
    "signature": "(self) -> DivisorRankLowerBound",
    "signatures": [
      "(self) -> DivisorRankLowerBound"
    ],
    "summary": "Compute divisor rank lower bound for this `SplitCyclicPullback`.",
    "summary_source": "derived projection vocabulary",
    "type_id": "noether/commalg/SplitCyclicPullback@1"
  },
  {
    "canonical_name": "commalg.SplitCyclicPullback.from_json",
    "determinism": "deterministic",
    "doc": "",
    "domain": "commalg",
    "file": "noether/commalg/__init__.pyi",
    "input_schema": {
      "additionalProperties": false,
      "properties": {
        "value": {
          "type": "string"
        }
      },
      "required": [
        "value"
      ],
      "type": "object"
    },
    "kind": "classmethod",
    "line": 2607,
    "module": "noether.commalg",
    "name": "from_json",
    "overloads": [
      {
        "input_schema": {
          "additionalProperties": false,
          "properties": {
            "value": {
              "type": "string"
            }
          },
          "required": [
            "value"
          ],
          "type": "object"
        },
        "parameters": [
          {
            "kind": "positional_or_keyword",
            "name": "value",
            "required": true,
            "schema": {
              "type": "string"
            }
          }
        ],
        "result_schema": {
          "x-python-type": "SplitCyclicPullback"
        },
        "signature": "(cls, value: str) -> SplitCyclicPullback"
      }
    ],
    "qualified_name": "SplitCyclicPullback.from_json",
    "releases_gil": false,
    "result_schema": {
      "x-python-type": "SplitCyclicPullback"
    },
    "signature": "(cls, value: str) -> SplitCyclicPullback",
    "signatures": [
      "(cls, value: str) -> SplitCyclicPullback"
    ],
    "summary": "Deserialize and revalidate a `SplitCyclicPullback` from typed JSON.",
    "summary_source": "derived projection vocabulary",
    "type_id": "noether/commalg/SplitCyclicPullback@1"
  },
  {
    "canonical_name": "commalg.SplitCyclicPullback.intersection_matrix",
    "determinism": "deterministic",
    "doc": "",
    "domain": "commalg",
    "file": "noether/commalg/__init__.pyi",
    "input_schema": {
      "additionalProperties": false,
      "properties": {},
      "type": "object"
    },
    "kind": "method",
    "line": 2602,
    "module": "noether.commalg",
    "name": "intersection_matrix",
    "overloads": [
      {
        "input_schema": {
          "additionalProperties": false,
          "properties": {},
          "type": "object"
        },
        "parameters": [],
        "result_schema": {
          "x-python-type": "Matrix[IntegerRing]"
        },
        "signature": "(self) -> Matrix[IntegerRing]"
      }
    ],
    "qualified_name": "SplitCyclicPullback.intersection_matrix",
    "releases_gil": false,
    "result_schema": {
      "x-python-type": "Matrix[IntegerRing]"
    },
    "signature": "(self) -> Matrix[IntegerRing]",
    "signatures": [
      "(self) -> Matrix[IntegerRing]"
    ],
    "summary": "Compute intersection matrix for this `SplitCyclicPullback`.",
    "summary_source": "derived projection vocabulary",
    "type_id": "noether/commalg/SplitCyclicPullback@1"
  },
  {
    "canonical_name": "commalg.SplitCyclicPullback.line_parameterization",
    "determinism": "deterministic",
    "doc": "",
    "domain": "commalg",
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    "doc": "",
    "domain": "commalg",
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    "doc": "",
    "domain": "commalg",
    "file": "noether/commalg/__init__.pyi",
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    "summary": "Return the module recorded by this `SymmetricAlgebra`.",
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    "canonical_name": "commalg.SymmetricAlgebra.to_json",
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    "domain": "commalg",
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    "domain": "commalg",
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    "summary": "Replay and verify the mathematical evidence carried by this `SymmetricAlgebra`.",
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    "doc": "One cyclotomic factor of the Tate-normalized Frobenius polynomial.",
    "domain": "commalg",
    "file": "noether/commalg/__init__.pyi",
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    "module": "noether.commalg",
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    "summary": "One cyclotomic factor of the Tate-normalized Frobenius polynomial.",
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    "canonical_name": "commalg.TateCyclotomicFactor.degree",
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    "doc": "",
    "domain": "commalg",
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    "summary": "Return the degree recorded by this `TateCyclotomicFactor`.",
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    "canonical_name": "commalg.TateCyclotomicFactor.index",
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    "domain": "commalg",
    "file": "noether/commalg/__init__.pyi",
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    "summary": "Return the index recorded by this `TateCyclotomicFactor`.",
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    "canonical_name": "commalg.TateCyclotomicFactor.multiplicity",
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    "domain": "commalg",
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    "summary": "Return the arity recorded by this `WeightedGrading`.",
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    "summary": "View this `WeightedGrading` as free abelian.",
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    "canonical_name": "commalg.WeightedGrading.degree",
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    "domain": "commalg",
    "file": "noether/commalg/__init__.pyi",
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    "canonical_name": "commalg.WeightedGrading.is_standard",
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    "domain": "commalg",
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      "properties": {},
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    },
    "kind": "property",
    "line": 2412,
    "module": "noether.commalg",
    "name": "ambient_quotient_strata",
    "overloads": [
      {
        "input_schema": {
          "additionalProperties": false,
          "properties": {},
          "type": "object"
        },
        "parameters": [],
        "result_schema": {
          "items": {
            "x-python-type": "WeightedProjectiveStratum"
          },
          "type": "array",
          "x-python-type": "list[WeightedProjectiveStratum]"
        },
        "signature": "(self) -> list[WeightedProjectiveStratum]"
      }
    ],
    "qualified_name": "WeightedHypersurfaceJacobianEvidence.ambient_quotient_strata",
    "releases_gil": false,
    "result_schema": {
      "items": {
        "x-python-type": "WeightedProjectiveStratum"
      },
      "type": "array",
      "x-python-type": "list[WeightedProjectiveStratum]"
    },
    "signature": "(self) -> list[WeightedProjectiveStratum]",
    "signatures": [
      "(self) -> list[WeightedProjectiveStratum]"
    ],
    "summary": "Return the ambient quotient strata recorded by this `WeightedHypersurfaceJacobianEvidence`.",
    "summary_source": "derived projection vocabulary"
  },
  {
    "canonical_name": "commalg.WeightedHypersurfaceJacobianEvidence.hypersurface",
    "determinism": "deterministic",
    "doc": "",
    "domain": "commalg",
    "file": "noether/commalg/__init__.pyi",
    "input_schema": {
      "additionalProperties": false,
      "properties": {},
      "type": "object"
    },
    "kind": "property",
    "line": 2407,
    "module": "noether.commalg",
    "name": "hypersurface",
    "overloads": [
      {
        "input_schema": {
          "additionalProperties": false,
          "properties": {},
          "type": "object"
        },
        "parameters": [],
        "result_schema": {
          "x-python-type": "WeightedProjectiveHypersurface[C]"
        },
        "signature": "(self) -> WeightedProjectiveHypersurface[C]"
      }
    ],
    "qualified_name": "WeightedHypersurfaceJacobianEvidence.hypersurface",
    "releases_gil": false,
    "result_schema": {
      "x-python-type": "WeightedProjectiveHypersurface[C]"
    },
    "signature": "(self) -> WeightedProjectiveHypersurface[C]",
    "signatures": [
      "(self) -> WeightedProjectiveHypersurface[C]"
    ],
    "summary": "Return the hypersurface recorded by this `WeightedHypersurfaceJacobianEvidence`.",
    "summary_source": "derived projection vocabulary"
  },
  {
    "canonical_name": "commalg.WeightedHypersurfaceJacobianEvidence.is_quasismooth",
    "determinism": "deterministic",
    "doc": "",
    "domain": "commalg",
    "file": "noether/commalg/__init__.pyi",
    "input_schema": {
      "additionalProperties": false,
      "properties": {},
      "type": "object"
    },
    "kind": "method",
    "line": 2410,
    "module": "noether.commalg",
    "name": "is_quasismooth",
    "overloads": [
      {
        "input_schema": {
          "additionalProperties": false,
          "properties": {},
          "type": "object"
        },
        "parameters": [],
        "result_schema": {
          "type": "boolean"
        },
        "signature": "(self) -> bool"
      }
    ],
    "qualified_name": "WeightedHypersurfaceJacobianEvidence.is_quasismooth",
    "releases_gil": false,
    "result_schema": {
      "type": "boolean"
    },
    "signature": "(self) -> bool",
    "signatures": [
      "(self) -> bool"
    ],
    "summary": "Test whether this `WeightedHypersurfaceJacobianEvidence` is quasismooth.",
    "summary_source": "derived projection vocabulary"
  },
  {
    "canonical_name": "commalg.WeightedHypersurfaceJacobianEvidence.saturated_critical_ideal",
    "determinism": "deterministic",
    "doc": "",
    "domain": "commalg",
    "file": "noether/commalg/__init__.pyi",
    "input_schema": {
      "additionalProperties": false,
      "properties": {},
      "type": "object"
    },
    "kind": "property",
    "line": 2409,
    "module": "noether.commalg",
    "name": "saturated_critical_ideal",
    "overloads": [
      {
        "input_schema": {
          "additionalProperties": false,
          "properties": {},
          "type": "object"
        },
        "parameters": [],
        "result_schema": {
          "x-python-type": "Ideal[PolynomialRing[C]]"
        },
        "signature": "(self) -> Ideal[PolynomialRing[C]]"
      }
    ],
    "qualified_name": "WeightedHypersurfaceJacobianEvidence.saturated_critical_ideal",
    "releases_gil": false,
    "result_schema": {
      "x-python-type": "Ideal[PolynomialRing[C]]"
    },
    "signature": "(self) -> Ideal[PolynomialRing[C]]",
    "signatures": [
      "(self) -> Ideal[PolynomialRing[C]]"
    ],
    "summary": "Return the saturated critical ideal recorded by this `WeightedHypersurfaceJacobianEvidence`.",
    "summary_source": "derived projection vocabulary"
  },
  {
    "canonical_name": "commalg.WeightedHypersurfaceJacobianEvidence.verify",
    "determinism": "deterministic",
    "doc": "",
    "domain": "commalg",
    "file": "noether/commalg/__init__.pyi",
    "input_schema": {
      "additionalProperties": false,
      "properties": {},
      "type": "object"
    },
    "kind": "method",
    "line": 2413,
    "module": "noether.commalg",
    "name": "verify",
    "overloads": [
      {
        "input_schema": {
          "additionalProperties": false,
          "properties": {},
          "type": "object"
        },
        "parameters": [],
        "result_schema": {
          "type": "null"
        },
        "signature": "(self) -> None"
      }
    ],
    "qualified_name": "WeightedHypersurfaceJacobianEvidence.verify",
    "releases_gil": false,
    "result_schema": {
      "type": "null"
    },
    "signature": "(self) -> None",
    "signatures": [
      "(self) -> None"
    ],
    "summary": "Replay and verify the mathematical evidence carried by this `WeightedHypersurfaceJacobianEvidence`.",
    "summary_source": "derived projection vocabulary"
  },
  {
    "canonical_name": "commalg.WeightedProjectiveChart",
    "determinism": "deterministic",
    "doc": "A standard affine cover together with its residual cyclic quotient action.",
    "domain": "commalg",
    "file": "noether/commalg/__init__.pyi",
    "input_schema": {},
    "kind": "class",
    "line": 2348,
    "module": "noether.commalg",
    "name": "WeightedProjectiveChart",
    "overloads": [],
    "qualified_name": "WeightedProjectiveChart",
    "releases_gil": false,
    "result_schema": {},
    "signature": "",
    "signatures": [],
    "summary": "A standard affine cover together with its residual cyclic quotient action.",
    "summary_source": "public docstring"
  },
  {
    "canonical_name": "commalg.WeightedProjectiveChart.action_weights",
    "determinism": "deterministic",
    "doc": "",
    "domain": "commalg",
    "file": "noether/commalg/__init__.pyi",
    "input_schema": {
      "additionalProperties": false,
      "properties": {},
      "type": "object"
    },
    "kind": "property",
    "line": 2358,
    "module": "noether.commalg",
    "name": "action_weights",
    "overloads": [
      {
        "input_schema": {
          "additionalProperties": false,
          "properties": {},
          "type": "object"
        },
        "parameters": [],
        "result_schema": {
          "items": {
            "type": "integer"
          },
          "type": "array",
          "x-python-type": "list[int]"
        },
        "signature": "(self) -> list[int]"
      }
    ],
    "qualified_name": "WeightedProjectiveChart.action_weights",
    "releases_gil": false,
    "result_schema": {
      "items": {
        "type": "integer"
      },
      "type": "array",
      "x-python-type": "list[int]"
    },
    "signature": "(self) -> list[int]",
    "signatures": [
      "(self) -> list[int]"
    ],
    "summary": "Return the action weights recorded by this `WeightedProjectiveChart`.",
    "summary_source": "derived projection vocabulary"
  },
  {
    "canonical_name": "commalg.WeightedProjectiveChart.affine_cover",
    "determinism": "deterministic",
    "doc": "",
    "domain": "commalg",
    "file": "noether/commalg/__init__.pyi",
    "input_schema": {
      "additionalProperties": false,
      "properties": {},
      "type": "object"
    },
    "kind": "property",
    "line": 2354,
    "module": "noether.commalg",
    "name": "affine_cover",
    "overloads": [
      {
        "input_schema": {
          "additionalProperties": false,
          "properties": {},
          "type": "object"
        },
        "parameters": [],
        "result_schema": {
          "x-python-type": "AffineSpace[C]"
        },
        "signature": "(self) -> AffineSpace[C]"
      }
    ],
    "qualified_name": "WeightedProjectiveChart.affine_cover",
    "releases_gil": false,
    "result_schema": {
      "x-python-type": "AffineSpace[C]"
    },
    "signature": "(self) -> AffineSpace[C]",
    "signatures": [
      "(self) -> AffineSpace[C]"
    ],
    "summary": "Return the affine cover recorded by this `WeightedProjectiveChart`.",
    "summary_source": "derived projection vocabulary"
  },
  {
    "canonical_name": "commalg.WeightedProjectiveChart.ambient",
    "determinism": "deterministic",
    "doc": "",
    "domain": "commalg",
    "file": "noether/commalg/__init__.pyi",
    "input_schema": {
      "additionalProperties": false,
      "properties": {},
      "type": "object"
    },
    "kind": "property",
    "line": 2350,
    "module": "noether.commalg",
    "name": "ambient",
    "overloads": [
      {
        "input_schema": {
          "additionalProperties": false,
          "properties": {},
          "type": "object"
        },
        "parameters": [],
        "result_schema": {
          "x-python-type": "WeightedProjectiveSpace[C]"
        },
        "signature": "(self) -> WeightedProjectiveSpace[C]"
      }
    ],
    "qualified_name": "WeightedProjectiveChart.ambient",
    "releases_gil": false,
    "result_schema": {
      "x-python-type": "WeightedProjectiveSpace[C]"
    },
    "signature": "(self) -> WeightedProjectiveSpace[C]",
    "signatures": [
      "(self) -> WeightedProjectiveSpace[C]"
    ],
    "summary": "Return the ambient recorded by this `WeightedProjectiveChart`.",
    "summary_source": "derived projection vocabulary"
  },
  {
    "canonical_name": "commalg.WeightedProjectiveChart.coordinate_index",
    "determinism": "deterministic",
    "doc": "",
    "domain": "commalg",
    "file": "noether/commalg/__init__.pyi",
    "input_schema": {
      "additionalProperties": false,
      "properties": {},
      "type": "object"
    },
    "kind": "property",
    "line": 2352,
    "module": "noether.commalg",
    "name": "coordinate_index",
    "overloads": [
      {
        "input_schema": {
          "additionalProperties": false,
          "properties": {},
          "type": "object"
        },
        "parameters": [],
        "result_schema": {
          "type": "integer"
        },
        "signature": "(self) -> int"
      }
    ],
    "qualified_name": "WeightedProjectiveChart.coordinate_index",
    "releases_gil": false,
    "result_schema": {
      "type": "integer"
    },
    "signature": "(self) -> int",
    "signatures": [
      "(self) -> int"
    ],
    "summary": "Return the coordinate index recorded by this `WeightedProjectiveChart`.",
    "summary_source": "derived projection vocabulary"
  },
  {
    "canonical_name": "commalg.WeightedProjectiveChart.is_affine_space",
    "determinism": "deterministic",
    "doc": "",
    "domain": "commalg",
    "file": "noether/commalg/__init__.pyi",
    "input_schema": {
      "additionalProperties": false,
      "properties": {},
      "type": "object"
    },
    "kind": "method",
    "line": 2359,
    "module": "noether.commalg",
    "name": "is_affine_space",
    "overloads": [
      {
        "input_schema": {
          "additionalProperties": false,
          "properties": {},
          "type": "object"
        },
        "parameters": [],
        "result_schema": {
          "type": "boolean"
        },
        "signature": "(self) -> bool"
      }
    ],
    "qualified_name": "WeightedProjectiveChart.is_affine_space",
    "releases_gil": false,
    "result_schema": {
      "type": "boolean"
    },
    "signature": "(self) -> bool",
    "signatures": [
      "(self) -> bool"
    ],
    "summary": "Test whether this `WeightedProjectiveChart` is affine space.",
    "summary_source": "derived projection vocabulary"
  },
  {
    "canonical_name": "commalg.WeightedProjectiveChart.stabilizer_order",
    "determinism": "deterministic",
    "doc": "",
    "domain": "commalg",
    "file": "noether/commalg/__init__.pyi",
    "input_schema": {
      "additionalProperties": false,
      "properties": {},
      "type": "object"
    },
    "kind": "property",
    "line": 2356,
    "module": "noether.commalg",
    "name": "stabilizer_order",
    "overloads": [
      {
        "input_schema": {
          "additionalProperties": false,
          "properties": {},
          "type": "object"
        },
        "parameters": [],
        "result_schema": {
          "type": "integer"
        },
        "signature": "(self) -> int"
      }
    ],
    "qualified_name": "WeightedProjectiveChart.stabilizer_order",
    "releases_gil": false,
    "result_schema": {
      "type": "integer"
    },
    "signature": "(self) -> int",
    "signatures": [
      "(self) -> int"
    ],
    "summary": "Return the stabilizer order recorded by this `WeightedProjectiveChart`.",
    "summary_source": "derived projection vocabulary"
  },
  {
    "canonical_name": "commalg.WeightedProjectiveHypersurface",
    "determinism": "deterministic",
    "doc": "A single weighted-homogeneous hypersurface.",
    "domain": "commalg",
    "file": "noether/commalg/__init__.pyi",
    "input_schema": {},
    "kind": "class",
    "line": 2415,
    "module": "noether.commalg",
    "name": "WeightedProjectiveHypersurface",
    "overloads": [],
    "qualified_name": "WeightedProjectiveHypersurface",
    "releases_gil": false,
    "result_schema": {},
    "signature": "",
    "signatures": [],
    "summary": "A single weighted-homogeneous hypersurface.",
    "summary_source": "public docstring",
    "type_id": "noether/commalg/WeightedProjectiveHypersurface@1"
  },
  {
    "canonical_name": "commalg.WeightedProjectiveHypersurface.ambient",
    "determinism": "deterministic",
    "doc": "",
    "domain": "commalg",
    "file": "noether/commalg/__init__.pyi",
    "input_schema": {
      "additionalProperties": false,
      "properties": {},
      "type": "object"
    },
    "kind": "property",
    "line": 2422,
    "module": "noether.commalg",
    "name": "ambient",
    "overloads": [
      {
        "input_schema": {
          "additionalProperties": false,
          "properties": {},
          "type": "object"
        },
        "parameters": [],
        "result_schema": {
          "x-python-type": "WeightedProjectiveSpace[C]"
        },
        "signature": "(self) -> WeightedProjectiveSpace[C]"
      }
    ],
    "qualified_name": "WeightedProjectiveHypersurface.ambient",
    "releases_gil": false,
    "result_schema": {
      "x-python-type": "WeightedProjectiveSpace[C]"
    },
    "signature": "(self) -> WeightedProjectiveSpace[C]",
    "signatures": [
      "(self) -> WeightedProjectiveSpace[C]"
    ],
    "summary": "Return the ambient recorded by this `WeightedProjectiveHypersurface`.",
    "summary_source": "derived projection vocabulary",
    "type_id": "noether/commalg/WeightedProjectiveHypersurface@1"
  },
  {
    "canonical_name": "commalg.WeightedProjectiveHypersurface.degree",
    "determinism": "deterministic",
    "doc": "",
    "domain": "commalg",
    "file": "noether/commalg/__init__.pyi",
    "input_schema": {
      "additionalProperties": false,
      "properties": {},
      "type": "object"
    },
    "kind": "property",
    "line": 2424,
    "module": "noether.commalg",
    "name": "degree",
    "overloads": [
      {
        "input_schema": {
          "additionalProperties": false,
          "properties": {},
          "type": "object"
        },
        "parameters": [],
        "result_schema": {
          "type": "integer"
        },
        "signature": "(self) -> int"
      }
    ],
    "qualified_name": "WeightedProjectiveHypersurface.degree",
    "releases_gil": false,
    "result_schema": {
      "type": "integer"
    },
    "signature": "(self) -> int",
    "signatures": [
      "(self) -> int"
    ],
    "summary": "Return the degree recorded by this `WeightedProjectiveHypersurface`.",
    "summary_source": "derived projection vocabulary",
    "type_id": "noether/commalg/WeightedProjectiveHypersurface@1"
  },
  {
    "canonical_name": "commalg.WeightedProjectiveHypersurface.equation",
    "determinism": "deterministic",
    "doc": "",
    "domain": "commalg",
    "file": "noether/commalg/__init__.pyi",
    "input_schema": {
      "additionalProperties": false,
      "properties": {},
      "type": "object"
    },
    "kind": "property",
    "line": 2420,
    "module": "noether.commalg",
    "name": "equation",
    "overloads": [
      {
        "input_schema": {
          "additionalProperties": false,
          "properties": {},
          "type": "object"
        },
        "parameters": [],
        "result_schema": {
          "x-python-type": "Polynomial[C]"
        },
        "signature": "(self) -> Polynomial[C]"
      }
    ],
    "qualified_name": "WeightedProjectiveHypersurface.equation",
    "releases_gil": false,
    "result_schema": {
      "x-python-type": "Polynomial[C]"
    },
    "signature": "(self) -> Polynomial[C]",
    "signatures": [
      "(self) -> Polynomial[C]"
    ],
    "summary": "Return the equation recorded by this `WeightedProjectiveHypersurface`.",
    "summary_source": "derived projection vocabulary",
    "type_id": "noether/commalg/WeightedProjectiveHypersurface@1"
  },
  {
    "canonical_name": "commalg.WeightedProjectiveHypersurface.from_json",
    "determinism": "deterministic",
    "doc": "",
    "domain": "commalg",
    "file": "noether/commalg/__init__.pyi",
    "input_schema": {
      "additionalProperties": false,
      "properties": {
        "value": {
          "type": "string"
        }
      },
      "required": [
        "value"
      ],
      "type": "object"
    },
    "kind": "classmethod",
    "line": 2429,
    "module": "noether.commalg",
    "name": "from_json",
    "overloads": [
      {
        "input_schema": {
          "additionalProperties": false,
          "properties": {
            "value": {
              "type": "string"
            }
          },
          "required": [
            "value"
          ],
          "type": "object"
        },
        "parameters": [
          {
            "kind": "positional_or_keyword",
            "name": "value",
            "required": true,
            "schema": {
              "type": "string"
            }
          }
        ],
        "result_schema": {
          "x-python-type": "WeightedProjectiveHypersurface[SchemeBase]"
        },
        "signature": "(cls, value: str) -> WeightedProjectiveHypersurface[SchemeBase]"
      }
    ],
    "qualified_name": "WeightedProjectiveHypersurface.from_json",
    "releases_gil": false,
    "result_schema": {
      "x-python-type": "WeightedProjectiveHypersurface[SchemeBase]"
    },
    "signature": "(cls, value: str) -> WeightedProjectiveHypersurface[SchemeBase]",
    "signatures": [
      "(cls, value: str) -> WeightedProjectiveHypersurface[SchemeBase]"
    ],
    "summary": "Deserialize and revalidate a `WeightedProjectiveHypersurface` from typed JSON.",
    "summary_source": "derived projection vocabulary",
    "type_id": "noether/commalg/WeightedProjectiveHypersurface@1"
  },
  {
    "canonical_name": "commalg.WeightedProjectiveHypersurface.is_quasismooth",
    "determinism": "deterministic",
    "doc": "",
    "domain": "commalg",
    "file": "noether/commalg/__init__.pyi",
    "input_schema": {
      "additionalProperties": false,
      "properties": {},
      "type": "object"
    },
    "kind": "method",
    "line": 2426,
    "module": "noether.commalg",
    "name": "is_quasismooth",
    "overloads": [
      {
        "input_schema": {
          "additionalProperties": false,
          "properties": {},
          "type": "object"
        },
        "parameters": [],
        "result_schema": {
          "type": "boolean"
        },
        "signature": "(self) -> bool"
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    ],
    "qualified_name": "WeightedProjectiveHypersurface.is_quasismooth",
    "releases_gil": false,
    "result_schema": {
      "type": "boolean"
    },
    "signature": "(self) -> bool",
    "signatures": [
      "(self) -> bool"
    ],
    "summary": "Test whether this `WeightedProjectiveHypersurface` is quasismooth.",
    "summary_source": "derived projection vocabulary",
    "type_id": "noether/commalg/WeightedProjectiveHypersurface@1"
  },
  {
    "canonical_name": "commalg.WeightedProjectiveHypersurface.jacobian_evidence",
    "determinism": "deterministic",
    "doc": "",
    "domain": "commalg",
    "file": "noether/commalg/__init__.pyi",
    "input_schema": {
      "additionalProperties": false,
      "properties": {},
      "type": "object"
    },
    "kind": "method",
    "line": 2425,
    "module": "noether.commalg",
    "name": "jacobian_evidence",
    "overloads": [
      {
        "input_schema": {
          "additionalProperties": false,
          "properties": {},
          "type": "object"
        },
        "parameters": [],
        "result_schema": {
          "x-python-type": "WeightedHypersurfaceJacobianEvidence[C]"
        },
        "signature": "(self) -> WeightedHypersurfaceJacobianEvidence[C]"
      }
    ],
    "qualified_name": "WeightedProjectiveHypersurface.jacobian_evidence",
    "releases_gil": false,
    "result_schema": {
      "x-python-type": "WeightedHypersurfaceJacobianEvidence[C]"
    },
    "signature": "(self) -> WeightedHypersurfaceJacobianEvidence[C]",
    "signatures": [
      "(self) -> WeightedHypersurfaceJacobianEvidence[C]"
    ],
    "summary": "Compute jacobian evidence for this `WeightedProjectiveHypersurface`.",
    "summary_source": "derived projection vocabulary",
    "type_id": "noether/commalg/WeightedProjectiveHypersurface@1"
  },
  {
    "canonical_name": "commalg.WeightedProjectiveHypersurface.scheme",
    "determinism": "deterministic",
    "doc": "",
    "domain": "commalg",
    "file": "noether/commalg/__init__.pyi",
    "input_schema": {
      "additionalProperties": false,
      "properties": {},
      "type": "object"
    },
    "kind": "property",
    "line": 2418,
    "module": "noether.commalg",
    "name": "scheme",
    "overloads": [
      {
        "input_schema": {
          "additionalProperties": false,
          "properties": {},
          "type": "object"
        },
        "parameters": [],
        "result_schema": {
          "x-python-type": "WeightedProjectiveScheme[C]"
        },
        "signature": "(self) -> WeightedProjectiveScheme[C]"
      }
    ],
    "qualified_name": "WeightedProjectiveHypersurface.scheme",
    "releases_gil": false,
    "result_schema": {
      "x-python-type": "WeightedProjectiveScheme[C]"
    },
    "signature": "(self) -> WeightedProjectiveScheme[C]",
    "signatures": [
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    "summary_source": "derived projection vocabulary",
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    "doc": "",
    "domain": "commalg",
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    "summary": "Compute coefficient for this `WeilDivisor`.",
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    "type_id": "noether/commalg/WeilDivisor@1"
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  {
    "canonical_name": "commalg.WeilDivisor.curve",
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    "doc": "",
    "domain": "commalg",
    "file": "noether/commalg/__init__.pyi",
    "input_schema": {
      "additionalProperties": false,
      "properties": {},
      "type": "object"
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    "kind": "property",
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    "qualified_name": "WeilDivisor.curve",
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    "signature": "(self) -> Curve[C]",
    "signatures": [
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    "summary": "Return the curve recorded by this `WeilDivisor`.",
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    "type_id": "noether/commalg/WeilDivisor@1"
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    "canonical_name": "commalg.WeilDivisor.degree",
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    "doc": "",
    "domain": "commalg",
    "file": "noether/commalg/__init__.pyi",
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      "properties": {},
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    "kind": "property",
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    "qualified_name": "WeilDivisor.degree",
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    "summary": "Return the degree recorded by this `WeilDivisor`.",
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    "canonical_name": "commalg.WeilDivisor.from_json",
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    "doc": "",
    "domain": "commalg",
    "file": "noether/commalg/__init__.pyi",
    "input_schema": {
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    "module": "noether.commalg",
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    "summary": "Deserialize and revalidate a `WeilDivisor` from typed JSON.",
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    "canonical_name": "commalg.WeilDivisor.negative_part",
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    "doc": "",
    "domain": "commalg",
    "file": "noether/commalg/__init__.pyi",
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      "properties": {},
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    "module": "noether.commalg",
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    "summary": "Compute negative part for this `WeilDivisor`.",
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    "canonical_name": "commalg.WeilDivisor.positive_part",
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    "doc": "",
    "domain": "commalg",
    "file": "noether/commalg/__init__.pyi",
    "input_schema": {
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      "properties": {},
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    "kind": "method",
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    "module": "noether.commalg",
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    "summary": "Compute positive part for this `WeilDivisor`.",
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    "domain": "commalg",
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    "module": "noether.commalg",
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    "summary": "Compute pullback for this `WeilDivisor`.",
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    "canonical_name": "commalg.WeilDivisor.riemann_roch_space",
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    "doc": "",
    "domain": "commalg",
    "file": "noether/commalg/__init__.pyi",
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      "properties": {},
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    "module": "noether.commalg",
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    "summary": "Compute riemann roch space for this `WeilDivisor`.",
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    "canonical_name": "commalg.WeilDivisor.scale",
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    "domain": "commalg",
    "file": "noether/commalg/__init__.pyi",
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      },
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    "module": "noether.commalg",
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    "summary": "Compute scale for this `WeilDivisor`.",
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    "canonical_name": "commalg.WeilDivisor.support",
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    "doc": "",
    "domain": "commalg",
    "file": "noether/commalg/__init__.pyi",
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      "properties": {},
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    "module": "noether.commalg",
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    "signature": "(self) -> list[CurveClosedPoint[C]]",
    "signatures": [
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    "summary": "Return the support recorded by this `WeilDivisor`.",
    "summary_source": "derived projection vocabulary",
    "type_id": "noether/commalg/WeilDivisor@1"
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    "canonical_name": "commalg.WeilDivisor.terms",
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    "domain": "commalg",
    "file": "noether/commalg/__init__.pyi",
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    "kind": "property",
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    "module": "noether.commalg",
    "name": "terms",
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    "signature": "(self) -> list[tuple[CurveClosedPoint[C], int]]",
    "signatures": [
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    "summary": "Return the terms recorded by this `WeilDivisor`.",
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    "canonical_name": "commalg.WeilDivisor.to_json",
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    "domain": "commalg",
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    "module": "noether.commalg",
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    "summary": "Serialize this `WeilDivisor` to its canonical typed JSON representation.",
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    "domain": "commalg",
    "file": "noether/commalg/__init__.pyi",
    "input_schema": {
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    "module": "noether.commalg",
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    "qualified_name": "WeilDivisor.verify",
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    "signature": "(self) -> None",
    "signatures": [
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    "summary": "Replay and verify the mathematical evidence carried by this `WeilDivisor`.",
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    "canonical_name": "commalg.ZetaFunction",
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    "file": "noether/commalg/__init__.pyi",
    "input_schema": {},
    "kind": "class",
    "line": 2564,
    "module": "noether.commalg",
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    "qualified_name": "ZetaFunction",
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    "summary": "Exact rational zeta function of a supported smooth projective surface.",
    "summary_source": "public docstring",
    "type_id": "noether/commalg/ZetaFunction@1"
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    "canonical_name": "commalg.ZetaFunction.denominator",
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    "domain": "commalg",
    "file": "noether/commalg/__init__.pyi",
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      "additionalProperties": false,
      "properties": {},
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    "kind": "property",
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    "module": "noether.commalg",
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        "parameters": [],
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        "signature": "(self) -> Polynomial[IntegerRing]"
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    ],
    "qualified_name": "ZetaFunction.denominator",
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    "result_schema": {
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    "summary": "Whether the cloud has no sites.",
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    "canonical_name": "geometry.PointCloud.points",
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    "domain": "geometry",
    "file": "noether/geometry/__init__.pyi",
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    "canonical_name": "geometry.PointCloud.to_json",
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    "canonical_name": "geometry.Polytope",
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    "canonical_name": "geometry.Polytope.ambient_dimension",
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    "file": "noether/geometry/__init__.pyi",
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    "doc": "Construct the exact convex hull of rational vertices.",
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        "parameters": [
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    "canonical_name": "geometry.Polytope.to_json",
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    "canonical_name": "geometry.Polytope.verify",
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    "domain": "geometry",
    "file": "noether/geometry/__init__.pyi",
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    "summary": "Recompute the exact hull and all derived combinatorics.",
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    "canonical_name": "geometry.Polytope.vertex_facet_incidence",
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    "doc": "Vertex indices lying on each facet.",
    "domain": "geometry",
    "file": "noether/geometry/__init__.pyi",
    "input_schema": {
      "additionalProperties": false,
      "properties": {},
      "type": "object"
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    "summary": "Vertex indices lying on each facet.",
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    "canonical_name": "geometry.Polytope.vertices",
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    "doc": "Exact vertices as `fractions.Fraction` values.",
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    "file": "noether/geometry/__init__.pyi",
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    "signature": "(self) -> int",
    "signatures": [
      "(self) -> int"
    ],
    "summary": "Short dimension alias retained for standard triangulation notation.",
    "summary_source": "public docstring",
    "type_id": "noether/geometry/Triangulation@1"
  },
  {
    "canonical_name": "geometry.Triangulation.dimension",
    "determinism": "deterministic",
    "doc": "Affine dimension actually triangulated.",
    "domain": "geometry",
    "file": "noether/geometry/__init__.pyi",
    "input_schema": {
      "additionalProperties": false,
      "properties": {},
      "type": "object"
    },
    "kind": "property",
    "line": 37,
    "module": "noether.geometry",
    "name": "dimension",
    "overloads": [
      {
        "input_schema": {
          "additionalProperties": false,
          "properties": {},
          "type": "object"
        },
        "parameters": [],
        "result_schema": {
          "type": "integer"
        },
        "signature": "(self) -> int"
      }
    ],
    "qualified_name": "Triangulation.dimension",
    "releases_gil": false,
    "result_schema": {
      "type": "integer"
    },
    "signature": "(self) -> int",
    "signatures": [
      "(self) -> int"
    ],
    "summary": "Affine dimension actually triangulated.",
    "summary_source": "public docstring",
    "type_id": "noether/geometry/Triangulation@1"
  },
  {
    "canonical_name": "geometry.Triangulation.from_json",
    "determinism": "deterministic",
    "doc": "Deserialize by reconstructing the deterministic triangulation.",
    "domain": "geometry",
    "file": "noether/geometry/__init__.pyi",
    "input_schema": {
      "additionalProperties": false,
      "properties": {
        "value": {
          "type": "string"
        }
      },
      "required": [
        "value"
      ],
      "type": "object"
    },
    "kind": "staticmethod",
    "line": 34,
    "module": "noether.geometry",
    "name": "from_json",
    "overloads": [
      {
        "input_schema": {
          "additionalProperties": false,
          "properties": {
            "value": {
              "type": "string"
            }
          },
          "required": [
            "value"
          ],
          "type": "object"
        },
        "parameters": [
          {
            "kind": "positional_or_keyword",
            "name": "value",
            "required": true,
            "schema": {
              "type": "string"
            }
          }
        ],
        "result_schema": {
          "x-python-type": "Triangulation"
        },
        "signature": "(value: str) -> Triangulation"
      }
    ],
    "qualified_name": "Triangulation.from_json",
    "releases_gil": false,
    "result_schema": {
      "x-python-type": "Triangulation"
    },
    "signature": "(value: str) -> Triangulation",
    "signatures": [
      "(value: str) -> Triangulation"
    ],
    "summary": "Deserialize by reconstructing the deterministic triangulation.",
    "summary_source": "public docstring",
    "type_id": "noether/geometry/Triangulation@1"
  },
  {
    "canonical_name": "geometry.Triangulation.n_cells",
    "determinism": "deterministic",
    "doc": "Number of maximal simplices.",
    "domain": "geometry",
    "file": "noether/geometry/__init__.pyi",
    "input_schema": {
      "additionalProperties": false,
      "properties": {},
      "type": "object"
    },
    "kind": "property",
    "line": 47,
    "module": "noether.geometry",
    "name": "n_cells",
    "overloads": [
      {
        "input_schema": {
          "additionalProperties": false,
          "properties": {},
          "type": "object"
        },
        "parameters": [],
        "result_schema": {
          "type": "integer"
        },
        "signature": "(self) -> int"
      }
    ],
    "qualified_name": "Triangulation.n_cells",
    "releases_gil": false,
    "result_schema": {
      "type": "integer"
    },
    "signature": "(self) -> int",
    "signatures": [
      "(self) -> int"
    ],
    "summary": "Number of maximal simplices.",
    "summary_source": "public docstring",
    "type_id": "noether/geometry/Triangulation@1"
  },
  {
    "canonical_name": "geometry.Triangulation.points",
    "determinism": "deterministic",
    "doc": "Sites in input order.",
    "domain": "geometry",
    "file": "noether/geometry/__init__.pyi",
    "input_schema": {
      "additionalProperties": false,
      "properties": {},
      "type": "object"
    },
    "kind": "property",
    "line": 41,
    "module": "noether.geometry",
    "name": "points",
    "overloads": [
      {
        "input_schema": {
          "additionalProperties": false,
          "properties": {},
          "type": "object"
        },
        "parameters": [],
        "result_schema": {
          "items": {
            "items": {
              "type": "number"
            },
            "type": "array",
            "x-python-type": "list[float]"
          },
          "type": "array",
          "x-python-type": "list[list[float]]"
        },
        "signature": "(self) -> list[list[float]]"
      }
    ],
    "qualified_name": "Triangulation.points",
    "releases_gil": false,
    "result_schema": {
      "items": {
        "items": {
          "type": "number"
        },
        "type": "array",
        "x-python-type": "list[float]"
      },
      "type": "array",
      "x-python-type": "list[list[float]]"
    },
    "signature": "(self) -> list[list[float]]",
    "signatures": [
      "(self) -> list[list[float]]"
    ],
    "summary": "Sites in input order.",
    "summary_source": "public docstring",
    "type_id": "noether/geometry/Triangulation@1"
  },
  {
    "canonical_name": "geometry.Triangulation.simplices",
    "determinism": "deterministic",
    "doc": "Maximal simplices as vertex-index lists, in canonical order.",
    "domain": "geometry",
    "file": "noether/geometry/__init__.pyi",
    "input_schema": {
      "additionalProperties": false,
      "properties": {},
      "type": "object"
    },
    "kind": "property",
    "line": 43,
    "module": "noether.geometry",
    "name": "simplices",
    "overloads": [
      {
        "input_schema": {
          "additionalProperties": false,
          "properties": {},
          "type": "object"
        },
        "parameters": [],
        "result_schema": {
          "items": {
            "items": {
              "type": "integer"
            },
            "type": "array",
            "x-python-type": "list[int]"
          },
          "type": "array",
          "x-python-type": "list[list[int]]"
        },
        "signature": "(self) -> list[list[int]]"
      }
    ],
    "qualified_name": "Triangulation.simplices",
    "releases_gil": false,
    "result_schema": {
      "items": {
        "items": {
          "type": "integer"
        },
        "type": "array",
        "x-python-type": "list[int]"
      },
      "type": "array",
      "x-python-type": "list[list[int]]"
    },
    "signature": "(self) -> list[list[int]]",
    "signatures": [
      "(self) -> list[list[int]]"
    ],
    "summary": "Maximal simplices as vertex-index lists, in canonical order.",
    "summary_source": "public docstring",
    "type_id": "noether/geometry/Triangulation@1"
  },
  {
    "canonical_name": "geometry.Triangulation.to_json",
    "determinism": "deterministic",
    "doc": "Serialize the checked sites and optional regular weights.",
    "domain": "geometry",
    "file": "noether/geometry/__init__.pyi",
    "input_schema": {
      "additionalProperties": false,
      "properties": {},
      "type": "object"
    },
    "kind": "method",
    "line": 35,
    "module": "noether.geometry",
    "name": "to_json",
    "overloads": [
      {
        "input_schema": {
          "additionalProperties": false,
          "properties": {},
          "type": "object"
        },
        "parameters": [],
        "result_schema": {
          "type": "string"
        },
        "signature": "(self) -> str"
      }
    ],
    "qualified_name": "Triangulation.to_json",
    "releases_gil": false,
    "result_schema": {
      "type": "string"
    },
    "signature": "(self) -> str",
    "signatures": [
      "(self) -> str"
    ],
    "summary": "Serialize the checked sites and optional regular weights.",
    "summary_source": "public docstring",
    "type_id": "noether/geometry/Triangulation@1"
  },
  {
    "canonical_name": "geometry.Triangulation.verify",
    "determinism": "deterministic",
    "doc": "Recompute the deterministic checked construction and compare its value.",
    "domain": "geometry",
    "file": "noether/geometry/__init__.pyi",
    "input_schema": {
      "additionalProperties": false,
      "properties": {},
      "type": "object"
    },
    "kind": "method",
    "line": 53,
    "module": "noether.geometry",
    "name": "verify",
    "overloads": [
      {
        "input_schema": {
          "additionalProperties": false,
          "properties": {},
          "type": "object"
        },
        "parameters": [],
        "result_schema": {
          "type": "null"
        },
        "signature": "(self) -> None"
      }
    ],
    "qualified_name": "Triangulation.verify",
    "releases_gil": true,
    "result_schema": {
      "type": "null"
    },
    "signature": "(self) -> None",
    "signatures": [
      "(self) -> None"
    ],
    "summary": "Recompute the deterministic checked construction and compare its value.",
    "summary_source": "public docstring",
    "type_id": "noether/geometry/Triangulation@1"
  },
  {
    "canonical_name": "geometry.Triangulation.weighted",
    "determinism": "deterministic",
    "doc": "Whether this is a regular rather than ordinary Delaunay triangulation.",
    "domain": "geometry",
    "file": "noether/geometry/__init__.pyi",
    "input_schema": {
      "additionalProperties": false,
      "properties": {},
      "type": "object"
    },
    "kind": "property",
    "line": 49,
    "module": "noether.geometry",
    "name": "weighted",
    "overloads": [
      {
        "input_schema": {
          "additionalProperties": false,
          "properties": {},
          "type": "object"
        },
        "parameters": [],
        "result_schema": {
          "type": "boolean"
        },
        "signature": "(self) -> bool"
      }
    ],
    "qualified_name": "Triangulation.weighted",
    "releases_gil": false,
    "result_schema": {
      "type": "boolean"
    },
    "signature": "(self) -> bool",
    "signatures": [
      "(self) -> bool"
    ],
    "summary": "Whether this is a regular rather than ordinary Delaunay triangulation.",
    "summary_source": "public docstring",
    "type_id": "noether/geometry/Triangulation@1"
  },
  {
    "canonical_name": "geometry.Triangulation.weights",
    "determinism": "deterministic",
    "doc": "Regular weights, or `None` for an ordinary Delaunay triangulation.",
    "domain": "geometry",
    "file": "noether/geometry/__init__.pyi",
    "input_schema": {
      "additionalProperties": false,
      "properties": {},
      "type": "object"
    },
    "kind": "property",
    "line": 51,
    "module": "noether.geometry",
    "name": "weights",
    "overloads": [
      {
        "input_schema": {
          "additionalProperties": false,
          "properties": {},
          "type": "object"
        },
        "parameters": [],
        "result_schema": {
          "anyOf": [
            {
              "items": {
                "type": "number"
              },
              "type": "array",
              "x-python-type": "list[float]"
            },
            {
              "type": "null"
            }
          ],
          "x-python-type": "list[float] | None"
        },
        "signature": "(self) -> list[float] | None"
      }
    ],
    "qualified_name": "Triangulation.weights",
    "releases_gil": false,
    "result_schema": {
      "anyOf": [
        {
          "items": {
            "type": "number"
          },
          "type": "array",
          "x-python-type": "list[float]"
        },
        {
          "type": "null"
        }
      ],
      "x-python-type": "list[float] | None"
    },
    "signature": "(self) -> list[float] | None",
    "signatures": [
      "(self) -> list[float] | None"
    ],
    "summary": "Regular weights, or `None` for an ordinary Delaunay triangulation.",
    "summary_source": "public docstring",
    "type_id": "noether/geometry/Triangulation@1"
  },
  {
    "algorithms": [
      "exact_rational_convex_hull"
    ],
    "backends": [
      "convex-hull",
      "flint"
    ],
    "canonical_name": "geometry.convex_hull",
    "determinism": "deterministic",
    "doc": "Return the exact rational convex hull of a finite point family.",
    "domain": "geometry",
    "errors": [
      "empty point family",
      "inconsistent ambient dimensions"
    ],
    "evidence": [
      "Polytope.verify"
    ],
    "example": "from noether.geometry import convex_hull\nresult = convex_hull([[0, 0], [1, 0], [0, 1]])\nresult.verify()",
    "file": "noether/geometry/__init__.pyi",
    "input_schema": {
      "additionalProperties": false,
      "properties": {
        "points": {
          "items": {
            "items": {
              "anyOf": [
                {
                  "type": "integer"
                },
                {
                  "x-python-type": "Fraction"
                }
              ],
              "x-python-type": "int | Fraction"
            },
            "type": "array",
            "x-python-type": "Iterable[int | Fraction]"
          },
          "type": "array",
          "x-python-type": "Iterable[Iterable[int | Fraction]]"
        }
      },
      "required": [
        "points"
      ],
      "type": "object"
    },
    "kind": "function",
    "line": 87,
    "module": "noether.geometry",
    "name": "convex_hull",
    "overloads": [
      {
        "input_schema": {
          "additionalProperties": false,
          "properties": {
            "points": {
              "items": {
                "items": {
                  "anyOf": [
                    {
                      "type": "integer"
                    },
                    {
                      "x-python-type": "Fraction"
                    }
                  ],
                  "x-python-type": "int | Fraction"
                },
                "type": "array",
                "x-python-type": "Iterable[int | Fraction]"
              },
              "type": "array",
              "x-python-type": "Iterable[Iterable[int | Fraction]]"
            }
          },
          "required": [
            "points"
          ],
          "type": "object"
        },
        "parameters": [
          {
            "kind": "positional_or_keyword",
            "name": "points",
            "required": true,
            "schema": {
              "items": {
                "items": {
                  "anyOf": [
                    {
                      "type": "integer"
                    },
                    {
                      "x-python-type": "Fraction"
                    }
                  ],
                  "x-python-type": "int | Fraction"
                },
                "type": "array",
                "x-python-type": "Iterable[int | Fraction]"
              },
              "type": "array",
              "x-python-type": "Iterable[Iterable[int | Fraction]]"
            }
          }
        ],
        "result_schema": {
          "x-python-type": "Polytope"
        },
        "signature": "(points: Iterable[Iterable[int | Fraction]]) -> Polytope"
      }
    ],
    "qualified_name": "convex_hull",
    "releases_gil": true,
    "result_schema": {
      "x-python-type": "Polytope"
    },
    "signature": "(points: Iterable[Iterable[int | Fraction]]) -> Polytope",
    "signatures": [
      "(points: Iterable[Iterable[int | Fraction]]) -> Polytope"
    ],
    "summary": "Return the exact rational convex hull of a finite point family.",
    "summary_source": "public docstring"
  },
  {
    "canonical_name": "geometry.delaunay",
    "determinism": "deterministic",
    "doc": "Return the exact-predicate Delaunay triangulation of ``points``.",
    "domain": "geometry",
    "file": "noether/geometry/__init__.pyi",
    "input_schema": {
      "additionalProperties": false,
      "properties": {
        "points": {
          "anyOf": [
            {
              "x-python-type": "PointCloud"
            },
            {
              "items": {
                "items": {
                  "type": "number"
                },
                "type": "array",
                "x-python-type": "Iterable[float]"
              },
              "type": "array",
              "x-python-type": "Iterable[Iterable[float]]"
            }
          ],
          "x-python-type": "PointCloud | Iterable[Iterable[float]]"
        }
      },
      "required": [
        "points"
      ],
      "type": "object"
    },
    "kind": "function",
    "line": 90,
    "module": "noether.geometry",
    "name": "delaunay",
    "overloads": [
      {
        "input_schema": {
          "additionalProperties": false,
          "properties": {
            "points": {
              "anyOf": [
                {
                  "x-python-type": "PointCloud"
                },
                {
                  "items": {
                    "items": {
                      "type": "number"
                    },
                    "type": "array",
                    "x-python-type": "Iterable[float]"
                  },
                  "type": "array",
                  "x-python-type": "Iterable[Iterable[float]]"
                }
              ],
              "x-python-type": "PointCloud | Iterable[Iterable[float]]"
            }
          },
          "required": [
            "points"
          ],
          "type": "object"
        },
        "parameters": [
          {
            "kind": "positional_or_keyword",
            "name": "points",
            "required": true,
            "schema": {
              "anyOf": [
                {
                  "x-python-type": "PointCloud"
                },
                {
                  "items": {
                    "items": {
                      "type": "number"
                    },
                    "type": "array",
                    "x-python-type": "Iterable[float]"
                  },
                  "type": "array",
                  "x-python-type": "Iterable[Iterable[float]]"
                }
              ],
              "x-python-type": "PointCloud | Iterable[Iterable[float]]"
            }
          }
        ],
        "result_schema": {
          "x-python-type": "Triangulation"
        },
        "signature": "(points: PointCloud | Iterable[Iterable[float]]) -> Triangulation"
      }
    ],
    "qualified_name": "delaunay",
    "releases_gil": false,
    "result_schema": {
      "x-python-type": "Triangulation"
    },
    "signature": "(points: PointCloud | Iterable[Iterable[float]]) -> Triangulation",
    "signatures": [
      "(points: PointCloud | Iterable[Iterable[float]]) -> Triangulation"
    ],
    "summary": "Return the exact-predicate Delaunay triangulation of ``points``.",
    "summary_source": "public docstring"
  },
  {
    "canonical_name": "geometry.regular_triangulation",
    "determinism": "deterministic",
    "doc": "Return the regular triangulation with one finite weight per site.",
    "domain": "geometry",
    "file": "noether/geometry/__init__.pyi",
    "input_schema": {
      "additionalProperties": false,
      "properties": {
        "points": {
          "anyOf": [
            {
              "x-python-type": "PointCloud"
            },
            {
              "items": {
                "items": {
                  "type": "number"
                },
                "type": "array",
                "x-python-type": "Iterable[float]"
              },
              "type": "array",
              "x-python-type": "Iterable[Iterable[float]]"
            }
          ],
          "x-python-type": "PointCloud | Iterable[Iterable[float]]"
        },
        "weights": {
          "items": {
            "type": "number"
          },
          "type": "array",
          "x-python-type": "Sequence[float]"
        }
      },
      "required": [
        "points",
        "weights"
      ],
      "type": "object"
    },
    "kind": "function",
    "line": 91,
    "module": "noether.geometry",
    "name": "regular_triangulation",
    "overloads": [
      {
        "input_schema": {
          "additionalProperties": false,
          "properties": {
            "points": {
              "anyOf": [
                {
                  "x-python-type": "PointCloud"
                },
                {
                  "items": {
                    "items": {
                      "type": "number"
                    },
                    "type": "array",
                    "x-python-type": "Iterable[float]"
                  },
                  "type": "array",
                  "x-python-type": "Iterable[Iterable[float]]"
                }
              ],
              "x-python-type": "PointCloud | Iterable[Iterable[float]]"
            },
            "weights": {
              "items": {
                "type": "number"
              },
              "type": "array",
              "x-python-type": "Sequence[float]"
            }
          },
          "required": [
            "points",
            "weights"
          ],
          "type": "object"
        },
        "parameters": [
          {
            "kind": "positional_or_keyword",
            "name": "points",
            "required": true,
            "schema": {
              "anyOf": [
                {
                  "x-python-type": "PointCloud"
                },
                {
                  "items": {
                    "items": {
                      "type": "number"
                    },
                    "type": "array",
                    "x-python-type": "Iterable[float]"
                  },
                  "type": "array",
                  "x-python-type": "Iterable[Iterable[float]]"
                }
              ],
              "x-python-type": "PointCloud | Iterable[Iterable[float]]"
            }
          },
          {
            "kind": "positional_or_keyword",
            "name": "weights",
            "required": true,
            "schema": {
              "items": {
                "type": "number"
              },
              "type": "array",
              "x-python-type": "Sequence[float]"
            }
          }
        ],
        "result_schema": {
          "x-python-type": "Triangulation"
        },
        "signature": "(points: PointCloud | Iterable[Iterable[float]], weights: Sequence[float]) -> Triangulation"
      }
    ],
    "qualified_name": "regular_triangulation",
    "releases_gil": false,
    "result_schema": {
      "x-python-type": "Triangulation"
    },
    "signature": "(points: PointCloud | Iterable[Iterable[float]], weights: Sequence[float]) -> Triangulation",
    "signatures": [
      "(points: PointCloud | Iterable[Iterable[float]], weights: Sequence[float]) -> Triangulation"
    ],
    "summary": "Return the regular triangulation with one finite weight per site.",
    "summary_source": "public docstring"
  },
  {
    "canonical_name": "groups.ConjugacyClass",
    "determinism": "deterministic",
    "doc": "One exact conjugacy class in a concrete permutation group.",
    "domain": "groups",
    "file": "noether/groups/__init__.pyi",
    "input_schema": {},
    "kind": "class",
    "line": 277,
    "module": "noether.groups",
    "name": "ConjugacyClass",
    "overloads": [],
    "qualified_name": "ConjugacyClass",
    "releases_gil": false,
    "result_schema": {},
    "signature": "",
    "signatures": [],
    "summary": "One exact conjugacy class in a concrete permutation group.",
    "summary_source": "public docstring",
    "type_id": "noether/groups/ConjugacyClass@1"
  },
  {
    "canonical_name": "groups.ConjugacyClass.centralizer",
    "determinism": "deterministic",
    "doc": "Centralizer of the representative inside the source group.",
    "domain": "groups",
    "file": "noether/groups/__init__.pyi",
    "input_schema": {
      "additionalProperties": false,
      "properties": {},
      "type": "object"
    },
    "kind": "method",
    "line": 287,
    "module": "noether.groups",
    "name": "centralizer",
    "overloads": [
      {
        "input_schema": {
          "additionalProperties": false,
          "properties": {},
          "type": "object"
        },
        "parameters": [],
        "result_schema": {
          "x-python-type": "PermutationGroup"
        },
        "signature": "(self) -> PermutationGroup"
      }
    ],
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    "canonical_name": "groups.PairOrbit.right_seed",
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    "canonical_name": "groups.PairOrbit.verify",
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    "canonical_name": "groups.Permutation.__call__",
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    "signature": "(images: Sequence[int]) -> Permutation",
    "signatures": [
      "(images: Sequence[int]) -> Permutation"
    ],
    "summary": "Build a permutation from its image vector.",
    "summary_source": "public docstring",
    "type_id": "noether/groups/Permutation@1"
  },
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    "canonical_name": "groups.Permutation.identity",
    "determinism": "deterministic",
    "doc": "The identity permutation on `0..degree`.",
    "domain": "groups",
    "file": "noether/groups/__init__.pyi",
    "input_schema": {
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      "properties": {
        "degree": {
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      "required": [
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    },
    "kind": "staticmethod",
    "line": 117,
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          "properties": {
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          ],
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            "schema": {
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        ],
        "result_schema": {
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        "signature": "(degree: int) -> Permutation"
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    ],
    "qualified_name": "Permutation.identity",
    "releases_gil": false,
    "result_schema": {
      "x-python-type": "Permutation"
    },
    "signature": "(degree: int) -> Permutation",
    "signatures": [
      "(degree: int) -> Permutation"
    ],
    "summary": "The identity permutation on `0..degree`.",
    "summary_source": "public docstring",
    "type_id": "noether/groups/Permutation@1"
  },
  {
    "canonical_name": "groups.Permutation.images",
    "determinism": "deterministic",
    "doc": "The image vector as a `list[int]` of length `degree`.",
    "domain": "groups",
    "file": "noether/groups/__init__.pyi",
    "input_schema": {
      "additionalProperties": false,
      "properties": {},
      "type": "object"
    },
    "kind": "property",
    "line": 125,
    "module": "noether.groups",
    "name": "images",
    "overloads": [
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          "properties": {},
          "type": "object"
        },
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    "qualified_name": "Permutation.images",
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      "x-python-type": "list[int]"
    },
    "signature": "(self) -> list[int]",
    "signatures": [
      "(self) -> list[int]"
    ],
    "summary": "The image vector as a `list[int]` of length `degree`.",
    "summary_source": "public docstring",
    "type_id": "noether/groups/Permutation@1"
  },
  {
    "canonical_name": "groups.Permutation.inverse",
    "determinism": "deterministic",
    "doc": "The inverse permutation.",
    "domain": "groups",
    "file": "noether/groups/__init__.pyi",
    "input_schema": {
      "additionalProperties": false,
      "properties": {},
      "type": "object"
    },
    "kind": "method",
    "line": 130,
    "module": "noether.groups",
    "name": "inverse",
    "overloads": [
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        "input_schema": {
          "additionalProperties": false,
          "properties": {},
          "type": "object"
        },
        "parameters": [],
        "result_schema": {
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        },
        "signature": "(self) -> Permutation"
      }
    ],
    "qualified_name": "Permutation.inverse",
    "releases_gil": false,
    "result_schema": {
      "x-python-type": "Permutation"
    },
    "signature": "(self) -> Permutation",
    "signatures": [
      "(self) -> Permutation"
    ],
    "summary": "The inverse permutation.",
    "summary_source": "public docstring",
    "type_id": "noether/groups/Permutation@1"
  },
  {
    "canonical_name": "groups.Permutation.is_identity",
    "determinism": "deterministic",
    "doc": "`True` iff this is the identity permutation on its domain.",
    "domain": "groups",
    "file": "noether/groups/__init__.pyi",
    "input_schema": {
      "additionalProperties": false,
      "properties": {},
      "type": "object"
    },
    "kind": "property",
    "line": 121,
    "module": "noether.groups",
    "name": "is_identity",
    "overloads": [
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        "input_schema": {
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          "properties": {},
          "type": "object"
        },
        "parameters": [],
        "result_schema": {
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        },
        "signature": "(self) -> bool"
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    "qualified_name": "Permutation.is_identity",
    "releases_gil": false,
    "result_schema": {
      "type": "boolean"
    },
    "signature": "(self) -> bool",
    "signatures": [
      "(self) -> bool"
    ],
    "summary": "`True` iff this is the identity permutation on its domain.",
    "summary_source": "public docstring",
    "type_id": "noether/groups/Permutation@1"
  },
  {
    "canonical_name": "groups.Permutation.order",
    "determinism": "deterministic",
    "doc": "The (multiplicative) order of this permutation — the smallest\n`k ≥ 1` with `σ^k = identity`. Equals the LCM of cycle lengths\nin the cycle decomposition.",
    "domain": "groups",
    "file": "noether/groups/__init__.pyi",
    "input_schema": {
      "additionalProperties": false,
      "properties": {},
      "type": "object"
    },
    "kind": "property",
    "line": 123,
    "module": "noether.groups",
    "name": "order",
    "overloads": [
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          "properties": {},
          "type": "object"
        },
        "parameters": [],
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        },
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    "qualified_name": "Permutation.order",
    "releases_gil": false,
    "result_schema": {
      "type": "integer"
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    "signature": "(self) -> int",
    "signatures": [
      "(self) -> int"
    ],
    "summary": "The (multiplicative) order of this permutation — the smallest `k ≥ 1` with `σ^k = identity`.",
    "summary_source": "public docstring",
    "type_id": "noether/groups/Permutation@1"
  },
  {
    "canonical_name": "groups.PermutationGroup",
    "determinism": "deterministic",
    "doc": "A permutation group given by generators, carrying its BSGS.\n\nConstruct via `PermGroup.from_generators` (deterministic\nSchreier–Sims) or `PermGroup.from_generators_randomized`\n(Monte-Carlo BSGS, **verified** by a deterministic final sweep, so an\nunverified probabilistic result is never returned). The constructor computes the base, strong generating set,\nstabilizer chain, per-level transversals, and group order; every\nquery downstream is a method on this object and runs in\nconstant-time-per-step.",
    "domain": "groups",
    "file": "noether/groups/__init__.pyi",
    "input_schema": {},
    "kind": "class",
    "line": 141,
    "module": "noether.groups",
    "name": "PermutationGroup",
    "overloads": [],
    "qualified_name": "PermutationGroup",
    "releases_gil": false,
    "result_schema": {},
    "signature": "",
    "signatures": [],
    "summary": "A permutation group given by generators, carrying its BSGS.",
    "summary_source": "public docstring",
    "type_id": "noether/groups/PermutationGroup@1"
  },
  {
    "canonical_name": "groups.PermutationGroup.__contains__",
    "determinism": "deterministic",
    "doc": "Return bool(key in self).",
    "domain": "groups",
    "file": "noether/groups/__init__.pyi",
    "input_schema": {
      "additionalProperties": false,
      "properties": {
        "element": {
          "x-python-type": "Permutation"
        }
      },
      "required": [
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      ],
      "type": "object"
    },
    "kind": "method",
    "line": 177,
    "module": "noether.groups",
    "name": "__contains__",
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              "x-python-type": "Permutation"
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          ],
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            "kind": "positional_or_keyword",
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            "required": true,
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              "x-python-type": "Permutation"
            }
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        ],
        "result_schema": {
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        "signature": "(self, element: Permutation) -> bool"
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    "qualified_name": "PermutationGroup.__contains__",
    "releases_gil": true,
    "result_schema": {
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    "signature": "(self, element: Permutation) -> bool",
    "signatures": [
      "(self, element: Permutation) -> bool"
    ],
    "summary": "Return bool(key in self).",
    "summary_source": "public docstring",
    "type_id": "noether/groups/PermutationGroup@1"
  },
  {
    "canonical_name": "groups.PermutationGroup.abelian_invariant_factors",
    "determinism": "deterministic",
    "doc": "Canonical invariant factors when this concrete group is finite abelian.",
    "domain": "groups",
    "file": "noether/groups/__init__.pyi",
    "input_schema": {
      "additionalProperties": false,
      "properties": {},
      "type": "object"
    },
    "kind": "method",
    "line": 185,
    "module": "noether.groups",
    "name": "abelian_invariant_factors",
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          "properties": {},
          "type": "object"
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    ],
    "qualified_name": "PermutationGroup.abelian_invariant_factors",
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    "signature": "(self) -> list[int]",
    "signatures": [
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    ],
    "summary": "Canonical invariant factors when this concrete group is finite abelian.",
    "summary_source": "public docstring",
    "type_id": "noether/groups/PermutationGroup@1"
  },
  {
    "canonical_name": "groups.PermutationGroup.abelianization_invariant_factors",
    "determinism": "deterministic",
    "doc": "Canonical invariant factors of this finite group's abelianization.",
    "domain": "groups",
    "file": "noether/groups/__init__.pyi",
    "input_schema": {
      "additionalProperties": false,
      "properties": {},
      "type": "object"
    },
    "kind": "method",
    "line": 186,
    "module": "noether.groups",
    "name": "abelianization_invariant_factors",
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        },
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    ],
    "qualified_name": "PermutationGroup.abelianization_invariant_factors",
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    "signatures": [
      "(self) -> list[int]"
    ],
    "summary": "Canonical invariant factors of this finite group's abelianization.",
    "summary_source": "public docstring",
    "type_id": "noether/groups/PermutationGroup@1"
  },
  {
    "canonical_name": "groups.PermutationGroup.base",
    "determinism": "deterministic",
    "doc": "The base — the ordered list of base points the stabilizer\nchain stabilises one level at a time. Empty `list[int]` for\nthe trivial group.",
    "domain": "groups",
    "file": "noether/groups/__init__.pyi",
    "input_schema": {
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      "properties": {},
      "type": "object"
    },
    "kind": "property",
    "line": 163,
    "module": "noether.groups",
    "name": "base",
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          "properties": {},
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    "qualified_name": "PermutationGroup.base",
    "releases_gil": false,
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    "signature": "(self) -> list[int]",
    "signatures": [
      "(self) -> list[int]"
    ],
    "summary": "The base — the ordered list of base points the stabilizer chain stabilises one level at a time.",
    "summary_source": "public docstring",
    "type_id": "noether/groups/PermutationGroup@1"
  },
  {
    "canonical_name": "groups.PermutationGroup.base_and_strong_generating_set",
    "determinism": "deterministic",
    "doc": "The base and strong generating set as a `(list[int],\nlist[Permutation])` pair, useful for callers\nthat want the BSGS without constructing the full\n`PermGroup` API on it twice.",
    "domain": "groups",
    "file": "noether/groups/__init__.pyi",
    "input_schema": {
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      "properties": {
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    "kind": "staticmethod",
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    "module": "noether.groups",
    "name": "base_and_strong_generating_set",
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          "properties": {
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              "type": "integer"
            },
            "generators": {
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          },
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        },
        "signature": "(degree: int, generators: Sequence[Permutation]) -> tuple[list[int], list[Permutation]]"
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    "qualified_name": "PermutationGroup.base_and_strong_generating_set",
    "releases_gil": true,
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    "signatures": [
      "(degree: int, generators: Sequence[Permutation]) -> tuple[list[int], list[Permutation]]"
    ],
    "summary": "The base and strong generating set as a `(list[int], list[Permutation])` pair, useful for callers that want the BSGS without constructing the full `PermGroup` API on it twice.",
    "summary_source": "public docstring",
    "type_id": "noether/groups/PermutationGroup@1"
  },
  {
    "canonical_name": "groups.PermutationGroup.complete_abelian_integral_cohomology",
    "determinism": "deterministic",
    "doc": "Complete closed-form H*(BG; Z) for a supported finite abelian group.",
    "domain": "groups",
    "file": "noether/groups/__init__.pyi",
    "input_schema": {
      "additionalProperties": false,
      "properties": {},
      "type": "object"
    },
    "kind": "method",
    "line": 190,
    "module": "noether.groups",
    "name": "complete_abelian_integral_cohomology",
    "overloads": [
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        "input_schema": {
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          "properties": {},
          "type": "object"
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        "parameters": [],
        "result_schema": {
          "x-python-type": "FiniteAbelianGroupIntegralCohomologyPresentation"
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    "qualified_name": "PermutationGroup.complete_abelian_integral_cohomology",
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    "signatures": [
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    "summary": "Complete closed-form H*(BG; Z) for a supported finite abelian group.",
    "summary_source": "public docstring",
    "type_id": "noether/groups/PermutationGroup@1"
  },
  {
    "canonical_name": "groups.PermutationGroup.complete_abelian_modular_cohomology",
    "determinism": "deterministic",
    "doc": "Complete closed-form H*(BG; F_p) for a finite abelian group.",
    "domain": "groups",
    "file": "noether/groups/__init__.pyi",
    "input_schema": {
      "additionalProperties": false,
      "properties": {
        "field": {
          "x-python-type": "FiniteField"
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      },
      "required": [
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      "type": "object"
    },
    "kind": "method",
    "line": 187,
    "module": "noether.groups",
    "name": "complete_abelian_modular_cohomology",
    "overloads": [
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          "properties": {
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            "schema": {
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        ],
        "result_schema": {
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    "qualified_name": "PermutationGroup.complete_abelian_modular_cohomology",
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    "result_schema": {
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    "signature": "(self, field: FiniteField) -> FiniteAbelianGroupCohomologyPresentation",
    "signatures": [
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    "summary": "Complete closed-form H*(BG; F_p) for a finite abelian group.",
    "summary_source": "public docstring",
    "type_id": "noether/groups/PermutationGroup@1"
  },
  {
    "canonical_name": "groups.PermutationGroup.complete_cyclic_integral_cohomology",
    "determinism": "deterministic",
    "doc": "Complete closed-form H*(BG; Z) for a finite cyclic group.",
    "domain": "groups",
    "file": "noether/groups/__init__.pyi",
    "input_schema": {
      "additionalProperties": false,
      "properties": {},
      "type": "object"
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    "kind": "method",
    "line": 193,
    "module": "noether.groups",
    "name": "complete_cyclic_integral_cohomology",
    "overloads": [
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          "properties": {},
          "type": "object"
        },
        "parameters": [],
        "result_schema": {
          "x-python-type": "FiniteCyclicGroupIntegralCohomologyPresentation"
        },
        "signature": "(self) -> FiniteCyclicGroupIntegralCohomologyPresentation"
      }
    ],
    "qualified_name": "PermutationGroup.complete_cyclic_integral_cohomology",
    "releases_gil": true,
    "result_schema": {
      "x-python-type": "FiniteCyclicGroupIntegralCohomologyPresentation"
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    "signature": "(self) -> FiniteCyclicGroupIntegralCohomologyPresentation",
    "signatures": [
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    "summary": "Complete closed-form H*(BG; Z) for a finite cyclic group.",
    "summary_source": "public docstring",
    "type_id": "noether/groups/PermutationGroup@1"
  },
  {
    "canonical_name": "groups.PermutationGroup.conjugacy_classes",
    "determinism": "deterministic",
    "doc": "Exact conjugacy classes, ordered by their canonical representatives.\n\nThe native correctness floor enumerates groups of order at most one\nmillion and fails before returning a partial result above that bound.",
    "domain": "groups",
    "file": "noether/groups/__init__.pyi",
    "input_schema": {
      "additionalProperties": false,
      "properties": {},
      "type": "object"
    },
    "kind": "method",
    "line": 218,
    "module": "noether.groups",
    "name": "conjugacy_classes",
    "overloads": [
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        "parameters": [],
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          "items": {
            "x-python-type": "ConjugacyClass"
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          "type": "array",
          "x-python-type": "list[ConjugacyClass]"
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        "signature": "(self) -> list[ConjugacyClass]"
      }
    ],
    "qualified_name": "PermutationGroup.conjugacy_classes",
    "releases_gil": true,
    "result_schema": {
      "items": {
        "x-python-type": "ConjugacyClass"
      },
      "type": "array",
      "x-python-type": "list[ConjugacyClass]"
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    "signature": "(self) -> list[ConjugacyClass]",
    "signatures": [
      "(self) -> list[ConjugacyClass]"
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    "summary": "Exact conjugacy classes, ordered by their canonical representatives.",
    "summary_source": "public docstring",
    "type_id": "noether/groups/PermutationGroup@1"
  },
  {
    "canonical_name": "groups.PermutationGroup.conjugating_element",
    "determinism": "deterministic",
    "doc": "A conjugating element `g` with `selfᵍ = other` (i.e.\n`g⁻¹ · self · g = other`), or `None` if the two groups are not\nconjugate in `S_n`. The witness is sift-verified before it is\nreturned. Raises `ValueError` on a degree mismatch, `OverflowError`\nif the backtrack exceeds its search budget.",
    "domain": "groups",
    "file": "noether/groups/__init__.pyi",
    "input_schema": {
      "additionalProperties": false,
      "properties": {
        "other": {
          "x-python-type": "PermutationGroup"
        }
      },
      "required": [
        "other"
      ],
      "type": "object"
    },
    "kind": "method",
    "line": 220,
    "module": "noether.groups",
    "name": "conjugating_element",
    "overloads": [
      {
        "input_schema": {
          "additionalProperties": false,
          "properties": {
            "other": {
              "x-python-type": "PermutationGroup"
            }
          },
          "required": [
            "other"
          ],
          "type": "object"
        },
        "parameters": [
          {
            "kind": "positional_or_keyword",
            "name": "other",
            "required": true,
            "schema": {
              "x-python-type": "PermutationGroup"
            }
          }
        ],
        "result_schema": {
          "anyOf": [
            {
              "x-python-type": "Permutation"
            },
            {
              "type": "null"
            }
          ],
          "x-python-type": "Permutation | None"
        },
        "signature": "(self, other: PermutationGroup) -> Permutation | None"
      }
    ],
    "qualified_name": "PermutationGroup.conjugating_element",
    "releases_gil": true,
    "result_schema": {
      "anyOf": [
        {
          "x-python-type": "Permutation"
        },
        {
          "type": "null"
        }
      ],
      "x-python-type": "Permutation | None"
    },
    "signature": "(self, other: PermutationGroup) -> Permutation | None",
    "signatures": [
      "(self, other: PermutationGroup) -> Permutation | None"
    ],
    "summary": "A conjugating element `g` with `selfᵍ = other` (i.e.",
    "summary_source": "public docstring",
    "type_id": "noether/groups/PermutationGroup@1"
  },
  {
    "canonical_name": "groups.PermutationGroup.contains",
    "determinism": "deterministic",
    "doc": "`True` iff `g` lies in the group. Sifts `g` down the stabilizer\nchain and checks the residue is the identity. Raises\n`ValueError` on a degree mismatch.",
    "domain": "groups",
    "file": "noether/groups/__init__.pyi",
    "input_schema": {
      "additionalProperties": false,
      "properties": {
        "element": {
          "x-python-type": "Permutation"
        }
      },
      "required": [
        "element"
      ],
      "type": "object"
    },
    "kind": "method",
    "line": 176,
    "module": "noether.groups",
    "name": "contains",
    "overloads": [
      {
        "input_schema": {
          "additionalProperties": false,
          "properties": {
            "element": {
              "x-python-type": "Permutation"
            }
          },
          "required": [
            "element"
          ],
          "type": "object"
        },
        "parameters": [
          {
            "kind": "positional_or_keyword",
            "name": "element",
            "required": true,
            "schema": {
              "x-python-type": "Permutation"
            }
          }
        ],
        "result_schema": {
          "type": "boolean"
        },
        "signature": "(self, element: Permutation) -> bool"
      }
    ],
    "qualified_name": "PermutationGroup.contains",
    "releases_gil": true,
    "result_schema": {
      "type": "boolean"
    },
    "signature": "(self, element: Permutation) -> bool",
    "signatures": [
      "(self, element: Permutation) -> bool"
    ],
    "summary": "`True` iff `g` lies in the group.",
    "summary_source": "public docstring",
    "type_id": "noether/groups/PermutationGroup@1"
  },
  {
    "canonical_name": "groups.PermutationGroup.degree",
    "determinism": "deterministic",
    "doc": "The degree — the size of the permutation domain `0..degree`.",
    "domain": "groups",
    "file": "noether/groups/__init__.pyi",
    "input_schema": {
      "additionalProperties": false,
      "properties": {},
      "type": "object"
    },
    "kind": "property",
    "line": 159,
    "module": "noether.groups",
    "name": "degree",
    "overloads": [
      {
        "input_schema": {
          "additionalProperties": false,
          "properties": {},
          "type": "object"
        },
        "parameters": [],
        "result_schema": {
          "type": "integer"
        },
        "signature": "(self) -> int"
      }
    ],
    "qualified_name": "PermutationGroup.degree",
    "releases_gil": false,
    "result_schema": {
      "type": "integer"
    },
    "signature": "(self) -> int",
    "signatures": [
      "(self) -> int"
    ],
    "summary": "The degree — the size of the permutation domain `0..degree`.",
    "summary_source": "public docstring",
    "type_id": "noether/groups/PermutationGroup@1"
  },
  {
    "canonical_name": "groups.PermutationGroup.double_coset_triangle",
    "determinism": "deterministic",
    "doc": "Lower-triangular exact double-coset table for several subgroups.",
    "domain": "groups",
    "file": "noether/groups/__init__.pyi",
    "input_schema": {
      "additionalProperties": false,
      "properties": {
        "subgroups": {
          "items": {
            "x-python-type": "PermutationGroup"
          },
          "type": "array",
          "x-python-type": "Sequence[PermutationGroup]"
        }
      },
      "required": [
        "subgroups"
      ],
      "type": "object"
    },
    "kind": "method",
    "line": 207,
    "module": "noether.groups",
    "name": "double_coset_triangle",
    "overloads": [
      {
        "input_schema": {
          "additionalProperties": false,
          "properties": {
            "subgroups": {
              "items": {
                "x-python-type": "PermutationGroup"
              },
              "type": "array",
              "x-python-type": "Sequence[PermutationGroup]"
            }
          },
          "required": [
            "subgroups"
          ],
          "type": "object"
        },
        "parameters": [
          {
            "kind": "positional_or_keyword",
            "name": "subgroups",
            "required": true,
            "schema": {
              "items": {
                "x-python-type": "PermutationGroup"
              },
              "type": "array",
              "x-python-type": "Sequence[PermutationGroup]"
            }
          }
        ],
        "result_schema": {
          "items": {
            "items": {
              "items": {
                "x-python-type": "DoubleCoset"
              },
              "type": "array",
              "x-python-type": "list[DoubleCoset]"
            },
            "type": "array",
            "x-python-type": "list[list[DoubleCoset]]"
          },
          "type": "array",
          "x-python-type": "list[list[list[DoubleCoset]]]"
        },
        "signature": "(self, subgroups: Sequence[PermutationGroup]) -> list[list[list[DoubleCoset]]]"
      }
    ],
    "qualified_name": "PermutationGroup.double_coset_triangle",
    "releases_gil": true,
    "result_schema": {
      "items": {
        "items": {
          "items": {
            "x-python-type": "DoubleCoset"
          },
          "type": "array",
          "x-python-type": "list[DoubleCoset]"
        },
        "type": "array",
        "x-python-type": "list[list[DoubleCoset]]"
      },
      "type": "array",
      "x-python-type": "list[list[list[DoubleCoset]]]"
    },
    "signature": "(self, subgroups: Sequence[PermutationGroup]) -> list[list[list[DoubleCoset]]]",
    "signatures": [
      "(self, subgroups: Sequence[PermutationGroup]) -> list[list[list[DoubleCoset]]]"
    ],
    "summary": "Lower-triangular exact double-coset table for several subgroups.",
    "summary_source": "public docstring",
    "type_id": "noether/groups/PermutationGroup@1"
  },
  {
    "canonical_name": "groups.PermutationGroup.double_cosets",
    "determinism": "deterministic",
    "doc": "Exact double cosets of two literal subgroups.",
    "domain": "groups",
    "file": "noether/groups/__init__.pyi",
    "input_schema": {
      "additionalProperties": false,
      "properties": {
        "left": {
          "x-python-type": "PermutationGroup"
        },
        "right": {
          "x-python-type": "PermutationGroup"
        }
      },
      "required": [
        "left",
        "right"
      ],
      "type": "object"
    },
    "kind": "method",
    "line": 204,
    "module": "noether.groups",
    "name": "double_cosets",
    "overloads": [
      {
        "input_schema": {
          "additionalProperties": false,
          "properties": {
            "left": {
              "x-python-type": "PermutationGroup"
            },
            "right": {
              "x-python-type": "PermutationGroup"
            }
          },
          "required": [
            "left",
            "right"
          ],
          "type": "object"
        },
        "parameters": [
          {
            "kind": "positional_or_keyword",
            "name": "left",
            "required": true,
            "schema": {
              "x-python-type": "PermutationGroup"
            }
          },
          {
            "kind": "positional_or_keyword",
            "name": "right",
            "required": true,
            "schema": {
              "x-python-type": "PermutationGroup"
            }
          }
        ],
        "result_schema": {
          "items": {
            "x-python-type": "DoubleCoset"
          },
          "type": "array",
          "x-python-type": "list[DoubleCoset]"
        },
        "signature": "(self, left: PermutationGroup, right: PermutationGroup) -> list[DoubleCoset]"
      }
    ],
    "qualified_name": "PermutationGroup.double_cosets",
    "releases_gil": true,
    "result_schema": {
      "items": {
        "x-python-type": "DoubleCoset"
      },
      "type": "array",
      "x-python-type": "list[DoubleCoset]"
    },
    "signature": "(self, left: PermutationGroup, right: PermutationGroup) -> list[DoubleCoset]",
    "signatures": [
      "(self, left: PermutationGroup, right: PermutationGroup) -> list[DoubleCoset]"
    ],
    "summary": "Exact double cosets of two literal subgroups.",
    "summary_source": "public docstring",
    "type_id": "noether/groups/PermutationGroup@1"
  },
  {
    "canonical_name": "groups.PermutationGroup.embeds_up_to_conjugacy",
    "determinism": "deterministic",
    "doc": "A witness `g` with `selfᵍ ≤ over` (some `S_n`-conjugate of `self`\nis a subgroup of `over`), or `None`. The witness is verified\nbefore it is returned. Raises `ValueError` on a degree mismatch,\n`OverflowError` on an exhausted search budget.",
    "domain": "groups",
    "file": "noether/groups/__init__.pyi",
    "input_schema": {
      "additionalProperties": false,
      "properties": {
        "over": {
          "x-python-type": "PermutationGroup"
        }
      },
      "required": [
        "over"
      ],
      "type": "object"
    },
    "kind": "method",
    "line": 221,
    "module": "noether.groups",
    "name": "embeds_up_to_conjugacy",
    "overloads": [
      {
        "input_schema": {
          "additionalProperties": false,
          "properties": {
            "over": {
              "x-python-type": "PermutationGroup"
            }
          },
          "required": [
            "over"
          ],
          "type": "object"
        },
        "parameters": [
          {
            "kind": "positional_or_keyword",
            "name": "over",
            "required": true,
            "schema": {
              "x-python-type": "PermutationGroup"
            }
          }
        ],
        "result_schema": {
          "anyOf": [
            {
              "x-python-type": "Permutation"
            },
            {
              "type": "null"
            }
          ],
          "x-python-type": "Permutation | None"
        },
        "signature": "(self, over: PermutationGroup) -> Permutation | None"
      }
    ],
    "qualified_name": "PermutationGroup.embeds_up_to_conjugacy",
    "releases_gil": true,
    "result_schema": {
      "anyOf": [
        {
          "x-python-type": "Permutation"
        },
        {
          "type": "null"
        }
      ],
      "x-python-type": "Permutation | None"
    },
    "signature": "(self, over: PermutationGroup) -> Permutation | None",
    "signatures": [
      "(self, over: PermutationGroup) -> Permutation | None"
    ],
    "summary": "A witness `g` with `selfᵍ ≤ over` (some `S_n`-conjugate of `self` is a subgroup of `over`), or `None`.",
    "summary_source": "public docstring",
    "type_id": "noether/groups/PermutationGroup@1"
  },
  {
    "algorithms": [
      "stabilizer_chain"
    ],
    "backends": [
      "perm-core"
    ],
    "canonical_name": "groups.PermutationGroup.from_generators",
    "determinism": "deterministic",
    "doc": "Construct a permutation group on `0..degree` from a generator\nlist, running deterministic Schreier–Sims.\n\nRaises `ValueError` on a degree mismatch or a non-permutation\ngenerator.",
    "domain": "groups",
    "errors": [
      "degree mismatch",
      "non-permutation generator"
    ],
    "example": "from noether.groups import Permutation, PermutationGroup\ntransposition = Permutation.from_images([1, 0, 2])\ncycle = Permutation.from_images([1, 2, 0])\nresult = PermutationGroup.from_generators(3, [transposition, cycle])\nassert result.order == 6",
    "file": "noether/groups/__init__.pyi",
    "input_schema": {
      "additionalProperties": false,
      "properties": {
        "degree": {
          "type": "integer"
        },
        "generators": {
          "items": {
            "x-python-type": "Permutation"
          },
          "type": "array",
          "x-python-type": "Sequence[Permutation]"
        }
      },
      "required": [
        "degree",
        "generators"
      ],
      "type": "object"
    },
    "kind": "staticmethod",
    "line": 144,
    "module": "noether.groups",
    "name": "from_generators",
    "overloads": [
      {
        "input_schema": {
          "additionalProperties": false,
          "properties": {
            "degree": {
              "type": "integer"
            },
            "generators": {
              "items": {
                "x-python-type": "Permutation"
              },
              "type": "array",
              "x-python-type": "Sequence[Permutation]"
            }
          },
          "required": [
            "degree",
            "generators"
          ],
          "type": "object"
        },
        "parameters": [
          {
            "kind": "positional_or_keyword",
            "name": "degree",
            "required": true,
            "schema": {
              "type": "integer"
            }
          },
          {
            "kind": "positional_or_keyword",
            "name": "generators",
            "required": true,
            "schema": {
              "items": {
                "x-python-type": "Permutation"
              },
              "type": "array",
              "x-python-type": "Sequence[Permutation]"
            }
          }
        ],
        "result_schema": {
          "x-python-type": "PermutationGroup"
        },
        "signature": "(degree: int, generators: Sequence[Permutation]) -> PermutationGroup"
      }
    ],
    "qualified_name": "PermutationGroup.from_generators",
    "releases_gil": true,
    "result_schema": {
      "x-python-type": "PermutationGroup"
    },
    "signature": "(degree: int, generators: Sequence[Permutation]) -> PermutationGroup",
    "signatures": [
      "(degree: int, generators: Sequence[Permutation]) -> PermutationGroup"
    ],
    "summary": "Construct a permutation group on `0..degree` from a generator list, running deterministic Schreier–Sims.",
    "summary_source": "public docstring",
    "type_id": "noether/groups/PermutationGroup@1"
  },
  {
    "canonical_name": "groups.PermutationGroup.from_generators_randomized",
    "determinism": "seeded",
    "doc": "Construct via the verified randomized Schreier–Sims variant.\n\nSims-style product replacement seeded with `seed` supplies\nrandom group elements; the chain stops growing after\n`stabilize` consecutive sift-to-identity events; the result\nis then verified by the deterministic Schreier-generator\nsweep. The `seed` is recorded on the result so the build is\nreproducible.",
    "domain": "groups",
    "file": "noether/groups/__init__.pyi",
    "input_schema": {
      "additionalProperties": false,
      "properties": {
        "degree": {
          "type": "integer"
        },
        "generators": {
          "items": {
            "x-python-type": "Permutation"
          },
          "type": "array",
          "x-python-type": "Sequence[Permutation]"
        },
        "seed": {
          "type": "integer"
        },
        "stabilize": {
          "type": "integer"
        }
      },
      "required": [
        "degree",
        "generators",
        "seed",
        "stabilize"
      ],
      "type": "object"
    },
    "kind": "staticmethod",
    "line": 146,
    "module": "noether.groups",
    "name": "from_generators_randomized",
    "overloads": [
      {
        "input_schema": {
          "additionalProperties": false,
          "properties": {
            "degree": {
              "type": "integer"
            },
            "generators": {
              "items": {
                "x-python-type": "Permutation"
              },
              "type": "array",
              "x-python-type": "Sequence[Permutation]"
            },
            "seed": {
              "type": "integer"
            },
            "stabilize": {
              "type": "integer"
            }
          },
          "required": [
            "degree",
            "generators",
            "seed",
            "stabilize"
          ],
          "type": "object"
        },
        "parameters": [
          {
            "kind": "positional_or_keyword",
            "name": "degree",
            "required": true,
            "schema": {
              "type": "integer"
            }
          },
          {
            "kind": "positional_or_keyword",
            "name": "generators",
            "required": true,
            "schema": {
              "items": {
                "x-python-type": "Permutation"
              },
              "type": "array",
              "x-python-type": "Sequence[Permutation]"
            }
          },
          {
            "kind": "positional_or_keyword",
            "name": "seed",
            "required": true,
            "schema": {
              "type": "integer"
            }
          },
          {
            "kind": "positional_or_keyword",
            "name": "stabilize",
            "required": true,
            "schema": {
              "type": "integer"
            }
          }
        ],
        "result_schema": {
          "x-python-type": "PermutationGroup"
        },
        "signature": "(degree: int, generators: Sequence[Permutation], seed: int, stabilize: int) -> PermutationGroup"
      }
    ],
    "qualified_name": "PermutationGroup.from_generators_randomized",
    "releases_gil": true,
    "result_schema": {
      "x-python-type": "PermutationGroup"
    },
    "signature": "(degree: int, generators: Sequence[Permutation], seed: int, stabilize: int) -> PermutationGroup",
    "signatures": [
      "(degree: int, generators: Sequence[Permutation], seed: int, stabilize: int) -> PermutationGroup"
    ],
    "summary": "Construct via the verified randomized Schreier–Sims variant.",
    "summary_source": "public docstring",
    "type_id": "noether/groups/PermutationGroup@1"
  },
  {
    "canonical_name": "groups.PermutationGroup.generators",
    "determinism": "deterministic",
    "doc": "The original generators, in input order, as a `list[\nPermutation]`.",
    "domain": "groups",
    "file": "noether/groups/__init__.pyi",
    "input_schema": {
      "additionalProperties": false,
      "properties": {},
      "type": "object"
    },
    "kind": "property",
    "line": 161,
    "module": "noether.groups",
    "name": "generators",
    "overloads": [
      {
        "input_schema": {
          "additionalProperties": false,
          "properties": {},
          "type": "object"
        },
        "parameters": [],
        "result_schema": {
          "items": {
            "x-python-type": "Permutation"
          },
          "type": "array",
          "x-python-type": "list[Permutation]"
        },
        "signature": "(self) -> list[Permutation]"
      }
    ],
    "qualified_name": "PermutationGroup.generators",
    "releases_gil": false,
    "result_schema": {
      "items": {
        "x-python-type": "Permutation"
      },
      "type": "array",
      "x-python-type": "list[Permutation]"
    },
    "signature": "(self) -> list[Permutation]",
    "signatures": [
      "(self) -> list[Permutation]"
    ],
    "summary": "The original generators, in input order, as a `list[ Permutation]`.",
    "summary_source": "public docstring",
    "type_id": "noether/groups/PermutationGroup@1"
  },
  {
    "canonical_name": "groups.PermutationGroup.group_order",
    "determinism": "deterministic",
    "doc": "`|G|` as a Python `int`. Conversion through decimal text preserves\narbitrarily large\norders without imposing a Rust primitive-integer limit.",
    "domain": "groups",
    "file": "noether/groups/__init__.pyi",
    "input_schema": {
      "additionalProperties": false,
      "properties": {
        "degree": {
          "type": "integer"
        },
        "generators": {
          "items": {
            "x-python-type": "Permutation"
          },
          "type": "array",
          "x-python-type": "Sequence[Permutation]"
        }
      },
      "required": [
        "degree",
        "generators"
      ],
      "type": "object"
    },
    "kind": "staticmethod",
    "line": 157,
    "module": "noether.groups",
    "name": "group_order",
    "overloads": [
      {
        "input_schema": {
          "additionalProperties": false,
          "properties": {
            "degree": {
              "type": "integer"
            },
            "generators": {
              "items": {
                "x-python-type": "Permutation"
              },
              "type": "array",
              "x-python-type": "Sequence[Permutation]"
            }
          },
          "required": [
            "degree",
            "generators"
          ],
          "type": "object"
        },
        "parameters": [
          {
            "kind": "positional_or_keyword",
            "name": "degree",
            "required": true,
            "schema": {
              "type": "integer"
            }
          },
          {
            "kind": "positional_or_keyword",
            "name": "generators",
            "required": true,
            "schema": {
              "items": {
                "x-python-type": "Permutation"
              },
              "type": "array",
              "x-python-type": "Sequence[Permutation]"
            }
          }
        ],
        "result_schema": {
          "type": "integer"
        },
        "signature": "(degree: int, generators: Sequence[Permutation]) -> int"
      }
    ],
    "qualified_name": "PermutationGroup.group_order",
    "releases_gil": true,
    "result_schema": {
      "type": "integer"
    },
    "signature": "(degree: int, generators: Sequence[Permutation]) -> int",
    "signatures": [
      "(degree: int, generators: Sequence[Permutation]) -> int"
    ],
    "summary": "`|G|` as a Python `int`.",
    "summary_source": "public docstring",
    "type_id": "noether/groups/PermutationGroup@1"
  },
  {
    "canonical_name": "groups.PermutationGroup.identity",
    "determinism": "deterministic",
    "doc": "The identity element in this permutation action.",
    "domain": "groups",
    "file": "noether/groups/__init__.pyi",
    "input_schema": {
      "additionalProperties": false,
      "properties": {},
      "type": "object"
    },
    "kind": "method",
    "line": 178,
    "module": "noether.groups",
    "name": "identity",
    "overloads": [
      {
        "input_schema": {
          "additionalProperties": false,
          "properties": {},
          "type": "object"
        },
        "parameters": [],
        "result_schema": {
          "x-python-type": "Permutation"
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        "signature": "(self) -> Permutation"
      }
    ],
    "qualified_name": "PermutationGroup.identity",
    "releases_gil": false,
    "result_schema": {
      "x-python-type": "Permutation"
    },
    "signature": "(self) -> Permutation",
    "signatures": [
      "(self) -> Permutation"
    ],
    "summary": "The identity element in this permutation action.",
    "summary_source": "public docstring",
    "type_id": "noether/groups/PermutationGroup@1"
  },
  {
    "canonical_name": "groups.PermutationGroup.integral_cohomology",
    "determinism": "deterministic",
    "doc": "Compute the exact integral cohomology ring through `max_degree`.",
    "domain": "groups",
    "file": "noether/groups/__init__.pyi",
    "input_schema": {
      "additionalProperties": false,
      "properties": {
        "max_degree": {
          "type": "integer"
        }
      },
      "required": [
        "max_degree"
      ],
      "type": "object"
    },
    "kind": "method",
    "line": 184,
    "module": "noether.groups",
    "name": "integral_cohomology",
    "overloads": [
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        "input_schema": {
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          "properties": {
            "max_degree": {
              "type": "integer"
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          "required": [
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          "type": "object"
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            "required": true,
            "schema": {
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        ],
        "result_schema": {
          "x-python-type": "FiniteGroupIntegralCohomologyTruncation"
        },
        "signature": "(self, max_degree: int) -> FiniteGroupIntegralCohomologyTruncation"
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    ],
    "qualified_name": "PermutationGroup.integral_cohomology",
    "releases_gil": true,
    "result_schema": {
      "x-python-type": "FiniteGroupIntegralCohomologyTruncation"
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    "signature": "(self, max_degree: int) -> FiniteGroupIntegralCohomologyTruncation",
    "signatures": [
      "(self, max_degree: int) -> FiniteGroupIntegralCohomologyTruncation"
    ],
    "summary": "Compute the exact integral cohomology ring through `max_degree`.",
    "summary_source": "public docstring",
    "type_id": "noether/groups/PermutationGroup@1"
  },
  {
    "canonical_name": "groups.PermutationGroup.intersection",
    "determinism": "deterministic",
    "doc": "Literal intersection with another group on the same point set.",
    "domain": "groups",
    "file": "noether/groups/__init__.pyi",
    "input_schema": {
      "additionalProperties": false,
      "properties": {
        "other": {
          "x-python-type": "PermutationGroup"
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      },
      "required": [
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      "type": "object"
    },
    "kind": "method",
    "line": 197,
    "module": "noether.groups",
    "name": "intersection",
    "overloads": [
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        "input_schema": {
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          "properties": {
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              "x-python-type": "PermutationGroup"
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          "required": [
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          ],
          "type": "object"
        },
        "parameters": [
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            "kind": "positional_or_keyword",
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        "result_schema": {
          "x-python-type": "PermutationGroup"
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    "signature": "(self, other: PermutationGroup) -> PermutationGroup",
    "signatures": [
      "(self, other: PermutationGroup) -> PermutationGroup"
    ],
    "summary": "Literal intersection with another group on the same point set.",
    "summary_source": "public docstring",
    "type_id": "noether/groups/PermutationGroup@1"
  },
  {
    "canonical_name": "groups.PermutationGroup.inverse",
    "determinism": "deterministic",
    "doc": "Checked inverse of an element of this group.",
    "domain": "groups",
    "file": "noether/groups/__init__.pyi",
    "input_schema": {
      "additionalProperties": false,
      "properties": {
        "value": {
          "x-python-type": "Permutation"
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      },
      "required": [
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      "type": "object"
    },
    "kind": "method",
    "line": 180,
    "module": "noether.groups",
    "name": "inverse",
    "overloads": [
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          "type": "object"
        },
        "parameters": [
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            "kind": "positional_or_keyword",
            "name": "value",
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          }
        ],
        "result_schema": {
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        "signature": "(self, value: Permutation) -> Permutation"
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    "qualified_name": "PermutationGroup.inverse",
    "releases_gil": false,
    "result_schema": {
      "x-python-type": "Permutation"
    },
    "signature": "(self, value: Permutation) -> Permutation",
    "signatures": [
      "(self, value: Permutation) -> Permutation"
    ],
    "summary": "Checked inverse of an element of this group.",
    "summary_source": "public docstring",
    "type_id": "noether/groups/PermutationGroup@1"
  },
  {
    "canonical_name": "groups.PermutationGroup.is_nilpotent",
    "determinism": "deterministic",
    "doc": "Whether the group is nilpotent, computed from its lower central series.",
    "domain": "groups",
    "file": "noether/groups/__init__.pyi",
    "input_schema": {
      "additionalProperties": false,
      "properties": {},
      "type": "object"
    },
    "kind": "property",
    "line": 173,
    "module": "noether.groups",
    "name": "is_nilpotent",
    "overloads": [
      {
        "input_schema": {
          "additionalProperties": false,
          "properties": {},
          "type": "object"
        },
        "parameters": [],
        "result_schema": {
          "type": "boolean"
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    "qualified_name": "PermutationGroup.is_nilpotent",
    "releases_gil": true,
    "result_schema": {
      "type": "boolean"
    },
    "signature": "(self) -> bool",
    "signatures": [
      "(self) -> bool"
    ],
    "summary": "Whether the group is nilpotent, computed from its lower central series.",
    "summary_source": "public docstring",
    "type_id": "noether/groups/PermutationGroup@1"
  },
  {
    "canonical_name": "groups.PermutationGroup.is_perfect",
    "determinism": "deterministic",
    "doc": "Whether the group equals its derived subgroup.",
    "domain": "groups",
    "file": "noether/groups/__init__.pyi",
    "input_schema": {
      "additionalProperties": false,
      "properties": {},
      "type": "object"
    },
    "kind": "property",
    "line": 175,
    "module": "noether.groups",
    "name": "is_perfect",
    "overloads": [
      {
        "input_schema": {
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          "properties": {},
          "type": "object"
        },
        "parameters": [],
        "result_schema": {
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        },
        "signature": "(self) -> bool"
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    ],
    "qualified_name": "PermutationGroup.is_perfect",
    "releases_gil": true,
    "result_schema": {
      "type": "boolean"
    },
    "signature": "(self) -> bool",
    "signatures": [
      "(self) -> bool"
    ],
    "summary": "Whether the group equals its derived subgroup.",
    "summary_source": "public docstring",
    "type_id": "noether/groups/PermutationGroup@1"
  },
  {
    "canonical_name": "groups.PermutationGroup.is_solvable",
    "determinism": "deterministic",
    "doc": "Whether the group is solvable, computed from its derived series.",
    "domain": "groups",
    "file": "noether/groups/__init__.pyi",
    "input_schema": {
      "additionalProperties": false,
      "properties": {},
      "type": "object"
    },
    "kind": "property",
    "line": 171,
    "module": "noether.groups",
    "name": "is_solvable",
    "overloads": [
      {
        "input_schema": {
          "additionalProperties": false,
          "properties": {},
          "type": "object"
        },
        "parameters": [],
        "result_schema": {
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        },
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    "qualified_name": "PermutationGroup.is_solvable",
    "releases_gil": true,
    "result_schema": {
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    },
    "signature": "(self) -> bool",
    "signatures": [
      "(self) -> bool"
    ],
    "summary": "Whether the group is solvable, computed from its derived series.",
    "summary_source": "public docstring",
    "type_id": "noether/groups/PermutationGroup@1"
  },
  {
    "canonical_name": "groups.PermutationGroup.is_transitive",
    "determinism": "deterministic",
    "doc": "Whether this action has one orbit. The empty action is not transitive.",
    "domain": "groups",
    "file": "noether/groups/__init__.pyi",
    "input_schema": {
      "additionalProperties": false,
      "properties": {},
      "type": "object"
    },
    "kind": "property",
    "line": 169,
    "module": "noether.groups",
    "name": "is_transitive",
    "overloads": [
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        "input_schema": {
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          "properties": {},
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        },
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        "signature": "(self) -> bool"
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    "qualified_name": "PermutationGroup.is_transitive",
    "releases_gil": true,
    "result_schema": {
      "type": "boolean"
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    "signature": "(self) -> bool",
    "signatures": [
      "(self) -> bool"
    ],
    "summary": "Whether this action has one orbit.",
    "summary_source": "public docstring",
    "type_id": "noether/groups/PermutationGroup@1"
  },
  {
    "canonical_name": "groups.PermutationGroup.iterate_elements",
    "determinism": "deterministic",
    "doc": "A lazy odometer iterator over every group element. Each iteration\nyields a fresh\n`Permutation`. The total number of yields equals\n`order`.",
    "domain": "groups",
    "file": "noether/groups/__init__.pyi",
    "input_schema": {
      "additionalProperties": false,
      "properties": {},
      "type": "object"
    },
    "kind": "method",
    "line": 217,
    "module": "noether.groups",
    "name": "iterate_elements",
    "overloads": [
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        "input_schema": {
          "additionalProperties": false,
          "properties": {},
          "type": "object"
        },
        "parameters": [],
        "result_schema": {
          "items": {
            "x-python-type": "Permutation"
          },
          "type": "array",
          "x-python-type": "Iterator[Permutation]"
        },
        "signature": "(self) -> Iterator[Permutation]"
      }
    ],
    "qualified_name": "PermutationGroup.iterate_elements",
    "releases_gil": true,
    "result_schema": {
      "items": {
        "x-python-type": "Permutation"
      },
      "type": "array",
      "x-python-type": "Iterator[Permutation]"
    },
    "signature": "(self) -> Iterator[Permutation]",
    "signatures": [
      "(self) -> Iterator[Permutation]"
    ],
    "summary": "A lazy odometer iterator over every group element.",
    "summary_source": "public docstring",
    "type_id": "noether/groups/PermutationGroup@1"
  },
  {
    "canonical_name": "groups.PermutationGroup.modular_cohomology",
    "determinism": "deterministic",
    "doc": "Compute the exact mod-p cohomology ring through `max_degree`.",
    "domain": "groups",
    "file": "noether/groups/__init__.pyi",
    "input_schema": {
      "additionalProperties": false,
      "properties": {
        "max_degree": {
          "type": "integer"
        },
        "prime": {
          "type": "integer"
        }
      },
      "required": [
        "prime",
        "max_degree"
      ],
      "type": "object"
    },
    "kind": "method",
    "line": 181,
    "module": "noether.groups",
    "name": "modular_cohomology",
    "overloads": [
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        "input_schema": {
          "additionalProperties": false,
          "properties": {
            "max_degree": {
              "type": "integer"
            },
            "prime": {
              "type": "integer"
            }
          },
          "required": [
            "prime",
            "max_degree"
          ],
          "type": "object"
        },
        "parameters": [
          {
            "kind": "positional_or_keyword",
            "name": "prime",
            "required": true,
            "schema": {
              "type": "integer"
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          },
          {
            "kind": "positional_or_keyword",
            "name": "max_degree",
            "required": true,
            "schema": {
              "type": "integer"
            }
          }
        ],
        "result_schema": {
          "x-python-type": "FiniteGroupCohomologyTruncation"
        },
        "signature": "(self, prime: int, max_degree: int) -> FiniteGroupCohomologyTruncation"
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    ],
    "qualified_name": "PermutationGroup.modular_cohomology",
    "releases_gil": true,
    "result_schema": {
      "x-python-type": "FiniteGroupCohomologyTruncation"
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    "signature": "(self, prime: int, max_degree: int) -> FiniteGroupCohomologyTruncation",
    "signatures": [
      "(self, prime: int, max_degree: int) -> FiniteGroupCohomologyTruncation"
    ],
    "summary": "Compute the exact mod-p cohomology ring through `max_degree`.",
    "summary_source": "public docstring",
    "type_id": "noether/groups/PermutationGroup@1"
  },
  {
    "canonical_name": "groups.PermutationGroup.multiply",
    "determinism": "deterministic",
    "doc": "Checked multiplication of two elements of this group.",
    "domain": "groups",
    "file": "noether/groups/__init__.pyi",
    "input_schema": {
      "additionalProperties": false,
      "properties": {
        "left": {
          "x-python-type": "Permutation"
        },
        "right": {
          "x-python-type": "Permutation"
        }
      },
      "required": [
        "left",
        "right"
      ],
      "type": "object"
    },
    "kind": "method",
    "line": 179,
    "module": "noether.groups",
    "name": "multiply",
    "overloads": [
      {
        "input_schema": {
          "additionalProperties": false,
          "properties": {
            "left": {
              "x-python-type": "Permutation"
            },
            "right": {
              "x-python-type": "Permutation"
            }
          },
          "required": [
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            "right"
          ],
          "type": "object"
        },
        "parameters": [
          {
            "kind": "positional_or_keyword",
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            "required": true,
            "schema": {
              "x-python-type": "Permutation"
            }
          },
          {
            "kind": "positional_or_keyword",
            "name": "right",
            "required": true,
            "schema": {
              "x-python-type": "Permutation"
            }
          }
        ],
        "result_schema": {
          "x-python-type": "Permutation"
        },
        "signature": "(self, left: Permutation, right: Permutation) -> Permutation"
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    ],
    "qualified_name": "PermutationGroup.multiply",
    "releases_gil": false,
    "result_schema": {
      "x-python-type": "Permutation"
    },
    "signature": "(self, left: Permutation, right: Permutation) -> Permutation",
    "signatures": [
      "(self, left: Permutation, right: Permutation) -> Permutation"
    ],
    "summary": "Checked multiplication of two elements of this group.",
    "summary_source": "public docstring",
    "type_id": "noether/groups/PermutationGroup@1"
  },
  {
    "canonical_name": "groups.PermutationGroup.normalizer",
    "determinism": "deterministic",
    "doc": "The normalizer of `h` inside `self`, `N_self(h) = { g ∈ self :\nhᵍ = h }`, as a `PermGroup`. Raises `ValueError` on a degree\nmismatch.",
    "domain": "groups",
    "file": "noether/groups/__init__.pyi",
    "input_schema": {
      "additionalProperties": false,
      "properties": {
        "subgroup": {
          "x-python-type": "PermutationGroup"
        }
      },
      "required": [
        "subgroup"
      ],
      "type": "object"
    },
    "kind": "method",
    "line": 223,
    "module": "noether.groups",
    "name": "normalizer",
    "overloads": [
      {
        "input_schema": {
          "additionalProperties": false,
          "properties": {
            "subgroup": {
              "x-python-type": "PermutationGroup"
            }
          },
          "required": [
            "subgroup"
          ],
          "type": "object"
        },
        "parameters": [
          {
            "kind": "positional_or_keyword",
            "name": "subgroup",
            "required": true,
            "schema": {
              "x-python-type": "PermutationGroup"
            }
          }
        ],
        "result_schema": {
          "x-python-type": "PermutationGroup"
        },
        "signature": "(self, subgroup: PermutationGroup) -> PermutationGroup"
      }
    ],
    "qualified_name": "PermutationGroup.normalizer",
    "releases_gil": true,
    "result_schema": {
      "x-python-type": "PermutationGroup"
    },
    "signature": "(self, subgroup: PermutationGroup) -> PermutationGroup",
    "signatures": [
      "(self, subgroup: PermutationGroup) -> PermutationGroup"
    ],
    "summary": "The normalizer of `h` inside `self`, `N_self(h) = { g ∈ self : hᵍ = h }`, as a `PermGroup`.",
    "summary_source": "public docstring",
    "type_id": "noether/groups/PermutationGroup@1"
  },
  {
    "canonical_name": "groups.PermutationGroup.normalizer_in_symmetric",
    "determinism": "deterministic",
    "doc": "The normalizer of `self` in the full symmetric group,\n`N_{S_n}(self)`, returned as a `PermGroup`.",
    "domain": "groups",
    "file": "noether/groups/__init__.pyi",
    "input_schema": {
      "additionalProperties": false,
      "properties": {},
      "type": "object"
    },
    "kind": "method",
    "line": 222,
    "module": "noether.groups",
    "name": "normalizer_in_symmetric",
    "overloads": [
      {
        "input_schema": {
          "additionalProperties": false,
          "properties": {},
          "type": "object"
        },
        "parameters": [],
        "result_schema": {
          "x-python-type": "PermutationGroup"
        },
        "signature": "(self) -> PermutationGroup"
      }
    ],
    "qualified_name": "PermutationGroup.normalizer_in_symmetric",
    "releases_gil": true,
    "result_schema": {
      "x-python-type": "PermutationGroup"
    },
    "signature": "(self) -> PermutationGroup",
    "signatures": [
      "(self) -> PermutationGroup"
    ],
    "summary": "The normalizer of `self` in the full symmetric group, `N_{S_n}(self)`, returned as a `PermGroup`.",
    "summary_source": "public docstring",
    "type_id": "noether/groups/PermutationGroup@1"
  },
  {
    "canonical_name": "groups.PermutationGroup.orbit",
    "determinism": "deterministic",
    "doc": "The orbit of `point` under the **full group**, returned as\n`list[int]` in BFS-from-`point` order.\nRaises `ValueError` if `point >= degree`.",
    "domain": "groups",
    "file": "noether/groups/__init__.pyi",
    "input_schema": {
      "additionalProperties": false,
      "properties": {
        "point": {
          "type": "integer"
        }
      },
      "required": [
        "point"
      ],
      "type": "object"
    },
    "kind": "method",
    "line": 199,
    "module": "noether.groups",
    "name": "orbit",
    "overloads": [
      {
        "input_schema": {
          "additionalProperties": false,
          "properties": {
            "point": {
              "type": "integer"
            }
          },
          "required": [
            "point"
          ],
          "type": "object"
        },
        "parameters": [
          {
            "kind": "positional_or_keyword",
            "name": "point",
            "required": true,
            "schema": {
              "type": "integer"
            }
          }
        ],
        "result_schema": {
          "items": {
            "type": "integer"
          },
          "type": "array",
          "x-python-type": "list[int]"
        },
        "signature": "(self, point: int) -> list[int]"
      }
    ],
    "qualified_name": "PermutationGroup.orbit",
    "releases_gil": true,
    "result_schema": {
      "items": {
        "type": "integer"
      },
      "type": "array",
      "x-python-type": "list[int]"
    },
    "signature": "(self, point: int) -> list[int]",
    "signatures": [
      "(self, point: int) -> list[int]"
    ],
    "summary": "The orbit of `point` under the **full group**, returned as `list[int]` in BFS-from-`point` order.",
    "summary_source": "public docstring",
    "type_id": "noether/groups/PermutationGroup@1"
  },
  {
    "canonical_name": "groups.PermutationGroup.orbit_of_subset",
    "determinism": "deterministic",
    "doc": "Exact orbit of one subset under the induced set action.",
    "domain": "groups",
    "file": "noether/groups/__init__.pyi",
    "input_schema": {
      "additionalProperties": false,
      "properties": {
        "points": {
          "items": {
            "type": "integer"
          },
          "type": "array",
          "x-python-type": "Sequence[int]"
        }
      },
      "required": [
        "points"
      ],
      "type": "object"
    },
    "kind": "method",
    "line": 200,
    "module": "noether.groups",
    "name": "orbit_of_subset",
    "overloads": [
      {
        "input_schema": {
          "additionalProperties": false,
          "properties": {
            "points": {
              "items": {
                "type": "integer"
              },
              "type": "array",
              "x-python-type": "Sequence[int]"
            }
          },
          "required": [
            "points"
          ],
          "type": "object"
        },
        "parameters": [
          {
            "kind": "positional_or_keyword",
            "name": "points",
            "required": true,
            "schema": {
              "items": {
                "type": "integer"
              },
              "type": "array",
              "x-python-type": "Sequence[int]"
            }
          }
        ],
        "result_schema": {
          "x-python-type": "SubsetOrbit"
        },
        "signature": "(self, points: Sequence[int]) -> SubsetOrbit"
      }
    ],
    "qualified_name": "PermutationGroup.orbit_of_subset",
    "releases_gil": true,
    "result_schema": {
      "x-python-type": "SubsetOrbit"
    },
    "signature": "(self, points: Sequence[int]) -> SubsetOrbit",
    "signatures": [
      "(self, points: Sequence[int]) -> SubsetOrbit"
    ],
    "summary": "Exact orbit of one subset under the induced set action.",
    "summary_source": "public docstring",
    "type_id": "noether/groups/PermutationGroup@1"
  },
  {
    "canonical_name": "groups.PermutationGroup.orbit_stabilizer",
    "determinism": "deterministic",
    "doc": "The (orbit, stabilizer) pair of `point` under `G`. Returns an\n[`OrbitStabilizer`]\ncarrying the orbit (`list[int]`) and the stabilizer\n(`PermGroup`).",
    "domain": "groups",
    "file": "noether/groups/__init__.pyi",
    "input_schema": {
      "additionalProperties": false,
      "properties": {
        "point": {
          "type": "integer"
        }
      },
      "required": [
        "point"
      ],
      "type": "object"
    },
    "kind": "method",
    "line": 215,
    "module": "noether.groups",
    "name": "orbit_stabilizer",
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          "properties": {
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              "type": "integer"
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            "required": true,
            "schema": {
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          "x-python-type": "OrbitStabilizer"
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        "signature": "(self, point: int) -> OrbitStabilizer"
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    "releases_gil": true,
    "result_schema": {
      "x-python-type": "OrbitStabilizer"
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    "signature": "(self, point: int) -> OrbitStabilizer",
    "signatures": [
      "(self, point: int) -> OrbitStabilizer"
    ],
    "summary": "The (orbit, stabilizer) pair of `point` under `G`.",
    "summary_source": "public docstring",
    "type_id": "noether/groups/PermutationGroup@1"
  },
  {
    "canonical_name": "groups.PermutationGroup.orbits_on_pairs",
    "determinism": "deterministic",
    "doc": "Exact orbit partition of unordered pairs of distinct points.",
    "domain": "groups",
    "file": "noether/groups/__init__.pyi",
    "input_schema": {
      "additionalProperties": false,
      "properties": {},
      "type": "object"
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    "kind": "method",
    "line": 202,
    "module": "noether.groups",
    "name": "orbits_on_pairs",
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          "properties": {},
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        "result_schema": {
          "items": {
            "x-python-type": "SubsetOrbit"
          },
          "type": "array",
          "x-python-type": "list[SubsetOrbit]"
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    },
    "signature": "(self) -> list[SubsetOrbit]",
    "signatures": [
      "(self) -> list[SubsetOrbit]"
    ],
    "summary": "Exact orbit partition of unordered pairs of distinct points.",
    "summary_source": "public docstring",
    "type_id": "noether/groups/PermutationGroup@1"
  },
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    "canonical_name": "groups.PermutationGroup.orbits_on_subsets",
    "determinism": "deterministic",
    "doc": "Exact orbit partition of all subsets with the requested cardinality.",
    "domain": "groups",
    "file": "noether/groups/__init__.pyi",
    "input_schema": {
      "additionalProperties": false,
      "properties": {
        "size": {
          "type": "integer"
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      },
      "required": [
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      "type": "object"
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    "kind": "method",
    "line": 201,
    "module": "noether.groups",
    "name": "orbits_on_subsets",
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    "signature": "(self, size: int) -> list[SubsetOrbit]",
    "signatures": [
      "(self, size: int) -> list[SubsetOrbit]"
    ],
    "summary": "Exact orbit partition of all subsets with the requested cardinality.",
    "summary_source": "public docstring",
    "type_id": "noether/groups/PermutationGroup@1"
  },
  {
    "canonical_name": "groups.PermutationGroup.order",
    "determinism": "deterministic",
    "doc": "`|G|` as a Python `int` — computed once at construction as\nthe product of the transversal lengths. Returned as a string\nthe Python side parses to `int`, since the order can exceed\nany primitive integer (e.g. `|S_512|` is ~1170 digits).",
    "domain": "groups",
    "file": "noether/groups/__init__.pyi",
    "input_schema": {
      "additionalProperties": false,
      "properties": {},
      "type": "object"
    },
    "kind": "property",
    "line": 167,
    "module": "noether.groups",
    "name": "order",
    "overloads": [
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        "result_schema": {
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        },
        "signature": "(self) -> int"
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    "qualified_name": "PermutationGroup.order",
    "releases_gil": false,
    "result_schema": {
      "type": "integer"
    },
    "signature": "(self) -> int",
    "signatures": [
      "(self) -> int"
    ],
    "summary": "`|G|` as a Python `int` — computed once at construction as the product of the transversal lengths.",
    "summary_source": "public docstring",
    "type_id": "noether/groups/PermutationGroup@1"
  },
  {
    "canonical_name": "groups.PermutationGroup.pair_orbits",
    "determinism": "deterministic",
    "doc": "Compact exact double-coset reduction of point pairs.",
    "domain": "groups",
    "file": "noether/groups/__init__.pyi",
    "input_schema": {
      "additionalProperties": false,
      "properties": {
        "left_seed": {
          "type": "integer"
        },
        "right_seed": {
          "type": "integer"
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      },
      "required": [
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        "right_seed"
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      "type": "object"
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    "kind": "method",
    "line": 203,
    "module": "noether.groups",
    "name": "pair_orbits",
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            "kind": "positional_or_keyword",
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    "signatures": [
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    "summary": "Compact exact double-coset reduction of point pairs.",
    "summary_source": "public docstring",
    "type_id": "noether/groups/PermutationGroup@1"
  },
  {
    "canonical_name": "groups.PermutationGroup.pointwise_stabilizer",
    "determinism": "deterministic",
    "doc": "The pointwise stabilizer of `points` in `G`. Returns a new\n`PermGroup` (the\nstabilizer itself, with its own BSGS pre-computed). Raises\n`ValueError` if any `points` entry is `>= degree`.",
    "domain": "groups",
    "file": "noether/groups/__init__.pyi",
    "input_schema": {
      "additionalProperties": false,
      "properties": {
        "points": {
          "items": {
            "type": "integer"
          },
          "type": "array",
          "x-python-type": "Sequence[int]"
        }
      },
      "required": [
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      ],
      "type": "object"
    },
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    "line": 214,
    "module": "noether.groups",
    "name": "pointwise_stabilizer",
    "overloads": [
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        "input_schema": {
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          "properties": {
            "points": {
              "items": {
                "type": "integer"
              },
              "type": "array",
              "x-python-type": "Sequence[int]"
            }
          },
          "required": [
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          ],
          "type": "object"
        },
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              "type": "array",
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        ],
        "result_schema": {
          "x-python-type": "PermutationGroup"
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        "signature": "(self, points: Sequence[int]) -> PermutationGroup"
      }
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    "qualified_name": "PermutationGroup.pointwise_stabilizer",
    "releases_gil": true,
    "result_schema": {
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    "signature": "(self, points: Sequence[int]) -> PermutationGroup",
    "signatures": [
      "(self, points: Sequence[int]) -> PermutationGroup"
    ],
    "summary": "The pointwise stabilizer of `points` in `G`.",
    "summary_source": "public docstring",
    "type_id": "noether/groups/PermutationGroup@1"
  },
  {
    "canonical_name": "groups.PermutationGroup.schreier_generators",
    "determinism": "deterministic",
    "doc": "The Schreier generators of `Stab_G(point)`. Returns\n`list[SchreierGenerator]`,\neach carrying the coset index, generator index, and the\nSchreier-generator permutation itself.",
    "domain": "groups",
    "file": "noether/groups/__init__.pyi",
    "input_schema": {
      "additionalProperties": false,
      "properties": {
        "point": {
          "type": "integer"
        }
      },
      "required": [
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      ],
      "type": "object"
    },
    "kind": "method",
    "line": 212,
    "module": "noether.groups",
    "name": "schreier_generators",
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          "properties": {
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          ],
          "type": "object"
        },
        "parameters": [
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          }
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        "result_schema": {
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          },
          "type": "array",
          "x-python-type": "list[SchreierGenerator]"
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    "qualified_name": "PermutationGroup.schreier_generators",
    "releases_gil": true,
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    "signature": "(self, point: int) -> list[SchreierGenerator]",
    "signatures": [
      "(self, point: int) -> list[SchreierGenerator]"
    ],
    "summary": "The Schreier generators of `Stab_G(point)`.",
    "summary_source": "public docstring",
    "type_id": "noether/groups/PermutationGroup@1"
  },
  {
    "canonical_name": "groups.PermutationGroup.schreier_tree",
    "determinism": "deterministic",
    "doc": "The Schreier tree of `point` under the full group. Returns a\n[`SchreierTree`] (BFS\norbit + predecessor table).",
    "domain": "groups",
    "file": "noether/groups/__init__.pyi",
    "input_schema": {
      "additionalProperties": false,
      "properties": {
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          "type": "integer"
        }
      },
      "required": [
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      "type": "object"
    },
    "kind": "method",
    "line": 211,
    "module": "noether.groups",
    "name": "schreier_tree",
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        "result_schema": {
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    "qualified_name": "PermutationGroup.schreier_tree",
    "releases_gil": true,
    "result_schema": {
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    },
    "signature": "(self, point: int) -> SchreierTree",
    "signatures": [
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    ],
    "summary": "The Schreier tree of `point` under the full group.",
    "summary_source": "public docstring",
    "type_id": "noether/groups/PermutationGroup@1"
  },
  {
    "canonical_name": "groups.PermutationGroup.set_stabilizer",
    "determinism": "deterministic",
    "doc": "The set stabilizer `Stab_self(points)` under the `OnSets`\naction — the subgroup fixing `points` setwise — as a\n`PermGroup`. Duplicate points are canonicalized (sort + dedup);\na point `>= degree` raises `ValueError`.",
    "domain": "groups",
    "file": "noether/groups/__init__.pyi",
    "input_schema": {
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      "properties": {
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        }
      },
      "required": [
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      ],
      "type": "object"
    },
    "kind": "method",
    "line": 224,
    "module": "noether.groups",
    "name": "set_stabilizer",
    "overloads": [
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          "properties": {
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              },
              "type": "array",
              "x-python-type": "Sequence[int]"
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          },
          "required": [
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    "releases_gil": true,
    "result_schema": {
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    },
    "signature": "(self, points: Sequence[int]) -> PermutationGroup",
    "signatures": [
      "(self, points: Sequence[int]) -> PermutationGroup"
    ],
    "summary": "The set stabilizer `Stab_self(points)` under the `OnSets` action — the subgroup fixing `points` setwise — as a `PermGroup`.",
    "summary_source": "public docstring",
    "type_id": "noether/groups/PermutationGroup@1"
  },
  {
    "canonical_name": "groups.PermutationGroup.sift",
    "determinism": "deterministic",
    "doc": "Sift `g` through the stabilizer chain. Returns the sift\nresidue and a `bool` indicating membership (residue is the\nidentity iff `g ∈ G`).",
    "domain": "groups",
    "file": "noether/groups/__init__.pyi",
    "input_schema": {
      "additionalProperties": false,
      "properties": {
        "element": {
          "x-python-type": "Permutation"
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      "type": "object"
    },
    "kind": "method",
    "line": 198,
    "module": "noether.groups",
    "name": "sift",
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          ],
          "type": "object"
        },
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            "required": true,
            "schema": {
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          "minItems": 2,
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    "signature": "(self, element: Permutation) -> tuple[Permutation, bool]",
    "signatures": [
      "(self, element: Permutation) -> tuple[Permutation, bool]"
    ],
    "summary": "Sift `g` through the stabilizer chain.",
    "summary_source": "public docstring",
    "type_id": "noether/groups/PermutationGroup@1"
  },
  {
    "canonical_name": "groups.PermutationGroup.stabilizer",
    "determinism": "deterministic",
    "doc": "The stabilizer of one point.",
    "domain": "groups",
    "file": "noether/groups/__init__.pyi",
    "input_schema": {
      "additionalProperties": false,
      "properties": {
        "point": {
          "type": "integer"
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      },
      "required": [
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      "type": "object"
    },
    "kind": "method",
    "line": 213,
    "module": "noether.groups",
    "name": "stabilizer",
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        ],
        "result_schema": {
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    "qualified_name": "PermutationGroup.stabilizer",
    "releases_gil": true,
    "result_schema": {
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    "signature": "(self, point: int) -> PermutationGroup",
    "signatures": [
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    "summary": "The stabilizer of one point.",
    "summary_source": "public docstring",
    "type_id": "noether/groups/PermutationGroup@1"
  },
  {
    "canonical_name": "groups.PermutationGroup.stabilizer_chain",
    "determinism": "deterministic",
    "doc": "The full stabilizer chain. Returns a [`StabilizerChain`] whose\n`steps` exposes per-level\nbase point, generators, orbit, and transversal.",
    "domain": "groups",
    "file": "noether/groups/__init__.pyi",
    "input_schema": {
      "additionalProperties": false,
      "properties": {},
      "type": "object"
    },
    "kind": "method",
    "line": 216,
    "module": "noether.groups",
    "name": "stabilizer_chain",
    "overloads": [
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          "properties": {},
          "type": "object"
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        "result_schema": {
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    ],
    "qualified_name": "PermutationGroup.stabilizer_chain",
    "releases_gil": true,
    "result_schema": {
      "x-python-type": "StabilizerChain"
    },
    "signature": "(self) -> StabilizerChain",
    "signatures": [
      "(self) -> StabilizerChain"
    ],
    "summary": "The full stabilizer chain.",
    "summary_source": "public docstring",
    "type_id": "noether/groups/PermutationGroup@1"
  },
  {
    "canonical_name": "groups.PermutationGroup.strong_generating_set",
    "determinism": "deterministic",
    "doc": "The strong generating set, returned as `list[Permutation]`\nin the order the generators were added (top of the chain\nfirst).",
    "domain": "groups",
    "file": "noether/groups/__init__.pyi",
    "input_schema": {
      "additionalProperties": false,
      "properties": {},
      "type": "object"
    },
    "kind": "property",
    "line": 165,
    "module": "noether.groups",
    "name": "strong_generating_set",
    "overloads": [
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          "type": "array",
          "x-python-type": "list[Permutation]"
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    "qualified_name": "PermutationGroup.strong_generating_set",
    "releases_gil": false,
    "result_schema": {
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      "x-python-type": "list[Permutation]"
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    "signature": "(self) -> list[Permutation]",
    "signatures": [
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    "canonical_name": "groups.PolynomialGlRepresentation.verify_second_chern_index",
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    "summary": "Compute verify second chern index for this `PolynomialGlRepresentation`.",
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    "canonical_name": "groups.PolynomialGlRepresentation.weyl_dimension",
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    "summary": "Return the weyl dimension recorded by this `PolynomialGlRepresentation`.",
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    "canonical_name": "groups.RandomElement",
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    "canonical_name": "groups.RandomElement.seed",
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    "doc": "The seed the element was generated from.",
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    "canonical_name": "groups.SchreierGenerator",
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    "doc": "One Schreier generator of a point stabilizer.",
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    "summary": "One Schreier generator of a point stabilizer.",
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    "canonical_name": "groups.SchreierGenerator.coset_index",
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    "file": "noether/groups/__init__.pyi",
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    "canonical_name": "groups.SchreierGenerator.perm",
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    "canonical_name": "groups.SchreierTree.contains",
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    "doc": "`True` iff `p` is in the orbit.",
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    "summary": "`True` iff `p` is in the orbit.",
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    "canonical_name": "groups.SchreierTree.generators_used",
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    "canonical_name": "groups.SubsetOrbit.elements",
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    "summary": "Return the basis convention recorded by this `WeightLattice`.",
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    "domain": "groups",
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    "type_id": "noether/groups/WeightLattice@1"
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    "canonical_name": "groups.WeightLattice.from_json",
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    "doc": "",
    "domain": "groups",
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    "summary": "Deserialize and revalidate a `WeightLattice` from typed JSON.",
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    "doc": "",
    "domain": "groups",
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    "module": "noether.groups",
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    "summary": "Compute homomorphism for this `WeightLattice`.",
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    "canonical_name": "groups.WeightLattice.identity_map",
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    "doc": "",
    "domain": "groups",
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    "summary": "Compute identity map for this `WeightLattice`.",
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    "canonical_name": "groups.WeightLattice.rank",
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    "domain": "groups",
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    "summary": "Return the rank recorded by this `WeightLattice`.",
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    "canonical_name": "groups.WeightLattice.standard",
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    "doc": "",
    "domain": "groups",
    "file": "noether/groups/__init__.pyi",
    "input_schema": {
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      "properties": {
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      "required": [
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    "qualified_name": "WeightLattice.standard",
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    "summary": "Construct a `WeightLattice` using standard.",
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    "domain": "groups",
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    "summary": "Serialize this `WeightLattice` to its canonical typed JSON representation.",
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    "canonical_name": "groups.WeightLattice.type_a",
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    "doc": "",
    "domain": "groups",
    "file": "noether/groups/__init__.pyi",
    "input_schema": {
      "additionalProperties": false,
      "properties": {
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      },
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    "summary": "Construct a `WeightLattice` using type a.",
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    "domain": "groups",
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    "summary": "Return the type a degree recorded by this `WeightLattice`.",
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    "summary": "Test mathematical equality with another value.",
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      "properties": {
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    "summary": "Return the basis images recorded by this `WeightLatticeMap`.",
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    "summary": "Deserialize and revalidate a `WeightLatticeMap` from typed JSON.",
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    "summary": "Return the source recorded by this `WeightLatticeMap`.",
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    "qualified_name": "WeightLatticeMap.target",
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    "result_schema": {
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    "summary": "Return the target recorded by this `WeightLatticeMap`.",
    "summary_source": "derived projection vocabulary",
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    "doc": "",
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    "canonical_name": "groups.WeightLatticeMap.to_json",
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          },
          "type": "array",
          "x-python-type": "list[int]"
        },
        "signature": "(self) -> list[int]"
      }
    ],
    "qualified_name": "ChainComplex.degrees",
    "releases_gil": false,
    "result_schema": {
      "items": {
        "type": "integer"
      },
      "type": "array",
      "x-python-type": "list[int]"
    },
    "signature": "(self) -> list[int]",
    "signatures": [
      "(self) -> list[int]"
    ],
    "summary": "Displayed degrees carrying an inferred free module.",
    "summary_source": "public docstring",
    "type_id": "noether/homalg/ChainComplex@1"
  },
  {
    "canonical_name": "homalg.ChainComplex.differential",
    "determinism": "deterministic",
    "doc": "Differential supplied at one displayed degree, if present.",
    "domain": "homalg",
    "file": "noether/homalg/__init__.pyi",
    "input_schema": {
      "additionalProperties": false,
      "properties": {
        "degree": {
          "type": "integer"
        }
      },
      "required": [
        "degree"
      ],
      "type": "object"
    },
    "kind": "method",
    "line": 1469,
    "module": "noether.homalg",
    "name": "differential",
    "overloads": [
      {
        "input_schema": {
          "additionalProperties": false,
          "properties": {
            "degree": {
              "type": "integer"
            }
          },
          "required": [
            "degree"
          ],
          "type": "object"
        },
        "parameters": [
          {
            "kind": "positional_or_keyword",
            "name": "degree",
            "required": true,
            "schema": {
              "type": "integer"
            }
          }
        ],
        "result_schema": {
          "anyOf": [
            {
              "x-python-type": "Matrix[R]"
            },
            {
              "type": "null"
            }
          ],
          "x-python-type": "Matrix[R] | None"
        },
        "signature": "(self, degree: int) -> Matrix[R] | None"
      }
    ],
    "qualified_name": "ChainComplex.differential",
    "releases_gil": false,
    "result_schema": {
      "anyOf": [
        {
          "x-python-type": "Matrix[R]"
        },
        {
          "type": "null"
        }
      ],
      "x-python-type": "Matrix[R] | None"
    },
    "signature": "(self, degree: int) -> Matrix[R] | None",
    "signatures": [
      "(self, degree: int) -> Matrix[R] | None"
    ],
    "summary": "Differential supplied at one displayed degree, if present.",
    "summary_source": "public docstring",
    "type_id": "noether/homalg/ChainComplex@1"
  },
  {
    "canonical_name": "homalg.ChainComplex.differential_layout",
    "determinism": "deterministic",
    "doc": "Retained storage layout of one represented differential.",
    "domain": "homalg",
    "file": "noether/homalg/__init__.pyi",
    "input_schema": {
      "additionalProperties": false,
      "properties": {
        "degree": {
          "type": "integer"
        }
      },
      "required": [
        "degree"
      ],
      "type": "object"
    },
    "kind": "method",
    "line": 1468,
    "module": "noether.homalg",
    "name": "differential_layout",
    "overloads": [
      {
        "input_schema": {
          "additionalProperties": false,
          "properties": {
            "degree": {
              "type": "integer"
            }
          },
          "required": [
            "degree"
          ],
          "type": "object"
        },
        "parameters": [
          {
            "kind": "positional_or_keyword",
            "name": "degree",
            "required": true,
            "schema": {
              "type": "integer"
            }
          }
        ],
        "result_schema": {
          "anyOf": [
            {
              "enum": [
                "dense",
                "sparse"
              ],
              "x-python-type": "Literal['dense', 'sparse']"
            },
            {
              "type": "null"
            }
          ],
          "x-python-type": "Literal['dense', 'sparse'] | None"
        },
        "signature": "(self, degree: int) -> Literal['dense', 'sparse'] | None"
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    ],
    "qualified_name": "ChainComplex.differential_layout",
    "releases_gil": false,
    "result_schema": {
      "anyOf": [
        {
          "enum": [
            "dense",
            "sparse"
          ],
          "x-python-type": "Literal['dense', 'sparse']"
        },
        {
          "type": "null"
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      ],
      "x-python-type": "Literal['dense', 'sparse'] | None"
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    "signature": "(self, degree: int) -> Literal['dense', 'sparse'] | None",
    "signatures": [
      "(self, degree: int) -> Literal['dense', 'sparse'] | None"
    ],
    "summary": "Retained storage layout of one represented differential.",
    "summary_source": "public docstring",
    "type_id": "noether/homalg/ChainComplex@1"
  },
  {
    "canonical_name": "homalg.ChainComplex.from_boundaries",
    "determinism": "deterministic",
    "doc": "Construct from `d_n: C_n -> C_(n-1)` matrices keyed by `n`.",
    "domain": "homalg",
    "evidence": [
      "GradedHomology.verify"
    ],
    "example": "from noether import Q\nfrom noether.homalg import ChainComplex\nfrom noether.linalg import Matrix\ncomplex_ = ChainComplex.from_boundaries(Q, {1: Matrix.from_rows(Q, [[0]])})\nresult = complex_.graded_homology()\nresult.verify()",
    "file": "noether/homalg/__init__.pyi",
    "input_schema": {
      "additionalProperties": false,
      "properties": {
        "boundaries": {
          "additionalProperties": {
            "anyOf": [
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                "x-python-type": "Matrix[_CoefficientContext]"
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                "x-python-type": "SparseMatrix[_CoefficientContext]"
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            "x-python-type": "Matrix[_CoefficientContext] | SparseMatrix[_CoefficientContext]"
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          "x-python-type": "Mapping[int, Matrix[_CoefficientContext] | SparseMatrix[_CoefficientContext]]"
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        "ring": {
          "x-python-type": "_CoefficientContext"
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      },
      "required": [
        "ring",
        "boundaries"
      ],
      "type": "object"
    },
    "kind": "classmethod",
    "line": 1452,
    "module": "noether.homalg",
    "name": "from_boundaries",
    "overloads": [
      {
        "input_schema": {
          "additionalProperties": false,
          "properties": {
            "boundaries": {
              "additionalProperties": {
                "anyOf": [
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                    "x-python-type": "Matrix[_CoefficientContext]"
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                    "x-python-type": "SparseMatrix[_CoefficientContext]"
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                ],
                "x-python-type": "Matrix[_CoefficientContext] | SparseMatrix[_CoefficientContext]"
              },
              "type": "object",
              "x-python-type": "Mapping[int, Matrix[_CoefficientContext] | SparseMatrix[_CoefficientContext]]"
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            "ring": {
              "x-python-type": "_CoefficientContext"
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          "required": [
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        "parameters": [
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            "kind": "positional_or_keyword",
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            "kind": "positional_or_keyword",
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            "required": true,
            "schema": {
              "additionalProperties": {
                "anyOf": [
                  {
                    "x-python-type": "Matrix[_CoefficientContext]"
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                  {
                    "x-python-type": "SparseMatrix[_CoefficientContext]"
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                ],
                "x-python-type": "Matrix[_CoefficientContext] | SparseMatrix[_CoefficientContext]"
              },
              "type": "object",
              "x-python-type": "Mapping[int, Matrix[_CoefficientContext] | SparseMatrix[_CoefficientContext]]"
            }
          }
        ],
        "result_schema": {
          "x-python-type": "ChainComplex[_CoefficientContext]"
        },
        "signature": "(cls, ring: _CoefficientContext, boundaries: Mapping[int, Matrix[_CoefficientContext] | SparseMatrix[_CoefficientContext]]) -> ChainComplex[_CoefficientContext]"
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    ],
    "qualified_name": "ChainComplex.from_boundaries",
    "releases_gil": false,
    "result_schema": {
      "x-python-type": "ChainComplex[_CoefficientContext]"
    },
    "signature": "(cls, ring: _CoefficientContext, boundaries: Mapping[int, Matrix[_CoefficientContext] | SparseMatrix[_CoefficientContext]]) -> ChainComplex[_CoefficientContext]",
    "signatures": [
      "(cls, ring: _CoefficientContext, boundaries: Mapping[int, Matrix[_CoefficientContext] | SparseMatrix[_CoefficientContext]]) -> ChainComplex[_CoefficientContext]"
    ],
    "summary": "Construct from `d_n: C_n -> C_(n-1)` matrices keyed by `n`.",
    "summary_source": "public docstring",
    "supported_rings": [
      "Z",
      "exact fields"
    ],
    "type_id": "noether/homalg/ChainComplex@1"
  },
  {
    "canonical_name": "homalg.ChainComplex.from_coboundaries",
    "determinism": "deterministic",
    "doc": "Construct from `delta^n: C^n -> C^(n+1)` matrices keyed by `n`.",
    "domain": "homalg",
    "file": "noether/homalg/__init__.pyi",
    "input_schema": {
      "additionalProperties": false,
      "properties": {
        "coboundaries": {
          "additionalProperties": {
            "anyOf": [
              {
                "x-python-type": "Matrix[_CoefficientContext]"
              },
              {
                "x-python-type": "SparseMatrix[_CoefficientContext]"
              }
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            "x-python-type": "Matrix[_CoefficientContext] | SparseMatrix[_CoefficientContext]"
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        "ring": {
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        }
      },
      "required": [
        "ring",
        "coboundaries"
      ],
      "type": "object"
    },
    "kind": "classmethod",
    "line": 1458,
    "module": "noether.homalg",
    "name": "from_coboundaries",
    "overloads": [
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        "input_schema": {
          "additionalProperties": false,
          "properties": {
            "coboundaries": {
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                ],
                "x-python-type": "Matrix[_CoefficientContext] | SparseMatrix[_CoefficientContext]"
              },
              "type": "object",
              "x-python-type": "Mapping[int, Matrix[_CoefficientContext] | SparseMatrix[_CoefficientContext]]"
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            "ring": {
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            "kind": "positional_or_keyword",
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              },
              "type": "object",
              "x-python-type": "Mapping[int, Matrix[_CoefficientContext] | SparseMatrix[_CoefficientContext]]"
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        ],
        "result_schema": {
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        "signature": "(cls, ring: _CoefficientContext, coboundaries: Mapping[int, Matrix[_CoefficientContext] | SparseMatrix[_CoefficientContext]]) -> ChainComplex[_CoefficientContext]"
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    "qualified_name": "ChainComplex.from_coboundaries",
    "releases_gil": false,
    "result_schema": {
      "x-python-type": "ChainComplex[_CoefficientContext]"
    },
    "signature": "(cls, ring: _CoefficientContext, coboundaries: Mapping[int, Matrix[_CoefficientContext] | SparseMatrix[_CoefficientContext]]) -> ChainComplex[_CoefficientContext]",
    "signatures": [
      "(cls, ring: _CoefficientContext, coboundaries: Mapping[int, Matrix[_CoefficientContext] | SparseMatrix[_CoefficientContext]]) -> ChainComplex[_CoefficientContext]"
    ],
    "summary": "Construct from `delta^n: C^n -> C^(n+1)` matrices keyed by `n`.",
    "summary_source": "public docstring",
    "type_id": "noether/homalg/ChainComplex@1"
  },
  {
    "canonical_name": "homalg.ChainComplex.graded_cohomology",
    "determinism": "deterministic",
    "doc": "Compute every explicitly represented cohomological degree.",
    "domain": "homalg",
    "file": "noether/homalg/__init__.pyi",
    "input_schema": {
      "additionalProperties": false,
      "properties": {},
      "type": "object"
    },
    "kind": "method",
    "line": 1547,
    "module": "noether.homalg",
    "name": "graded_cohomology",
    "overloads": [
      {
        "input_schema": {
          "additionalProperties": false,
          "properties": {},
          "type": "object"
        },
        "parameters": [],
        "result_schema": {
          "x-python-type": "GradedHomology[R]"
        },
        "signature": "(self) -> GradedHomology[R]"
      }
    ],
    "qualified_name": "ChainComplex.graded_cohomology",
    "releases_gil": true,
    "result_schema": {
      "x-python-type": "GradedHomology[R]"
    },
    "signature": "(self) -> GradedHomology[R]",
    "signatures": [
      "(self) -> GradedHomology[R]"
    ],
    "summary": "Compute every explicitly represented cohomological degree.",
    "summary_source": "public docstring",
    "type_id": "noether/homalg/ChainComplex@1"
  },
  {
    "canonical_name": "homalg.ChainComplex.graded_cohomology_invariants",
    "determinism": "deterministic",
    "doc": "Return lean integral or p-adic cohomology invariants keyed by degree.",
    "domain": "homalg",
    "file": "noether/homalg/__init__.pyi",
    "input_schema": {
      "additionalProperties": false,
      "properties": {},
      "type": "object"
    },
    "kind": "method",
    "line": 1531,
    "module": "noether.homalg",
    "name": "graded_cohomology_invariants",
    "overloads": [
      {
        "input_schema": {
          "additionalProperties": false,
          "properties": {},
          "type": "object"
        },
        "parameters": [],
        "result_schema": {
          "additionalProperties": {
            "x-python-type": "FinitelyGeneratedAbelianGroup"
          },
          "type": "object",
          "x-python-type": "dict[int, FinitelyGeneratedAbelianGroup]"
        },
        "signature": "(self: ChainComplex[IntegerRing]) -> dict[int, FinitelyGeneratedAbelianGroup]"
      },
      {
        "input_schema": {
          "additionalProperties": false,
          "properties": {},
          "type": "object"
        },
        "parameters": [],
        "result_schema": {
          "additionalProperties": {
            "x-python-type": "PadicModuleInvariants"
          },
          "type": "object",
          "x-python-type": "dict[int, PadicModuleInvariants]"
        },
        "signature": "(self: ChainComplex[PadicRing]) -> dict[int, PadicModuleInvariants]"
      }
    ],
    "qualified_name": "ChainComplex.graded_cohomology_invariants",
    "releases_gil": true,
    "result_schema": {
      "additionalProperties": {
        "x-python-type": "FinitelyGeneratedAbelianGroup"
      },
      "type": "object",
      "x-python-type": "dict[int, FinitelyGeneratedAbelianGroup]"
    },
    "signature": "(self: ChainComplex[IntegerRing]) -> dict[int, FinitelyGeneratedAbelianGroup]",
    "signatures": [
      "(self: ChainComplex[IntegerRing]) -> dict[int, FinitelyGeneratedAbelianGroup]",
      "(self: ChainComplex[PadicRing]) -> dict[int, PadicModuleInvariants]"
    ],
    "summary": "Return lean integral or p-adic cohomology invariants keyed by degree.",
    "summary_source": "public docstring",
    "type_id": "noether/homalg/ChainComplex@1"
  },
  {
    "canonical_name": "homalg.ChainComplex.graded_cohomology_invariants_with_integer_lift",
    "determinism": "deterministic",
    "doc": "Certify graded p-adic cohomology using an exact integral cochain lift.",
    "domain": "homalg",
    "file": "noether/homalg/__init__.pyi",
    "input_schema": {
      "additionalProperties": false,
      "properties": {
        "integer_lift": {
          "x-python-type": "ChainComplex[IntegerRing]"
        }
      },
      "required": [
        "integer_lift"
      ],
      "type": "object"
    },
    "kind": "method",
    "line": 1541,
    "module": "noether.homalg",
    "name": "graded_cohomology_invariants_with_integer_lift",
    "overloads": [
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        "input_schema": {
          "additionalProperties": false,
          "properties": {
            "integer_lift": {
              "x-python-type": "ChainComplex[IntegerRing]"
            }
          },
          "required": [
            "integer_lift"
          ],
          "type": "object"
        },
        "parameters": [
          {
            "kind": "positional_or_keyword",
            "name": "integer_lift",
            "required": true,
            "schema": {
              "x-python-type": "ChainComplex[IntegerRing]"
            }
          }
        ],
        "result_schema": {
          "additionalProperties": {
            "x-python-type": "PadicModuleInvariants"
          },
          "type": "object",
          "x-python-type": "dict[int, PadicModuleInvariants]"
        },
        "signature": "(self: ChainComplex[PadicRing], integer_lift: ChainComplex[IntegerRing]) -> dict[int, PadicModuleInvariants]"
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    ],
    "qualified_name": "ChainComplex.graded_cohomology_invariants_with_integer_lift",
    "releases_gil": true,
    "result_schema": {
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        "x-python-type": "PadicModuleInvariants"
      },
      "type": "object",
      "x-python-type": "dict[int, PadicModuleInvariants]"
    },
    "signature": "(self: ChainComplex[PadicRing], integer_lift: ChainComplex[IntegerRing]) -> dict[int, PadicModuleInvariants]",
    "signatures": [
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    ],
    "summary": "Certify graded p-adic cohomology using an exact integral cochain lift.",
    "summary_source": "public docstring",
    "type_id": "noether/homalg/ChainComplex@1"
  },
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    "canonical_name": "homalg.ChainComplex.graded_cohomology_invariants_with_ranks",
    "determinism": "deterministic",
    "doc": "Resolve graded p-adic cohomology with ranks keyed by coboundary degree.",
    "domain": "homalg",
    "file": "noether/homalg/__init__.pyi",
    "input_schema": {
      "additionalProperties": false,
      "properties": {
        "differential_ranks": {
          "additionalProperties": {
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          "type": "object",
          "x-python-type": "Mapping[int, int]"
        }
      },
      "required": [
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      "type": "object"
    },
    "kind": "method",
    "line": 1538,
    "module": "noether.homalg",
    "name": "graded_cohomology_invariants_with_ranks",
    "overloads": [
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        "input_schema": {
          "additionalProperties": false,
          "properties": {
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              "x-python-type": "Mapping[int, int]"
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          },
          "required": [
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        "parameters": [
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            "kind": "positional_or_keyword",
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    "qualified_name": "ChainComplex.graded_cohomology_invariants_with_ranks",
    "releases_gil": true,
    "result_schema": {
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      "type": "object",
      "x-python-type": "dict[int, PadicModuleInvariants]"
    },
    "signature": "(self: ChainComplex[PadicRing], differential_ranks: Mapping[int, int]) -> dict[int, PadicModuleInvariants]",
    "signatures": [
      "(self: ChainComplex[PadicRing], differential_ranks: Mapping[int, int]) -> dict[int, PadicModuleInvariants]"
    ],
    "summary": "Resolve graded p-adic cohomology with ranks keyed by coboundary degree.",
    "summary_source": "public docstring",
    "type_id": "noether/homalg/ChainComplex@1"
  },
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    "canonical_name": "homalg.ChainComplex.graded_homology",
    "determinism": "deterministic",
    "doc": "Compute every explicitly represented homological degree.",
    "domain": "homalg",
    "file": "noether/homalg/__init__.pyi",
    "input_schema": {
      "additionalProperties": false,
      "properties": {},
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    "kind": "method",
    "line": 1546,
    "module": "noether.homalg",
    "name": "graded_homology",
    "overloads": [
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        "input_schema": {
          "additionalProperties": false,
          "properties": {},
          "type": "object"
        },
        "parameters": [],
        "result_schema": {
          "x-python-type": "GradedHomology[R]"
        },
        "signature": "(self) -> GradedHomology[R]"
      }
    ],
    "qualified_name": "ChainComplex.graded_homology",
    "releases_gil": true,
    "result_schema": {
      "x-python-type": "GradedHomology[R]"
    },
    "signature": "(self) -> GradedHomology[R]",
    "signatures": [
      "(self) -> GradedHomology[R]"
    ],
    "summary": "Compute every explicitly represented homological degree.",
    "summary_source": "public docstring",
    "type_id": "noether/homalg/ChainComplex@1"
  },
  {
    "canonical_name": "homalg.ChainComplex.graded_homology_invariants",
    "determinism": "deterministic",
    "doc": "Return lean integral or p-adic homology invariants keyed by degree.",
    "domain": "homalg",
    "file": "noether/homalg/__init__.pyi",
    "input_schema": {
      "additionalProperties": false,
      "properties": {},
      "type": "object"
    },
    "kind": "method",
    "line": 1498,
    "module": "noether.homalg",
    "name": "graded_homology_invariants",
    "overloads": [
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        "input_schema": {
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    "summary": "Return the relation coefficient recorded by this `FiniteCyclicGroupIntegralCohomologyPresentation`.",
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    "summary": "Replay and verify the mathematical evidence carried by this `FiniteCyclicGroupIntegralCohomologyPresentation`.",
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    "summary": "Replayable evidence for Symonds's parameter completion criterion.",
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    "summary": "Return the maximum module generator degree recorded by this `FiniteGroupCohomologySymondsParameterCertificate`.",
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    "domain": "homalg",
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    "summary": "Return the parameter generator indices recorded by this `FiniteGroupCohomologySymondsParameterCertificate`.",
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    "domain": "homalg",
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    "summary": "Exact initial segment of a finite group's mod-p cohomology ring.",
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    "summary": "Compute symmetric power for this `FreeModule`.",
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    "type_id": "noether/homalg/FreeModule@1"
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  {
    "canonical_name": "homalg.FreeModule.tensor",
    "determinism": "deterministic",
    "doc": "",
    "domain": "homalg",
    "file": "noether/homalg/__init__.pyi",
    "input_schema": {
      "additionalProperties": false,
      "properties": {
        "rhs": {
          "x-python-type": "FreeModule[_BasicModuleRing]"
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      },
      "required": [
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      "type": "object"
    },
    "kind": "method",
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    "module": "noether.homalg",
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          "properties": {
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          "required": [
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          "type": "object"
        },
        "parameters": [
          {
            "kind": "positional_or_keyword",
            "name": "rhs",
            "required": true,
            "schema": {
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        ],
        "result_schema": {
          "x-python-type": "FreeModule[_BasicModuleRing]"
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        "signature": "(self, rhs: FreeModule[_BasicModuleRing]) -> FreeModule[_BasicModuleRing]"
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    "qualified_name": "FreeModule.tensor",
    "releases_gil": true,
    "result_schema": {
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    "signature": "(self, rhs: FreeModule[_BasicModuleRing]) -> FreeModule[_BasicModuleRing]",
    "signatures": [
      "(self, rhs: FreeModule[_BasicModuleRing]) -> FreeModule[_BasicModuleRing]"
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    "summary": "Compute tensor for this `FreeModule`.",
    "summary_source": "derived projection vocabulary",
    "type_id": "noether/homalg/FreeModule@1"
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  {
    "canonical_name": "homalg.FreeModule.to_json",
    "determinism": "deterministic",
    "doc": "",
    "domain": "homalg",
    "file": "noether/homalg/__init__.pyi",
    "input_schema": {
      "additionalProperties": false,
      "properties": {},
      "type": "object"
    },
    "kind": "method",
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    "module": "noether.homalg",
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    "summary": "Serialize this `FreeModule` to its canonical typed JSON representation.",
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    "canonical_name": "homalg.FreeModulePower",
    "determinism": "deterministic",
    "doc": "A canonical symmetric or exterior power of a based free module.",
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    "file": "noether/homalg/__init__.pyi",
    "input_schema": {},
    "kind": "class",
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    "module": "noether.homalg",
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    "signatures": [],
    "summary": "A canonical symmetric or exterior power of a based free module.",
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    "type_id": "noether/homalg/FreeModulePower@1"
  },
  {
    "canonical_name": "homalg.FreeModulePower.basis_exponents",
    "determinism": "deterministic",
    "doc": "",
    "domain": "homalg",
    "file": "noether/homalg/__init__.pyi",
    "input_schema": {
      "additionalProperties": false,
      "properties": {},
      "type": "object"
    },
    "kind": "property",
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    "module": "noether.homalg",
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    "qualified_name": "FreeModulePower.basis_exponents",
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    "result_schema": {
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    "signature": "(self) -> list[list[int]]",
    "signatures": [
      "(self) -> list[list[int]]"
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    "summary": "Return the basis exponents recorded by this `FreeModulePower`.",
    "summary_source": "derived projection vocabulary",
    "type_id": "noether/homalg/FreeModulePower@1"
  },
  {
    "canonical_name": "homalg.FreeModulePower.degree",
    "determinism": "deterministic",
    "doc": "",
    "domain": "homalg",
    "file": "noether/homalg/__init__.pyi",
    "input_schema": {
      "additionalProperties": false,
      "properties": {},
      "type": "object"
    },
    "kind": "property",
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    "module": "noether.homalg",
    "name": "degree",
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    "qualified_name": "FreeModulePower.degree",
    "releases_gil": false,
    "result_schema": {
      "type": "integer"
    },
    "signature": "(self) -> int",
    "signatures": [
      "(self) -> int"
    ],
    "summary": "Return the degree recorded by this `FreeModulePower`.",
    "summary_source": "derived projection vocabulary",
    "type_id": "noether/homalg/FreeModulePower@1"
  },
  {
    "canonical_name": "homalg.FreeModulePower.from_json",
    "determinism": "deterministic",
    "doc": "",
    "domain": "homalg",
    "file": "noether/homalg/__init__.pyi",
    "input_schema": {
      "additionalProperties": false,
      "properties": {
        "value": {
          "type": "string"
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      },
      "required": [
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      "type": "object"
    },
    "kind": "classmethod",
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    "module": "noether.homalg",
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          "properties": {
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          },
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          "type": "object"
        },
        "parameters": [
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            "kind": "positional_or_keyword",
            "name": "value",
            "required": true,
            "schema": {
              "type": "string"
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          }
        ],
        "result_schema": {
          "x-python-type": "FreeModulePower[_BasicModuleRing]"
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        "signature": "(cls, value: str) -> FreeModulePower[_BasicModuleRing]"
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    ],
    "qualified_name": "FreeModulePower.from_json",
    "releases_gil": true,
    "result_schema": {
      "x-python-type": "FreeModulePower[_BasicModuleRing]"
    },
    "signature": "(cls, value: str) -> FreeModulePower[_BasicModuleRing]",
    "signatures": [
      "(cls, value: str) -> FreeModulePower[_BasicModuleRing]"
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    "summary": "Deserialize and revalidate a `FreeModulePower` from typed JSON.",
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    "type_id": "noether/homalg/FreeModulePower@1"
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  {
    "canonical_name": "homalg.FreeModulePower.kind",
    "determinism": "deterministic",
    "doc": "",
    "domain": "homalg",
    "file": "noether/homalg/__init__.pyi",
    "input_schema": {
      "additionalProperties": false,
      "properties": {},
      "type": "object"
    },
    "kind": "property",
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    "module": "noether.homalg",
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          "properties": {},
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        "parameters": [],
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        "signature": "(self) -> ModulePowerKind"
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    ],
    "qualified_name": "FreeModulePower.kind",
    "releases_gil": false,
    "result_schema": {
      "x-python-type": "ModulePowerKind"
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    "signature": "(self) -> ModulePowerKind",
    "signatures": [
      "(self) -> ModulePowerKind"
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    "summary": "Return the kind recorded by this `FreeModulePower`.",
    "summary_source": "derived projection vocabulary",
    "type_id": "noether/homalg/FreeModulePower@1"
  },
  {
    "canonical_name": "homalg.FreeModulePower.module",
    "determinism": "deterministic",
    "doc": "",
    "domain": "homalg",
    "file": "noether/homalg/__init__.pyi",
    "input_schema": {
      "additionalProperties": false,
      "properties": {},
      "type": "object"
    },
    "kind": "property",
    "line": 158,
    "module": "noether.homalg",
    "name": "module",
    "overloads": [
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        "input_schema": {
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          "properties": {},
          "type": "object"
        },
        "parameters": [],
        "result_schema": {
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        "signature": "(self) -> FreeModule[_BasicModuleRing]"
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    ],
    "qualified_name": "FreeModulePower.module",
    "releases_gil": true,
    "result_schema": {
      "x-python-type": "FreeModule[_BasicModuleRing]"
    },
    "signature": "(self) -> FreeModule[_BasicModuleRing]",
    "signatures": [
      "(self) -> FreeModule[_BasicModuleRing]"
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    "summary": "Return the module recorded by this `FreeModulePower`.",
    "summary_source": "derived projection vocabulary",
    "type_id": "noether/homalg/FreeModulePower@1"
  },
  {
    "canonical_name": "homalg.FreeModulePower.source",
    "determinism": "deterministic",
    "doc": "",
    "domain": "homalg",
    "file": "noether/homalg/__init__.pyi",
    "input_schema": {
      "additionalProperties": false,
      "properties": {},
      "type": "object"
    },
    "kind": "property",
    "line": 150,
    "module": "noether.homalg",
    "name": "source",
    "overloads": [
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          "properties": {},
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        "parameters": [],
        "result_schema": {
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        "signature": "(self) -> FreeModule[_BasicModuleRing]"
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    ],
    "qualified_name": "FreeModulePower.source",
    "releases_gil": true,
    "result_schema": {
      "x-python-type": "FreeModule[_BasicModuleRing]"
    },
    "signature": "(self) -> FreeModule[_BasicModuleRing]",
    "signatures": [
      "(self) -> FreeModule[_BasicModuleRing]"
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    "summary": "Return the source recorded by this `FreeModulePower`.",
    "summary_source": "derived projection vocabulary",
    "type_id": "noether/homalg/FreeModulePower@1"
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  {
    "canonical_name": "homalg.FreeModulePower.to_json",
    "determinism": "deterministic",
    "doc": "",
    "domain": "homalg",
    "file": "noether/homalg/__init__.pyi",
    "input_schema": {
      "additionalProperties": false,
      "properties": {},
      "type": "object"
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    "kind": "method",
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    "module": "noether.homalg",
    "name": "to_json",
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    "qualified_name": "FreeModulePower.to_json",
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    "result_schema": {
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    "signature": "(self) -> str",
    "signatures": [
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    "summary": "Serialize this `FreeModulePower` to its canonical typed JSON representation.",
    "summary_source": "derived projection vocabulary",
    "type_id": "noether/homalg/FreeModulePower@1"
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    "canonical_name": "homalg.FreeModulePower.verify",
    "determinism": "deterministic",
    "doc": "",
    "domain": "homalg",
    "file": "noether/homalg/__init__.pyi",
    "input_schema": {
      "additionalProperties": false,
      "properties": {},
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    "kind": "method",
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    "module": "noether.homalg",
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    "qualified_name": "FreeModulePower.verify",
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    "result_schema": {
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    "signature": "(self) -> None",
    "signatures": [
      "(self) -> None"
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    "summary": "Replay and verify the mathematical evidence carried by this `FreeModulePower`.",
    "summary_source": "derived projection vocabulary",
    "type_id": "noether/homalg/FreeModulePower@1"
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    "canonical_name": "homalg.FreeResolution",
    "determinism": "deterministic",
    "doc": "A verified finite free resolution over a supported polynomial field.",
    "domain": "homalg",
    "file": "noether/homalg/__init__.pyi",
    "input_schema": {},
    "kind": "class",
    "line": 1165,
    "module": "noether.homalg",
    "name": "FreeResolution",
    "overloads": [],
    "qualified_name": "FreeResolution",
    "releases_gil": false,
    "result_schema": {},
    "signature": "",
    "signatures": [],
    "summary": "A verified finite free resolution over a supported polynomial field.",
    "summary_source": "public docstring",
    "type_id": "noether/homalg/FreeResolution@1"
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  {
    "canonical_name": "homalg.FreeResolution.augmentation",
    "determinism": "deterministic",
    "doc": "",
    "domain": "homalg",
    "file": "noether/homalg/__init__.pyi",
    "input_schema": {
      "additionalProperties": false,
      "properties": {},
      "type": "object"
    },
    "kind": "property",
    "line": 1184,
    "module": "noether.homalg",
    "name": "augmentation",
    "overloads": [
      {
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          "properties": {},
          "type": "object"
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        "parameters": [],
        "result_schema": {
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        "signature": "(self) -> ModuleHomomorphism[_GroebnerModuleRing]"
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    ],
    "qualified_name": "FreeResolution.augmentation",
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    "result_schema": {
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    "signature": "(self) -> ModuleHomomorphism[_GroebnerModuleRing]",
    "signatures": [
      "(self) -> ModuleHomomorphism[_GroebnerModuleRing]"
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    "summary": "Return the augmentation recorded by this `FreeResolution`.",
    "summary_source": "derived projection vocabulary",
    "type_id": "noether/homalg/FreeResolution@1"
  },
  {
    "canonical_name": "homalg.FreeResolution.augmentation_inverse",
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    "doc": "",
    "domain": "homalg",
    "file": "noether/homalg/__init__.pyi",
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      "properties": {},
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    "module": "noether.homalg",
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        "signature": "(self) -> ModuleHomomorphism[_GroebnerModuleRing]"
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    ],
    "qualified_name": "FreeResolution.augmentation_inverse",
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    "result_schema": {
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    "signature": "(self) -> ModuleHomomorphism[_GroebnerModuleRing]",
    "signatures": [
      "(self) -> ModuleHomomorphism[_GroebnerModuleRing]"
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    "summary": "Return the augmentation inverse recorded by this `FreeResolution`.",
    "summary_source": "derived projection vocabulary",
    "type_id": "noether/homalg/FreeResolution@1"
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  {
    "canonical_name": "homalg.FreeResolution.completeness",
    "determinism": "deterministic",
    "doc": "",
    "domain": "homalg",
    "file": "noether/homalg/__init__.pyi",
    "input_schema": {
      "additionalProperties": false,
      "properties": {},
      "type": "object"
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    "kind": "property",
    "line": 1180,
    "module": "noether.homalg",
    "name": "completeness",
    "overloads": [
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        "parameters": [],
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    ],
    "qualified_name": "FreeResolution.completeness",
    "releases_gil": false,
    "result_schema": {
      "x-python-type": "ResolutionCompleteness"
    },
    "signature": "(self) -> ResolutionCompleteness",
    "signatures": [
      "(self) -> ResolutionCompleteness"
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    "summary": "Return the completeness recorded by this `FreeResolution`.",
    "summary_source": "derived projection vocabulary",
    "type_id": "noether/homalg/FreeResolution@1"
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  {
    "canonical_name": "homalg.FreeResolution.differential",
    "determinism": "deterministic",
    "doc": "",
    "domain": "homalg",
    "file": "noether/homalg/__init__.pyi",
    "input_schema": {
      "additionalProperties": false,
      "properties": {
        "degree": {
          "type": "integer"
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      },
      "required": [
        "degree"
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      "type": "object"
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    "kind": "method",
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    "module": "noether.homalg",
    "name": "differential",
    "overloads": [
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        "input_schema": {
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          "properties": {
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          },
          "required": [
            "degree"
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          "type": "object"
        },
        "parameters": [
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            "kind": "positional_or_keyword",
            "name": "degree",
            "required": true,
            "schema": {
              "type": "integer"
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          }
        ],
        "result_schema": {
          "anyOf": [
            {
              "x-python-type": "ModuleHomomorphism[_GroebnerModuleRing]"
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            {
              "type": "null"
            }
          ],
          "x-python-type": "ModuleHomomorphism[_GroebnerModuleRing] | None"
        },
        "signature": "(self, degree: int) -> ModuleHomomorphism[_GroebnerModuleRing] | None"
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    ],
    "qualified_name": "FreeResolution.differential",
    "releases_gil": false,
    "result_schema": {
      "anyOf": [
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          "x-python-type": "ModuleHomomorphism[_GroebnerModuleRing]"
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        {
          "type": "null"
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      ],
      "x-python-type": "ModuleHomomorphism[_GroebnerModuleRing] | None"
    },
    "signature": "(self, degree: int) -> ModuleHomomorphism[_GroebnerModuleRing] | None",
    "signatures": [
      "(self, degree: int) -> ModuleHomomorphism[_GroebnerModuleRing] | None"
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    "summary": "Compute differential for this `FreeResolution`.",
    "summary_source": "derived projection vocabulary",
    "type_id": "noether/homalg/FreeResolution@1"
  },
  {
    "canonical_name": "homalg.FreeResolution.ext",
    "determinism": "deterministic",
    "doc": "",
    "domain": "homalg",
    "file": "noether/homalg/__init__.pyi",
    "input_schema": {
      "additionalProperties": false,
      "properties": {
        "coefficient_module": {
          "x-python-type": "PresentedModule[_GroebnerModuleRing]"
        },
        "degree": {
          "type": "integer"
        },
        "order": {
          "x-python-type": "ModuleMonomialOrder"
        }
      },
      "required": [
        "coefficient_module",
        "degree",
        "order"
      ],
      "type": "object"
    },
    "kind": "method",
    "line": 1202,
    "module": "noether.homalg",
    "name": "ext",
    "overloads": [
      {
        "input_schema": {
          "additionalProperties": false,
          "properties": {
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              "x-python-type": "PresentedModule[_GroebnerModuleRing]"
            },
            "degree": {
              "type": "integer"
            },
            "order": {
              "x-python-type": "ModuleMonomialOrder"
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          },
          "required": [
            "coefficient_module",
            "degree",
            "order"
          ],
          "type": "object"
        },
        "parameters": [
          {
            "kind": "positional_or_keyword",
            "name": "coefficient_module",
            "required": true,
            "schema": {
              "x-python-type": "PresentedModule[_GroebnerModuleRing]"
            }
          },
          {
            "kind": "positional_or_keyword",
            "name": "degree",
            "required": true,
            "schema": {
              "type": "integer"
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          },
          {
            "kind": "positional_or_keyword",
            "name": "order",
            "required": true,
            "schema": {
              "x-python-type": "ModuleMonomialOrder"
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          }
        ],
        "result_schema": {
          "x-python-type": "DerivedModule[_GroebnerModuleRing]"
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        "signature": "(self, coefficient_module: PresentedModule[_GroebnerModuleRing], degree: int, order: ModuleMonomialOrder) -> DerivedModule[_GroebnerModuleRing]"
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    ],
    "qualified_name": "FreeResolution.ext",
    "releases_gil": true,
    "result_schema": {
      "x-python-type": "DerivedModule[_GroebnerModuleRing]"
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    "signature": "(self, coefficient_module: PresentedModule[_GroebnerModuleRing], degree: int, order: ModuleMonomialOrder) -> DerivedModule[_GroebnerModuleRing]",
    "signatures": [
      "(self, coefficient_module: PresentedModule[_GroebnerModuleRing], degree: int, order: ModuleMonomialOrder) -> DerivedModule[_GroebnerModuleRing]"
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    "summary": "Compute ext for this `FreeResolution`.",
    "summary_source": "derived projection vocabulary",
    "type_id": "noether/homalg/FreeResolution@1"
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    "doc": "",
    "domain": "homalg",
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    "domain": "homalg",
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    "domain": "homalg",
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    "summary": "Return the length recorded by this `FreeResolution`.",
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    "canonical_name": "homalg.FreeResolution.minimality",
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    "domain": "homalg",
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    "summary": "Return the minimality recorded by this `FreeResolution`.",
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    "canonical_name": "homalg.FreeResolution.minimalize",
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    "domain": "homalg",
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    "doc": "",
    "domain": "homalg",
    "file": "noether/homalg/__init__.pyi",
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      "additionalProperties": false,
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    "doc": "",
    "domain": "homalg",
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      "additionalProperties": false,
      "properties": {},
      "type": "object"
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    "module": "noether.homalg",
    "name": "to_json",
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    "domain": "homalg",
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    "signatures": [
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    "module": "noether.homalg",
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    "summary": "Canonical exact multigraded Betti numbers.",
    "summary_source": "public docstring",
    "type_id": "noether/homalg/GradedBettiTable@1"
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    "canonical_name": "homalg.GradedBettiTable.completeness",
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    "domain": "homalg",
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    "summary_source": "derived projection vocabulary",
    "type_id": "noether/homalg/GradedBettiTable@1"
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    "canonical_name": "homalg.GradedBettiTable.entries",
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    "domain": "homalg",
    "file": "noether/homalg/__init__.pyi",
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      "properties": {},
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    "canonical_name": "homalg.ModuleGroebnerBasis.contains",
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    "doc": "",
    "domain": "homalg",
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      "properties": {},
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    "domain": "homalg",
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    "domain": "homalg",
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      "properties": {},
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    "domain": "homalg",
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    "doc": "",
    "domain": "homalg",
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      "properties": {},
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    "domain": "homalg",
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    "module": "noether.homalg",
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          "x-python-type": "ModuleElement[_GroebnerModuleRing]"
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        "signature": "(self, value: ModuleElement[_GroebnerModuleRing]) -> ModuleElement[_GroebnerModuleRing]"
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    ],
    "qualified_name": "ModuleGroebnerBasis.normal_form",
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    "result_schema": {
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    "signature": "(self, value: ModuleElement[_GroebnerModuleRing]) -> ModuleElement[_GroebnerModuleRing]",
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    "canonical_name": "homalg.ModuleGroebnerBasis.order",
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    "domain": "homalg",
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    "domain": "homalg",
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    "summary": "Compute lift for this `ModuleReduction`.",
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    "type_id": "noether/homalg/ModuleReduction@1"
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    "canonical_name": "homalg.ModuleReduction.quotient",
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    "domain": "homalg",
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    "summary": "Return the quotient recorded by this `ModuleReduction`.",
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    "type_id": "noether/homalg/ModuleReduction@1"
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    "canonical_name": "homalg.ModuleReduction.remainder",
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    "domain": "homalg",
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    "module": "noether.homalg",
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    "type_id": "noether/homalg/ModuleReduction@1"
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    "domain": "homalg",
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    "module": "noether.homalg",
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    "qualified_name": "ModuleReduction.value",
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    "summary": "Return the value recorded by this `ModuleReduction`.",
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    "canonical_name": "homalg.ModuleReduction.verify",
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    "domain": "homalg",
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    "doc": "One certified p-adic-unit pivot removed from a based complex.",
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    "file": "noether/homalg/__init__.pyi",
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    "summary": "One certified p-adic-unit pivot removed from a based complex.",
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    "canonical_name": "homalg.PadicChainCancellation.column",
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    "domain": "homalg",
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    "canonical_name": "homalg.PadicChainCancellation.degree",
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    "qualified_name": "PadicChainCancellation.degree",
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    "summary": "Return the degree recorded by this `PadicChainCancellation`.",
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    "canonical_name": "homalg.PadicChainCancellation.pivot",
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    "domain": "homalg",
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    "qualified_name": "PadicChainCancellation.pivot",
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    "summary": "Return the pivot recorded by this `PadicChainCancellation`.",
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    "canonical_name": "homalg.PadicChainCancellation.row",
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    "domain": "homalg",
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    "canonical_name": "homalg.PadicChainReduction",
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    "summary": "Replayable algebraic Morse reduction by certified p-adic units.",
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    "canonical_name": "homalg.PadicChainReduction.cancellations",
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    "domain": "homalg",
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    "canonical_name": "homalg.PadicChainReduction.reduced",
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    "domain": "homalg",
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    "canonical_name": "homalg.PadicChainReduction.source",
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    "domain": "homalg",
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    "module": "noether.homalg",
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    "qualified_name": "PadicChainReduction.source",
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    "canonical_name": "homalg.PadicChainReduction.verify",
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    "qualified_name": "PadicChainReduction.verify",
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    "summary": "Replay and verify the mathematical evidence carried by this `PadicChainReduction`.",
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    "canonical_name": "homalg.PadicCyclicSummand",
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      "(self: PresentedModule[PolynomialRing[FiniteField]]) -> ModuleAssociatedPrimeSet[FiniteField]"
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    "summary": "Compute associated primes for this `PresentedModule`.",
    "summary_source": "derived projection vocabulary",
    "type_id": "noether/homalg/PresentedModule@1"
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    "canonical_name": "homalg.PresentedModule.ext",
    "determinism": "deterministic",
    "doc": "",
    "domain": "homalg",
    "file": "noether/homalg/__init__.pyi",
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      },
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      "type": "object"
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    "kind": "method",
    "line": 570,
    "module": "noether.homalg",
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            "order": {
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          },
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          },
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            "schema": {
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          },
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        "signature": "(self: PresentedModule[_GroebnerModuleRing], target: PresentedModule[_GroebnerModuleRing], degree: int, order: ModuleMonomialOrder) -> DerivedModule[_GroebnerModuleRing]"
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    "signature": "(self: PresentedModule[_GroebnerModuleRing], target: PresentedModule[_GroebnerModuleRing], degree: int, order: ModuleMonomialOrder) -> DerivedModule[_GroebnerModuleRing]",
    "signatures": [
      "(self: PresentedModule[_GroebnerModuleRing], target: PresentedModule[_GroebnerModuleRing], degree: int, order: ModuleMonomialOrder) -> DerivedModule[_GroebnerModuleRing]"
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    "summary": "Compute ext for this `PresentedModule`.",
    "summary_source": "derived projection vocabulary",
    "type_id": "noether/homalg/PresentedModule@1"
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    "canonical_name": "homalg.PresentedModule.exterior_power",
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    "doc": "",
    "domain": "homalg",
    "file": "noether/homalg/__init__.pyi",
    "input_schema": {
      "additionalProperties": false,
      "properties": {
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        }
      },
      "required": [
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      "type": "object"
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          "properties": {
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          "required": [
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          "type": "object"
        },
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            "kind": "positional_or_keyword",
            "name": "degree",
            "required": true,
            "schema": {
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          }
        ],
        "result_schema": {
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        "signature": "(self, degree: int) -> PresentedModulePower[_BasicModuleRing]"
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    "qualified_name": "PresentedModule.exterior_power",
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    "signature": "(self, degree: int) -> PresentedModulePower[_BasicModuleRing]",
    "signatures": [
      "(self, degree: int) -> PresentedModulePower[_BasicModuleRing]"
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    "summary": "Compute exterior power for this `PresentedModule`.",
    "summary_source": "derived projection vocabulary",
    "type_id": "noether/homalg/PresentedModule@1"
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    "canonical_name": "homalg.PresentedModule.fitting_ideal",
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    "doc": "",
    "domain": "homalg",
    "file": "noether/homalg/__init__.pyi",
    "input_schema": {
      "additionalProperties": false,
      "properties": {
        "index": {
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      },
      "required": [
        "index"
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      "type": "object"
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    "kind": "method",
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    "module": "noether.homalg",
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          "properties": {
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          "required": [
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          "type": "object"
        },
        "parameters": [
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            "kind": "positional_or_keyword",
            "name": "index",
            "required": true,
            "schema": {
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        ],
        "result_schema": {
          "x-python-type": "Ideal[_FieldModuleRing]"
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        "signature": "(self: PresentedModule[_FieldModuleRing], index: int) -> Ideal[_FieldModuleRing]"
      }
    ],
    "qualified_name": "PresentedModule.fitting_ideal",
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    "result_schema": {
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    "signature": "(self: PresentedModule[_FieldModuleRing], index: int) -> Ideal[_FieldModuleRing]",
    "signatures": [
      "(self: PresentedModule[_FieldModuleRing], index: int) -> Ideal[_FieldModuleRing]"
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    "summary": "Compute fitting ideal for this `PresentedModule`.",
    "summary_source": "derived projection vocabulary",
    "type_id": "noether/homalg/PresentedModule@1"
  },
  {
    "canonical_name": "homalg.PresentedModule.free_resolution",
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    "doc": "",
    "domain": "homalg",
    "file": "noether/homalg/__init__.pyi",
    "input_schema": {
      "additionalProperties": false,
      "properties": {
        "order": {
          "x-python-type": "ModuleMonomialOrder"
        }
      },
      "required": [
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      "type": "object"
    },
    "kind": "method",
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    "module": "noether.homalg",
    "name": "free_resolution",
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          "properties": {
            "order": {
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          "type": "object"
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        "parameters": [
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            "kind": "positional_or_keyword",
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            "schema": {
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          }
        ],
        "result_schema": {
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        "signature": "(self: PresentedModule[_FieldModuleRing], order: ModuleMonomialOrder) -> FreeResolution[_FieldModuleRing]"
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    "qualified_name": "PresentedModule.free_resolution",
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    "result_schema": {
      "x-python-type": "FreeResolution[_FieldModuleRing]"
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    "signature": "(self: PresentedModule[_FieldModuleRing], order: ModuleMonomialOrder) -> FreeResolution[_FieldModuleRing]",
    "signatures": [
      "(self: PresentedModule[_FieldModuleRing], order: ModuleMonomialOrder) -> FreeResolution[_FieldModuleRing]"
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    "summary": "Compute free resolution for this `PresentedModule`.",
    "summary_source": "derived projection vocabulary",
    "type_id": "noether/homalg/PresentedModule@1"
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    "canonical_name": "homalg.PresentedModule.free_resolution_through",
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    "doc": "",
    "domain": "homalg",
    "file": "noether/homalg/__init__.pyi",
    "input_schema": {
      "additionalProperties": false,
      "properties": {
        "degree": {
          "type": "integer"
        },
        "order": {
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      },
      "required": [
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      "type": "object"
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    "kind": "method",
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    "module": "noether.homalg",
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          },
          "required": [
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          "type": "object"
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        "parameters": [
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            "kind": "positional_or_keyword",
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            "schema": {
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          },
          {
            "kind": "positional_or_keyword",
            "name": "degree",
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            "schema": {
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        "result_schema": {
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        "signature": "(self: PresentedModule[_GroebnerModuleRing], order: ModuleMonomialOrder, degree: int) -> FreeResolution[_GroebnerModuleRing]"
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    ],
    "qualified_name": "PresentedModule.free_resolution_through",
    "releases_gil": true,
    "result_schema": {
      "x-python-type": "FreeResolution[_GroebnerModuleRing]"
    },
    "signature": "(self: PresentedModule[_GroebnerModuleRing], order: ModuleMonomialOrder, degree: int) -> FreeResolution[_GroebnerModuleRing]",
    "signatures": [
      "(self: PresentedModule[_GroebnerModuleRing], order: ModuleMonomialOrder, degree: int) -> FreeResolution[_GroebnerModuleRing]"
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    "summary": "Compute free resolution through for this `PresentedModule`.",
    "summary_source": "derived projection vocabulary",
    "type_id": "noether/homalg/PresentedModule@1"
  },
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    "canonical_name": "homalg.PresentedModule.from_json",
    "determinism": "deterministic",
    "doc": "",
    "domain": "homalg",
    "file": "noether/homalg/__init__.pyi",
    "input_schema": {
      "additionalProperties": false,
      "properties": {
        "value": {
          "type": "string"
        }
      },
      "required": [
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      "type": "object"
    },
    "kind": "classmethod",
    "line": 600,
    "module": "noether.homalg",
    "name": "from_json",
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          "properties": {
            "value": {
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          "required": [
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          ],
          "type": "object"
        },
        "parameters": [
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            "kind": "positional_or_keyword",
            "name": "value",
            "required": true,
            "schema": {
              "type": "string"
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          }
        ],
        "result_schema": {
          "x-python-type": "PresentedModule[_BasicModuleRing]"
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        "signature": "(cls, value: str) -> PresentedModule[_BasicModuleRing]"
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    ],
    "qualified_name": "PresentedModule.from_json",
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    "result_schema": {
      "x-python-type": "PresentedModule[_BasicModuleRing]"
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    "signature": "(cls, value: str) -> PresentedModule[_BasicModuleRing]",
    "signatures": [
      "(cls, value: str) -> PresentedModule[_BasicModuleRing]"
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    "summary": "Deserialize and revalidate a `PresentedModule` from typed JSON.",
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    "canonical_name": "homalg.PresentedModule.generator_count",
    "determinism": "deterministic",
    "doc": "",
    "domain": "homalg",
    "file": "noether/homalg/__init__.pyi",
    "input_schema": {
      "additionalProperties": false,
      "properties": {},
      "type": "object"
    },
    "kind": "property",
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    "module": "noether.homalg",
    "name": "generator_count",
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        "input_schema": {
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    ],
    "qualified_name": "PresentedModule.generator_count",
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    "result_schema": {
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    "signatures": [
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    "summary": "Return the generator count recorded by this `PresentedModule`.",
    "summary_source": "derived projection vocabulary",
    "type_id": "noether/homalg/PresentedModule@1"
  },
  {
    "canonical_name": "homalg.PresentedModule.generator_module",
    "determinism": "deterministic",
    "doc": "",
    "domain": "homalg",
    "file": "noether/homalg/__init__.pyi",
    "input_schema": {
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      "properties": {},
      "type": "object"
    },
    "kind": "property",
    "line": 512,
    "module": "noether.homalg",
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        "input_schema": {
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          "properties": {},
          "type": "object"
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        "signature": "(self) -> FreeModule[_BasicModuleRing]"
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    ],
    "qualified_name": "PresentedModule.generator_module",
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    "result_schema": {
      "x-python-type": "FreeModule[_BasicModuleRing]"
    },
    "signature": "(self) -> FreeModule[_BasicModuleRing]",
    "signatures": [
      "(self) -> FreeModule[_BasicModuleRing]"
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    "summary": "Return the generator module recorded by this `PresentedModule`.",
    "summary_source": "derived projection vocabulary",
    "type_id": "noether/homalg/PresentedModule@1"
  },
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    "canonical_name": "homalg.PresentedModule.graded_betti_table",
    "determinism": "deterministic",
    "doc": "",
    "domain": "homalg",
    "file": "noether/homalg/__init__.pyi",
    "input_schema": {
      "additionalProperties": false,
      "properties": {
        "order": {
          "x-python-type": "ModuleMonomialOrder"
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      },
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      "type": "object"
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    "module": "noether.homalg",
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        "input_schema": {
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          "type": "object"
        },
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            "schema": {
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          }
        ],
        "result_schema": {
          "x-python-type": "GradedBettiTable"
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        "signature": "(self: PresentedModule[_FieldModuleRing], order: ModuleMonomialOrder) -> GradedBettiTable"
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    ],
    "qualified_name": "PresentedModule.graded_betti_table",
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    "signatures": [
      "(self: PresentedModule[_FieldModuleRing], order: ModuleMonomialOrder) -> GradedBettiTable"
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    "summary": "Compute graded betti table for this `PresentedModule`.",
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    "type_id": "noether/homalg/PresentedModule@1"
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    "canonical_name": "homalg.PresentedModule.graded_invariants",
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    "doc": "",
    "domain": "homalg",
    "file": "noether/homalg/__init__.pyi",
    "input_schema": {
      "additionalProperties": false,
      "properties": {
        "order": {
          "x-python-type": "ModuleMonomialOrder"
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      },
      "required": [
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      "type": "object"
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    "kind": "method",
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    "module": "noether.homalg",
    "name": "graded_invariants",
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          "required": [
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          "type": "object"
        },
        "parameters": [
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            "kind": "positional_or_keyword",
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            "required": true,
            "schema": {
              "x-python-type": "ModuleMonomialOrder"
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          }
        ],
        "result_schema": {
          "x-python-type": "GradedModuleInvariants[_FieldModuleRing]"
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        "signature": "(self: PresentedModule[_FieldModuleRing], order: ModuleMonomialOrder) -> GradedModuleInvariants[_FieldModuleRing]"
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    "qualified_name": "PresentedModule.graded_invariants",
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    "result_schema": {
      "x-python-type": "GradedModuleInvariants[_FieldModuleRing]"
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    "signature": "(self: PresentedModule[_FieldModuleRing], order: ModuleMonomialOrder) -> GradedModuleInvariants[_FieldModuleRing]",
    "signatures": [
      "(self: PresentedModule[_FieldModuleRing], order: ModuleMonomialOrder) -> GradedModuleInvariants[_FieldModuleRing]"
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    "summary": "Compute graded invariants for this `PresentedModule`.",
    "summary_source": "derived projection vocabulary",
    "type_id": "noether/homalg/PresentedModule@1"
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  {
    "canonical_name": "homalg.PresentedModule.hom",
    "determinism": "deterministic",
    "doc": "",
    "domain": "homalg",
    "file": "noether/homalg/__init__.pyi",
    "input_schema": {
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      "properties": {
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      "required": [
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        "order"
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      "type": "object"
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    "kind": "method",
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    "module": "noether.homalg",
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              "x-python-type": "PresentedModule[_GroebnerModuleRing]"
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          "required": [
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        "parameters": [
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            "kind": "positional_or_keyword",
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        "result_schema": {
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        "signature": "(self: PresentedModule[_GroebnerModuleRing], target: PresentedModule[_GroebnerModuleRing], order: ModuleMonomialOrder) -> PresentedModule[_GroebnerModuleRing]"
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    ],
    "qualified_name": "PresentedModule.hom",
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    "result_schema": {
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    "signature": "(self: PresentedModule[_GroebnerModuleRing], target: PresentedModule[_GroebnerModuleRing], order: ModuleMonomialOrder) -> PresentedModule[_GroebnerModuleRing]",
    "signatures": [
      "(self: PresentedModule[_GroebnerModuleRing], target: PresentedModule[_GroebnerModuleRing], order: ModuleMonomialOrder) -> PresentedModule[_GroebnerModuleRing]"
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    "summary": "Compute hom for this `PresentedModule`.",
    "summary_source": "derived projection vocabulary",
    "type_id": "noether/homalg/PresentedModule@1"
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    "canonical_name": "homalg.PresentedModule.hom_inclusion",
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    "doc": "",
    "domain": "homalg",
    "file": "noether/homalg/__init__.pyi",
    "input_schema": {
      "additionalProperties": false,
      "properties": {
        "order": {
          "x-python-type": "ModuleMonomialOrder"
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        "target": {
          "x-python-type": "PresentedModule[_GroebnerModuleRing]"
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      },
      "required": [
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        "order"
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      "type": "object"
    },
    "kind": "method",
    "line": 559,
    "module": "noether.homalg",
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              "x-python-type": "PresentedModule[_GroebnerModuleRing]"
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          },
          "required": [
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          "type": "object"
        },
        "parameters": [
          {
            "kind": "positional_or_keyword",
            "name": "target",
            "required": true,
            "schema": {
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          },
          {
            "kind": "positional_or_keyword",
            "name": "order",
            "required": true,
            "schema": {
              "x-python-type": "ModuleMonomialOrder"
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          }
        ],
        "result_schema": {
          "x-python-type": "PresentedModuleHomomorphism[_GroebnerModuleRing]"
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        "signature": "(self: PresentedModule[_GroebnerModuleRing], target: PresentedModule[_GroebnerModuleRing], order: ModuleMonomialOrder) -> PresentedModuleHomomorphism[_GroebnerModuleRing]"
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    ],
    "qualified_name": "PresentedModule.hom_inclusion",
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    "result_schema": {
      "x-python-type": "PresentedModuleHomomorphism[_GroebnerModuleRing]"
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    "signature": "(self: PresentedModule[_GroebnerModuleRing], target: PresentedModule[_GroebnerModuleRing], order: ModuleMonomialOrder) -> PresentedModuleHomomorphism[_GroebnerModuleRing]",
    "signatures": [
      "(self: PresentedModule[_GroebnerModuleRing], target: PresentedModule[_GroebnerModuleRing], order: ModuleMonomialOrder) -> PresentedModuleHomomorphism[_GroebnerModuleRing]"
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    "summary": "Compute hom inclusion for this `PresentedModule`.",
    "summary_source": "derived projection vocabulary",
    "type_id": "noether/homalg/PresentedModule@1"
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    "summary": "Exact completeness or homological truncation of a free resolution.",
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    "domain": "homalg",
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    "summary": "Exact minimality claim of a graded free resolution.",
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    "domain": "homalg",
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    "summary": "Compute contains for this `StrongModuleGroebnerBasis`.",
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    "qualified_name": "StrongModuleGroebnerBasis.from_json",
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      "x-python-type": "StrongModuleGroebnerBasis"
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    "signatures": [
      "(cls, value: str) -> StrongModuleGroebnerBasis"
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    "summary": "Deserialize and revalidate a `StrongModuleGroebnerBasis` from typed JSON.",
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    "canonical_name": "homalg.StrongModuleGroebnerBasis.initial_submodule",
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    "domain": "homalg",
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    "canonical_name": "homalg.StrongModuleGroebnerBasis.input",
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    "canonical_name": "homalg.StrongModuleGroebnerBasis.is_complete",
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    "qualified_name": "StrongModuleGroebnerBasis.is_complete",
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    "canonical_name": "homalg.StrongModuleGroebnerBasis.leading_terms",
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    "doc": "",
    "domain": "homalg",
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      "properties": {},
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    "module": "noether.homalg",
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          "items": {
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    "summary": "Return the leading terms recorded by this `StrongModuleGroebnerBasis`.",
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    "canonical_name": "homalg.StrongModuleGroebnerBasis.lift",
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    "qualified_name": "StrongModuleGroebnerBasis.normal_form",
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    "summary": "Compute normal form for this `StrongModuleGroebnerBasis`.",
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    "canonical_name": "homalg.StrongModuleGroebnerBasis.order",
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    "domain": "homalg",
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    "module": "noether.homalg",
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    "qualified_name": "StrongModuleGroebnerBasis.order",
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    "summary": "Return the order recorded by this `StrongModuleGroebnerBasis`.",
    "summary_source": "derived projection vocabulary",
    "type_id": "noether/homalg/StrongModuleGroebnerBasis@1"
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    "canonical_name": "homalg.StrongModuleGroebnerBasis.reduce",
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    "module": "noether.homalg",
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        "result_schema": {
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    "summary": "Compute reduce for this `StrongModuleGroebnerBasis`.",
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    "canonical_name": "homalg.StrongModuleGroebnerBasis.syzygies",
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    "domain": "homalg",
    "file": "noether/homalg/__init__.pyi",
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      "properties": {},
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    "module": "noether.homalg",
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    "qualified_name": "StrongModuleGroebnerBasis.syzygies",
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    "signature": "(self) -> Submodule[_IntegralModuleRing]",
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    "summary": "Compute syzygies for this `StrongModuleGroebnerBasis`.",
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    "type_id": "noether/homalg/StrongModuleGroebnerBasis@1"
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  {
    "canonical_name": "homalg.StrongModuleGroebnerBasis.to_json",
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    "domain": "homalg",
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      "properties": {},
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    "kind": "method",
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    "module": "noether.homalg",
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    ],
    "qualified_name": "StrongModuleGroebnerBasis.to_json",
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    "summary_source": "derived projection vocabulary",
    "type_id": "noether/homalg/StrongModuleGroebnerBasis@1"
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    "canonical_name": "homalg.StrongModuleGroebnerBasis.transformation",
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          "x-python-type": "ModuleHomomorphism[_IntegralModuleRing]"
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    "qualified_name": "StrongModuleGroebnerBasis.transformation",
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    "signature": "(self) -> ModuleHomomorphism[_IntegralModuleRing]",
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    "summary": "Return the transformation recorded by this `StrongModuleGroebnerBasis`.",
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    "type_id": "noether/homalg/StrongModuleGroebnerBasis@1"
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    "canonical_name": "homalg.StrongModuleGroebnerBasis.verify",
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    "summary": "Replay and verify the mathematical evidence carried by this `StrongModuleGroebnerBasis`.",
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    "type_id": "noether/homalg/StrongModuleGroebnerBasis@1"
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    "canonical_name": "homalg.StrongModuleReduction",
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    "file": "noether/homalg/__init__.pyi",
    "input_schema": {},
    "kind": "class",
    "line": 1070,
    "module": "noether.homalg",
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    "qualified_name": "StrongModuleReduction",
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    "summary": "Exact quotient, remainder, and lift evidence over an integral strong module basis.",
    "summary_source": "public docstring",
    "type_id": "noether/homalg/StrongModuleReduction@1"
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    "canonical_name": "homalg.StrongModuleReduction.basis",
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    "input_schema": {
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      "properties": {},
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    "kind": "property",
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    "module": "noether.homalg",
    "name": "basis",
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        "input_schema": {
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    "qualified_name": "StrongModuleReduction.basis",
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    "result_schema": {
      "x-python-type": "StrongModuleGroebnerBasis"
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    "signature": "(self) -> StrongModuleGroebnerBasis",
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    "summary": "Return the basis recorded by this `StrongModuleReduction`.",
    "summary_source": "derived projection vocabulary",
    "type_id": "noether/homalg/StrongModuleReduction@1"
  },
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    "canonical_name": "homalg.StrongModuleReduction.from_json",
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    "doc": "",
    "domain": "homalg",
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    "input_schema": {
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    "module": "noether.homalg",
    "name": "from_json",
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    "qualified_name": "StrongModuleReduction.from_json",
    "releases_gil": false,
    "result_schema": {
      "x-python-type": "StrongModuleReduction"
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    "signature": "(cls, value: str) -> StrongModuleReduction",
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      "(cls, value: str) -> StrongModuleReduction"
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    "summary": "Deserialize and revalidate a `StrongModuleReduction` from typed JSON.",
    "summary_source": "derived projection vocabulary",
    "type_id": "noether/homalg/StrongModuleReduction@1"
  },
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    "canonical_name": "homalg.StrongModuleReduction.is_member",
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    "domain": "homalg",
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      "properties": {},
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      "(self: Matrix[_FieldContext]) -> SolutionSpace[_FieldContext]"
    ],
    "summary": "Column image as a checked linear subspace.",
    "summary_source": "public docstring",
    "type_id": "noether/linalg/Matrix@1"
  },
  {
    "canonical_name": "linalg.Matrix.input_dimension",
    "determinism": "deterministic",
    "doc": "",
    "domain": "linalg",
    "file": "noether/linalg/__init__.pyi",
    "input_schema": {
      "additionalProperties": false,
      "properties": {},
      "type": "object"
    },
    "kind": "property",
    "line": 1072,
    "module": "noether.linalg",
    "name": "input_dimension",
    "overloads": [
      {
        "input_schema": {
          "additionalProperties": false,
          "properties": {},
          "type": "object"
        },
        "parameters": [],
        "result_schema": {
          "type": "integer"
        },
        "signature": "(self) -> int"
      }
    ],
    "qualified_name": "Matrix.input_dimension",
    "releases_gil": false,
    "result_schema": {
      "type": "integer"
    },
    "signature": "(self) -> int",
    "signatures": [
      "(self) -> int"
    ],
    "summary": "Return the input dimension recorded by this `Matrix`.",
    "summary_source": "derived projection vocabulary",
    "type_id": "noether/linalg/Matrix@1"
  },
  {
    "canonical_name": "linalg.Matrix.kernel",
    "determinism": "deterministic",
    "doc": "Complete right kernel as a checked linear subspace.",
    "domain": "linalg",
    "file": "noether/linalg/__init__.pyi",
    "input_schema": {
      "additionalProperties": false,
      "properties": {},
      "type": "object"
    },
    "kind": "method",
    "line": 1211,
    "module": "noether.linalg",
    "name": "kernel",
    "overloads": [
      {
        "input_schema": {
          "additionalProperties": false,
          "properties": {},
          "type": "object"
        },
        "parameters": [],
        "result_schema": {
          "x-python-type": "SolutionSpace[_FieldContext]"
        },
        "signature": "(self: Matrix[_FieldContext]) -> SolutionSpace[_FieldContext]"
      }
    ],
    "qualified_name": "Matrix.kernel",
    "releases_gil": true,
    "result_schema": {
      "x-python-type": "SolutionSpace[_FieldContext]"
    },
    "signature": "(self: Matrix[_FieldContext]) -> SolutionSpace[_FieldContext]",
    "signatures": [
      "(self: Matrix[_FieldContext]) -> SolutionSpace[_FieldContext]"
    ],
    "summary": "Complete right kernel as a checked linear subspace.",
    "summary_source": "public docstring",
    "type_id": "noether/linalg/Matrix@1"
  },
  {
    "canonical_name": "linalg.Matrix.matmul",
    "determinism": "deterministic",
    "doc": "",
    "domain": "linalg",
    "file": "noether/linalg/__init__.pyi",
    "input_schema": {
      "additionalProperties": false,
      "properties": {
        "other": {
          "x-python-type": "Matrix[R]"
        }
      },
      "required": [
        "other"
      ],
      "type": "object"
    },
    "kind": "method",
    "line": 1194,
    "module": "noether.linalg",
    "name": "matmul",
    "overloads": [
      {
        "input_schema": {
          "additionalProperties": false,
          "properties": {
            "other": {
              "x-python-type": "Matrix[R]"
            }
          },
          "required": [
            "other"
          ],
          "type": "object"
        },
        "parameters": [
          {
            "kind": "positional_or_keyword",
            "name": "other",
            "required": true,
            "schema": {
              "x-python-type": "Matrix[R]"
            }
          }
        ],
        "result_schema": {
          "x-python-type": "Matrix[R]"
        },
        "signature": "(self, other: Matrix[R]) -> Matrix[R]"
      }
    ],
    "qualified_name": "Matrix.matmul",
    "releases_gil": true,
    "result_schema": {
      "x-python-type": "Matrix[R]"
    },
    "signature": "(self, other: Matrix[R]) -> Matrix[R]",
    "signatures": [
      "(self, other: Matrix[R]) -> Matrix[R]"
    ],
    "summary": "Compute matmul for this `Matrix`.",
    "summary_source": "derived projection vocabulary",
    "type_id": "noether/linalg/Matrix@1"
  },
  {
    "canonical_name": "linalg.Matrix.output_dimension",
    "determinism": "deterministic",
    "doc": "",
    "domain": "linalg",
    "file": "noether/linalg/__init__.pyi",
    "input_schema": {
      "additionalProperties": false,
      "properties": {},
      "type": "object"
    },
    "kind": "property",
    "line": 1074,
    "module": "noether.linalg",
    "name": "output_dimension",
    "overloads": [
      {
        "input_schema": {
          "additionalProperties": false,
          "properties": {},
          "type": "object"
        },
        "parameters": [],
        "result_schema": {
          "type": "integer"
        },
        "signature": "(self) -> int"
      }
    ],
    "qualified_name": "Matrix.output_dimension",
    "releases_gil": false,
    "result_schema": {
      "type": "integer"
    },
    "signature": "(self) -> int",
    "signatures": [
      "(self) -> int"
    ],
    "summary": "Return the output dimension recorded by this `Matrix`.",
    "summary_source": "derived projection vocabulary",
    "type_id": "noether/linalg/Matrix@1"
  },
  {
    "canonical_name": "linalg.Matrix.padic_smith_invariants",
    "determinism": "deterministic",
    "doc": "Smith invariant valuations visible below a p-adic ring's precision cap.",
    "domain": "linalg",
    "file": "noether/linalg/__init__.pyi",
    "input_schema": {
      "additionalProperties": false,
      "properties": {},
      "type": "object"
    },
    "kind": "method",
    "line": 1268,
    "module": "noether.linalg",
    "name": "padic_smith_invariants",
    "overloads": [
      {
        "input_schema": {
          "additionalProperties": false,
          "properties": {},
          "type": "object"
        },
        "parameters": [],
        "result_schema": {
          "x-python-type": "PadicSmithInvariants"
        },
        "signature": "(self: Matrix[PadicRing]) -> PadicSmithInvariants"
      }
    ],
    "qualified_name": "Matrix.padic_smith_invariants",
    "releases_gil": true,
    "result_schema": {
      "x-python-type": "PadicSmithInvariants"
    },
    "signature": "(self: Matrix[PadicRing]) -> PadicSmithInvariants",
    "signatures": [
      "(self: Matrix[PadicRing]) -> PadicSmithInvariants"
    ],
    "summary": "Smith invariant valuations visible below a p-adic ring's precision cap.",
    "summary_source": "public docstring",
    "type_id": "noether/linalg/Matrix@1"
  },
  {
    "canonical_name": "linalg.Matrix.pluq",
    "determinism": "deterministic",
    "doc": "Rank-revealing exact PLUQ factorization over a field context.",
    "domain": "linalg",
    "file": "noether/linalg/__init__.pyi",
    "input_schema": {
      "additionalProperties": false,
      "properties": {},
      "type": "object"
    },
    "kind": "method",
    "line": 1199,
    "module": "noether.linalg",
    "name": "pluq",
    "overloads": [
      {
        "input_schema": {
          "additionalProperties": false,
          "properties": {},
          "type": "object"
        },
        "parameters": [],
        "result_schema": {
          "x-python-type": "PLUQFactorization[_FieldContext]"
        },
        "signature": "(self: Matrix[_FieldContext]) -> PLUQFactorization[_FieldContext]"
      }
    ],
    "qualified_name": "Matrix.pluq",
    "releases_gil": true,
    "result_schema": {
      "x-python-type": "PLUQFactorization[_FieldContext]"
    },
    "signature": "(self: Matrix[_FieldContext]) -> PLUQFactorization[_FieldContext]",
    "signatures": [
      "(self: Matrix[_FieldContext]) -> PLUQFactorization[_FieldContext]"
    ],
    "summary": "Rank-revealing exact PLUQ factorization over a field context.",
    "summary_source": "public docstring",
    "type_id": "noether/linalg/Matrix@1"
  },
  {
    "canonical_name": "linalg.Matrix.rank",
    "determinism": "deterministic",
    "doc": "Rank over a field context.",
    "domain": "linalg",
    "file": "noether/linalg/__init__.pyi",
    "input_schema": {
      "additionalProperties": false,
      "properties": {},
      "type": "object"
    },
    "kind": "method",
    "line": 1197,
    "module": "noether.linalg",
    "name": "rank",
    "overloads": [
      {
        "input_schema": {
          "additionalProperties": false,
          "properties": {},
          "type": "object"
        },
        "parameters": [],
        "result_schema": {
          "type": "integer"
        },
        "signature": "(self: Matrix[_FieldContext]) -> int"
      }
    ],
    "qualified_name": "Matrix.rank",
    "releases_gil": true,
    "result_schema": {
      "type": "integer"
    },
    "signature": "(self: Matrix[_FieldContext]) -> int",
    "signatures": [
      "(self: Matrix[_FieldContext]) -> int"
    ],
    "summary": "Rank over a field context.",
    "summary_source": "public docstring",
    "type_id": "noether/linalg/Matrix@1"
  },
  {
    "canonical_name": "linalg.Matrix.ring",
    "determinism": "deterministic",
    "doc": "",
    "domain": "linalg",
    "file": "noether/linalg/__init__.pyi",
    "input_schema": {
      "additionalProperties": false,
      "properties": {},
      "type": "object"
    },
    "kind": "property",
    "line": 1068,
    "module": "noether.linalg",
    "name": "ring",
    "overloads": [
      {
        "input_schema": {
          "additionalProperties": false,
          "properties": {},
          "type": "object"
        },
        "parameters": [],
        "result_schema": {
          "x-python-type": "R"
        },
        "signature": "(self) -> R"
      }
    ],
    "qualified_name": "Matrix.ring",
    "releases_gil": false,
    "result_schema": {
      "x-python-type": "R"
    },
    "signature": "(self) -> R",
    "signatures": [
      "(self) -> R"
    ],
    "summary": "Return the ring recorded by this `Matrix`.",
    "summary_source": "derived projection vocabulary",
    "type_id": "noether/linalg/Matrix@1"
  },
  {
    "canonical_name": "linalg.Matrix.rref",
    "determinism": "deterministic",
    "doc": "Reduced row-echelon form over a field context.",
    "domain": "linalg",
    "file": "noether/linalg/__init__.pyi",
    "input_schema": {
      "additionalProperties": false,
      "properties": {},
      "type": "object"
    },
    "kind": "method",
    "line": 1198,
    "module": "noether.linalg",
    "name": "rref",
    "overloads": [
      {
        "input_schema": {
          "additionalProperties": false,
          "properties": {},
          "type": "object"
        },
        "parameters": [],
        "result_schema": {
          "x-python-type": "Matrix[_FieldContext]"
        },
        "signature": "(self: Matrix[_FieldContext]) -> Matrix[_FieldContext]"
      }
    ],
    "qualified_name": "Matrix.rref",
    "releases_gil": true,
    "result_schema": {
      "x-python-type": "Matrix[_FieldContext]"
    },
    "signature": "(self: Matrix[_FieldContext]) -> Matrix[_FieldContext]",
    "signatures": [
      "(self: Matrix[_FieldContext]) -> Matrix[_FieldContext]"
    ],
    "summary": "Reduced row-echelon form over a field context.",
    "summary_source": "public docstring",
    "type_id": "noether/linalg/Matrix@1"
  },
  {
    "canonical_name": "linalg.Matrix.shape",
    "determinism": "deterministic",
    "doc": "",
    "domain": "linalg",
    "file": "noether/linalg/__init__.pyi",
    "input_schema": {
      "additionalProperties": false,
      "properties": {},
      "type": "object"
    },
    "kind": "property",
    "line": 1070,
    "module": "noether.linalg",
    "name": "shape",
    "overloads": [
      {
        "input_schema": {
          "additionalProperties": false,
          "properties": {},
          "type": "object"
        },
        "parameters": [],
        "result_schema": {
          "maxItems": 2,
          "minItems": 2,
          "prefixItems": [
            {
              "type": "integer"
            },
            {
              "type": "integer"
            }
          ],
          "type": "array",
          "x-python-type": "tuple[int, int]"
        },
        "signature": "(self) -> tuple[int, int]"
      }
    ],
    "qualified_name": "Matrix.shape",
    "releases_gil": false,
    "result_schema": {
      "maxItems": 2,
      "minItems": 2,
      "prefixItems": [
        {
          "type": "integer"
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        {
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        }
      ],
      "type": "array",
      "x-python-type": "tuple[int, int]"
    },
    "signature": "(self) -> tuple[int, int]",
    "signatures": [
      "(self) -> tuple[int, int]"
    ],
    "summary": "Return the shape recorded by this `Matrix`.",
    "summary_source": "derived projection vocabulary",
    "type_id": "noether/linalg/Matrix@1"
  },
  {
    "algorithms": [
      "exact",
      "smooth_local"
    ],
    "backends": [
      "flint",
      "native"
    ],
    "canonical_name": "linalg.Matrix.smith_normal_form",
    "determinism": "deterministic",
    "doc": "Smith normal form over the integers, without transformation matrices.\n\n`algorithm=\"auto\"` and `\"exact\"` use the unconditional fast exact\nroute. `algorithm=\"smooth\"` requires an inclusive `smooth_bound` and\nreturns its exact smoothness assumption unless the recorded\ndeterministic bound is fully covered.",
    "domain": "linalg",
    "errors": [
      "invalid smooth bound",
      "unsupported word precision"
    ],
    "evidence": [
      "SmithVerification"
    ],
    "example": "from noether import Z\nfrom noether.linalg import Matrix\nresult = Matrix.from_rows(Z, [[2, 4], [6, 6]]).smith_normal_form()\nassert result.invariant_factors == [2, 6]\nresult.verify()",
    "file": "noether/linalg/__init__.pyi",
    "input_schema": {
      "additionalProperties": false,
      "properties": {
        "algorithm": {
          "enum": [
            "auto",
            "exact"
          ],
          "x-python-type": "Literal['auto', 'exact']"
        },
        "smooth_bound": {
          "type": "null"
        }
      },
      "type": "object"
    },
    "kind": "method",
    "line": 1252,
    "module": "noether.linalg",
    "name": "smith_normal_form",
    "overloads": [
      {
        "input_schema": {
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          "properties": {
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              "x-python-type": "Literal['auto', 'exact']"
            },
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          },
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        },
        "parameters": [
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            "default": "'auto'",
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              "enum": [
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              ],
              "x-python-type": "Literal['auto', 'exact']"
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          },
          {
            "default": "None",
            "kind": "keyword_only",
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          }
        ],
        "result_schema": {
          "x-python-type": "SmithNormalForm"
        },
        "signature": "(self: Matrix[IntegerRing], *, algorithm: Literal['auto', 'exact']='auto', smooth_bound: None=None) -> SmithNormalForm"
      },
      {
        "input_schema": {
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          "properties": {
            "algorithm": {
              "enum": [
                "smooth"
              ],
              "x-python-type": "Literal['smooth']"
            },
            "smooth_bound": {
              "type": "integer"
            }
          },
          "required": [
            "algorithm",
            "smooth_bound"
          ],
          "type": "object"
        },
        "parameters": [
          {
            "kind": "keyword_only",
            "name": "algorithm",
            "required": true,
            "schema": {
              "enum": [
                "smooth"
              ],
              "x-python-type": "Literal['smooth']"
            }
          },
          {
            "kind": "keyword_only",
            "name": "smooth_bound",
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          }
        ],
        "result_schema": {
          "x-python-type": "SmithNormalForm"
        },
        "signature": "(self: Matrix[IntegerRing], *, algorithm: Literal['smooth'], smooth_bound: int) -> SmithNormalForm"
      }
    ],
    "qualified_name": "Matrix.smith_normal_form",
    "releases_gil": true,
    "result_schema": {
      "x-python-type": "SmithNormalForm"
    },
    "signature": "(self: Matrix[IntegerRing], *, algorithm: Literal['auto', 'exact']='auto', smooth_bound: None=None) -> SmithNormalForm",
    "signatures": [
      "(self: Matrix[IntegerRing], *, algorithm: Literal['auto', 'exact']='auto', smooth_bound: None=None) -> SmithNormalForm",
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    ],
    "summary": "Smith normal form over the integers, without transformation matrices.",
    "summary_source": "public docstring",
    "supported_rings": [
      "Z"
    ],
    "type_id": "noether/linalg/Matrix@1",
    "validity": [
      "unconditional",
      "conditional:TorsionOrderIsSmooth"
    ]
  },
  {
    "canonical_name": "linalg.Matrix.smith_normal_form_with_transforms",
    "determinism": "deterministic",
    "doc": "Smith normal form with explicit unimodular transformation matrices.",
    "domain": "linalg",
    "file": "noether/linalg/__init__.pyi",
    "input_schema": {
      "additionalProperties": false,
      "properties": {},
      "type": "object"
    },
    "kind": "method",
    "line": 1265,
    "module": "noether.linalg",
    "name": "smith_normal_form_with_transforms",
    "overloads": [
      {
        "input_schema": {
          "additionalProperties": false,
          "properties": {},
          "type": "object"
        },
        "parameters": [],
        "result_schema": {
          "x-python-type": "SmithNormalFormWithTransforms"
        },
        "signature": "(self: Matrix[IntegerRing]) -> SmithNormalFormWithTransforms"
      }
    ],
    "qualified_name": "Matrix.smith_normal_form_with_transforms",
    "releases_gil": true,
    "result_schema": {
      "x-python-type": "SmithNormalFormWithTransforms"
    },
    "signature": "(self: Matrix[IntegerRing]) -> SmithNormalFormWithTransforms",
    "signatures": [
      "(self: Matrix[IntegerRing]) -> SmithNormalFormWithTransforms"
    ],
    "summary": "Smith normal form with explicit unimodular transformation matrices.",
    "summary_source": "public docstring",
    "type_id": "noether/linalg/Matrix@1"
  },
  {
    "canonical_name": "linalg.Matrix.solve",
    "determinism": "deterministic",
    "doc": "Solve `self @ x = rhs` and return the complete affine solution set.",
    "domain": "linalg",
    "file": "noether/linalg/__init__.pyi",
    "input_schema": {
      "additionalProperties": false,
      "properties": {
        "rhs": {
          "x-python-type": "Vector[_FieldContext]"
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      },
      "required": [
        "rhs"
      ],
      "type": "object"
    },
    "kind": "method",
    "line": 1200,
    "module": "noether.linalg",
    "name": "solve",
    "overloads": [
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        "input_schema": {
          "additionalProperties": false,
          "properties": {
            "rhs": {
              "x-python-type": "Vector[_FieldContext]"
            }
          },
          "required": [
            "rhs"
          ],
          "type": "object"
        },
        "parameters": [
          {
            "kind": "positional_or_keyword",
            "name": "rhs",
            "required": true,
            "schema": {
              "x-python-type": "Vector[_FieldContext]"
            }
          }
        ],
        "result_schema": {
          "x-python-type": "SolutionSpace[_FieldContext]"
        },
        "signature": "(self: Matrix[_FieldContext], rhs: Vector[_FieldContext]) -> SolutionSpace[_FieldContext]"
      }
    ],
    "qualified_name": "Matrix.solve",
    "releases_gil": true,
    "result_schema": {
      "x-python-type": "SolutionSpace[_FieldContext]"
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    "signature": "(self: Matrix[_FieldContext], rhs: Vector[_FieldContext]) -> SolutionSpace[_FieldContext]",
    "signatures": [
      "(self: Matrix[_FieldContext], rhs: Vector[_FieldContext]) -> SolutionSpace[_FieldContext]"
    ],
    "summary": "Solve `self @ x = rhs` and return the complete affine solution set.",
    "summary_source": "public docstring",
    "type_id": "noether/linalg/Matrix@1"
  },
  {
    "algorithms": [
      "Smith normal form",
      "integer lift and Howell form"
    ],
    "backends": [
      "flint",
      "pwvla"
    ],
    "canonical_name": "linalg.Matrix.solve_exact",
    "determinism": "deterministic",
    "doc": "Solve exactly over `Z` or arbitrary `Z/nZ`, retaining complete evidence.",
    "domain": "linalg",
    "errors": [
      "dimension mismatch",
      "incompatible coefficient context"
    ],
    "evidence": [
      "IntegerLinearSystemResult.verify",
      "CongruenceSystemResult.verify"
    ],
    "file": "noether/linalg/__init__.pyi",
    "input_schema": {
      "additionalProperties": false,
      "properties": {
        "rhs": {
          "x-python-type": "Vector[IntegerRing]"
        }
      },
      "required": [
        "rhs"
      ],
      "type": "object"
    },
    "kind": "method",
    "line": 1204,
    "module": "noether.linalg",
    "name": "solve_exact",
    "overloads": [
      {
        "input_schema": {
          "additionalProperties": false,
          "properties": {
            "rhs": {
              "x-python-type": "Vector[IntegerRing]"
            }
          },
          "required": [
            "rhs"
          ],
          "type": "object"
        },
        "parameters": [
          {
            "kind": "positional_or_keyword",
            "name": "rhs",
            "required": true,
            "schema": {
              "x-python-type": "Vector[IntegerRing]"
            }
          }
        ],
        "result_schema": {
          "x-python-type": "IntegerLinearSystemResult"
        },
        "signature": "(self: Matrix[IntegerRing], rhs: Vector[IntegerRing]) -> IntegerLinearSystemResult"
      },
      {
        "input_schema": {
          "additionalProperties": false,
          "properties": {
            "rhs": {
              "x-python-type": "Vector[ResidueRing]"
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          },
          "required": [
            "rhs"
          ],
          "type": "object"
        },
        "parameters": [
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            "kind": "positional_or_keyword",
            "name": "rhs",
            "required": true,
            "schema": {
              "x-python-type": "Vector[ResidueRing]"
            }
          }
        ],
        "result_schema": {
          "x-python-type": "CongruenceSystemResult"
        },
        "signature": "(self: Matrix[ResidueRing], rhs: Vector[ResidueRing]) -> CongruenceSystemResult"
      }
    ],
    "qualified_name": "Matrix.solve_exact",
    "releases_gil": true,
    "result_schema": {
      "x-python-type": "IntegerLinearSystemResult"
    },
    "signature": "(self: Matrix[IntegerRing], rhs: Vector[IntegerRing]) -> IntegerLinearSystemResult",
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              "x-python-type": "Iterable[Polynomial[_PolynomialCoefficientContext] | PolynomialScalar]"
            }
          }
        ],
        "result_schema": {
          "x-python-type": "Vector[PolynomialRing[_PolynomialCoefficientContext]]"
        },
        "signature": "(cls, ring: PolynomialRing[_PolynomialCoefficientContext], values: Iterable[Polynomial[_PolynomialCoefficientContext] | PolynomialScalar]) -> Vector[PolynomialRing[_PolynomialCoefficientContext]]"
      },
      {
        "input_schema": {
          "additionalProperties": false,
          "properties": {
            "ring": {
              "x-python-type": "QuotientRing[C]"
            },
            "values": {
              "items": {
                "anyOf": [
                  {
                    "x-python-type": "QuotientElement[C]"
                  },
                  {
                    "x-python-type": "Polynomial[C]"
                  }
                ],
                "x-python-type": "QuotientElement[C] | Polynomial[C]"
              },
              "type": "array",
              "x-python-type": "Iterable[QuotientElement[C] | Polynomial[C]]"
            }
          },
          "required": [
            "ring",
            "values"
          ],
          "type": "object"
        },
        "parameters": [
          {
            "kind": "positional_or_keyword",
            "name": "ring",
            "required": true,
            "schema": {
              "x-python-type": "QuotientRing[C]"
            }
          },
          {
            "kind": "positional_or_keyword",
            "name": "values",
            "required": true,
            "schema": {
              "items": {
                "anyOf": [
                  {
                    "x-python-type": "QuotientElement[C]"
                  },
                  {
                    "x-python-type": "Polynomial[C]"
                  }
                ],
                "x-python-type": "QuotientElement[C] | Polynomial[C]"
              },
              "type": "array",
              "x-python-type": "Iterable[QuotientElement[C] | Polynomial[C]]"
            }
          }
        ],
        "result_schema": {
          "x-python-type": "Vector[QuotientRing[C]]"
        },
        "signature": "(cls, ring: QuotientRing[C], values: Iterable[QuotientElement[C] | Polynomial[C]]) -> Vector[QuotientRing[C]]"
      }
    ],
    "qualified_name": "Vector.from_values",
    "releases_gil": false,
    "result_schema": {
      "x-python-type": "Vector[IntegerRing]"
    },
    "signature": "(cls, ring: IntegerRing, values: Iterable[int]) -> Vector[IntegerRing]",
    "signatures": [
      "(cls, ring: IntegerRing, values: Iterable[int]) -> Vector[IntegerRing]",
      "(cls, ring: RationalField, values: Iterable[int | _Fraction]) -> Vector[RationalField]",
      "(cls, ring: ResidueRing, values: Iterable[int | ResidueClass]) -> Vector[ResidueRing]",
      "(cls, ring: FiniteField, values: Iterable[int | FiniteFieldElement]) -> Vector[FiniteField]",
      "(cls, ring: RationalFunctionField[RationalField], values: Iterable[int | _Fraction | RationalFunction[RationalField]]) -> Vector[RationalFunctionField[RationalField]]",
      "(cls, ring: _NumberFieldContext, values: Iterable[int | _Fraction | NumberFieldElement | Iterable[int | _Fraction]]) -> Vector[_NumberFieldContext]",
      "(cls, ring: _PadicContext, values: Iterable[int | PadicElement]) -> Vector[_PadicContext]",
      "(cls, ring: _QadicContext, values: Iterable[int | QadicElement]) -> Vector[_QadicContext]",
      "(cls, ring: _PadicLocalIntegralContext, values: Iterable[int | PadicLocalFieldElement]) -> Vector[_PadicLocalIntegralContext]",
      "(cls, ring: RealBallField, values: Iterable[int | _Fraction | RealBall]) -> Vector[RealBallField]",
      "(cls, ring: ComplexBallField, values: Iterable[int | _Fraction | ComplexBall]) -> Vector[ComplexBallField]",
      "(cls, ring: PowerSeriesRing[C], values: Iterable[PowerSeries[C] | SeriesScalar]) -> Vector[PowerSeriesRing[C]]",
      "(cls, ring: PolynomialRing[_PolynomialCoefficientContext], values: Iterable[Polynomial[_PolynomialCoefficientContext] | PolynomialScalar]) -> Vector[PolynomialRing[_PolynomialCoefficientContext]]",
      "(cls, ring: QuotientRing[C], values: Iterable[QuotientElement[C] | Polynomial[C]]) -> Vector[QuotientRing[C]]"
    ],
    "summary": "Construct a `Vector` using values.",
    "summary_source": "derived projection vocabulary",
    "type_id": "noether/linalg/Vector@1"
  },
  {
    "canonical_name": "linalg.Vector.ring",
    "determinism": "deterministic",
    "doc": "",
    "domain": "linalg",
    "file": "noether/linalg/__init__.pyi",
    "input_schema": {
      "additionalProperties": false,
      "properties": {},
      "type": "object"
    },
    "kind": "property",
    "line": 449,
    "module": "noether.linalg",
    "name": "ring",
    "overloads": [
      {
        "input_schema": {
          "additionalProperties": false,
          "properties": {},
          "type": "object"
        },
        "parameters": [],
        "result_schema": {
          "x-python-type": "R"
        },
        "signature": "(self) -> R"
      }
    ],
    "qualified_name": "Vector.ring",
    "releases_gil": false,
    "result_schema": {
      "x-python-type": "R"
    },
    "signature": "(self) -> R",
    "signatures": [
      "(self) -> R"
    ],
    "summary": "Return the ring recorded by this `Vector`.",
    "summary_source": "derived projection vocabulary",
    "type_id": "noether/linalg/Vector@1"
  },
  {
    "canonical_name": "linalg.Vector.to_column_matrix",
    "determinism": "deterministic",
    "doc": "",
    "domain": "linalg",
    "file": "noether/linalg/__init__.pyi",
    "input_schema": {
      "additionalProperties": false,
      "properties": {},
      "type": "object"
    },
    "kind": "method",
    "line": 561,
    "module": "noether.linalg",
    "name": "to_column_matrix",
    "overloads": [
      {
        "input_schema": {
          "additionalProperties": false,
          "properties": {},
          "type": "object"
        },
        "parameters": [],
        "result_schema": {
          "x-python-type": "Matrix[R]"
        },
        "signature": "(self) -> Matrix[R]"
      }
    ],
    "qualified_name": "Vector.to_column_matrix",
    "releases_gil": false,
    "result_schema": {
      "x-python-type": "Matrix[R]"
    },
    "signature": "(self) -> Matrix[R]",
    "signatures": [
      "(self) -> Matrix[R]"
    ],
    "summary": "Convert this `Vector` to column matrix.",
    "summary_source": "derived projection vocabulary",
    "type_id": "noether/linalg/Vector@1"
  },
  {
    "canonical_name": "linalg.Vector.to_json",
    "determinism": "deterministic",
    "doc": "Serialize the vector and its complete coefficient context.",
    "domain": "linalg",
    "file": "noether/linalg/__init__.pyi",
    "input_schema": {
      "additionalProperties": false,
      "properties": {},
      "type": "object"
    },
    "kind": "method",
    "line": 363,
    "module": "noether.linalg",
    "name": "to_json",
    "overloads": [
      {
        "input_schema": {
          "additionalProperties": false,
          "properties": {},
          "type": "object"
        },
        "parameters": [],
        "result_schema": {
          "type": "string"
        },
        "signature": "(self) -> str"
      }
    ],
    "qualified_name": "Vector.to_json",
    "releases_gil": false,
    "result_schema": {
      "type": "string"
    },
    "signature": "(self) -> str",
    "signatures": [
      "(self) -> str"
    ],
    "summary": "Serialize the vector and its complete coefficient context.",
    "summary_source": "public docstring",
    "type_id": "noether/linalg/Vector@1"
  },
  {
    "canonical_name": "linalg.Vector.to_row_matrix",
    "determinism": "deterministic",
    "doc": "",
    "domain": "linalg",
    "file": "noether/linalg/__init__.pyi",
    "input_schema": {
      "additionalProperties": false,
      "properties": {},
      "type": "object"
    },
    "kind": "method",
    "line": 562,
    "module": "noether.linalg",
    "name": "to_row_matrix",
    "overloads": [
      {
        "input_schema": {
          "additionalProperties": false,
          "properties": {},
          "type": "object"
        },
        "parameters": [],
        "result_schema": {
          "x-python-type": "Matrix[R]"
        },
        "signature": "(self) -> Matrix[R]"
      }
    ],
    "qualified_name": "Vector.to_row_matrix",
    "releases_gil": false,
    "result_schema": {
      "x-python-type": "Matrix[R]"
    },
    "signature": "(self) -> Matrix[R]",
    "signatures": [
      "(self) -> Matrix[R]"
    ],
    "summary": "Convert this `Vector` to row matrix.",
    "summary_source": "derived projection vocabulary",
    "type_id": "noether/linalg/Vector@1"
  },
  {
    "canonical_name": "linalg.Vector.to_values",
    "determinism": "deterministic",
    "doc": "",
    "domain": "linalg",
    "file": "noether/linalg/__init__.pyi",
    "input_schema": {
      "additionalProperties": false,
      "properties": {},
      "type": "object"
    },
    "kind": "method",
    "line": 526,
    "module": "noether.linalg",
    "name": "to_values",
    "overloads": [
      {
        "input_schema": {
          "additionalProperties": false,
          "properties": {},
          "type": "object"
        },
        "parameters": [],
        "result_schema": {
          "items": {
            "type": "integer"
          },
          "type": "array",
          "x-python-type": "list[int]"
        },
        "signature": "(self: Vector[IntegerRing]) -> list[int]"
      },
      {
        "input_schema": {
          "additionalProperties": false,
          "properties": {},
          "type": "object"
        },
        "parameters": [],
        "result_schema": {
          "items": {
            "x-python-type": "_Fraction"
          },
          "type": "array",
          "x-python-type": "list[_Fraction]"
        },
        "signature": "(self: Vector[RationalField]) -> list[_Fraction]"
      },
      {
        "input_schema": {
          "additionalProperties": false,
          "properties": {},
          "type": "object"
        },
        "parameters": [],
        "result_schema": {
          "items": {
            "x-python-type": "ResidueClass"
          },
          "type": "array",
          "x-python-type": "list[ResidueClass]"
        },
        "signature": "(self: Vector[ResidueRing]) -> list[ResidueClass]"
      },
      {
        "input_schema": {
          "additionalProperties": false,
          "properties": {},
          "type": "object"
        },
        "parameters": [],
        "result_schema": {
          "items": {
            "x-python-type": "FiniteFieldElement"
          },
          "type": "array",
          "x-python-type": "list[FiniteFieldElement]"
        },
        "signature": "(self: Vector[FiniteField]) -> list[FiniteFieldElement]"
      },
      {
        "input_schema": {
          "additionalProperties": false,
          "properties": {},
          "type": "object"
        },
        "parameters": [],
        "result_schema": {
          "items": {
            "x-python-type": "RationalFunction[RationalField]"
          },
          "type": "array",
          "x-python-type": "list[RationalFunction[RationalField]]"
        },
        "signature": "(self: Vector[RationalFunctionField[RationalField]]) -> list[RationalFunction[RationalField]]"
      },
      {
        "input_schema": {
          "additionalProperties": false,
          "properties": {},
          "type": "object"
        },
        "parameters": [],
        "result_schema": {
          "items": {
            "x-python-type": "NumberFieldElement"
          },
          "type": "array",
          "x-python-type": "list[NumberFieldElement]"
        },
        "signature": "(self: Vector[_NumberFieldContext]) -> list[NumberFieldElement]"
      },
      {
        "input_schema": {
          "additionalProperties": false,
          "properties": {},
          "type": "object"
        },
        "parameters": [],
        "result_schema": {
          "items": {
            "x-python-type": "PadicElement"
          },
          "type": "array",
          "x-python-type": "list[PadicElement]"
        },
        "signature": "(self: Vector[_PadicContext]) -> list[PadicElement]"
      },
      {
        "input_schema": {
          "additionalProperties": false,
          "properties": {},
          "type": "object"
        },
        "parameters": [],
        "result_schema": {
          "items": {
            "x-python-type": "QadicElement"
          },
          "type": "array",
          "x-python-type": "list[QadicElement]"
        },
        "signature": "(self: Vector[_QadicContext]) -> list[QadicElement]"
      },
      {
        "input_schema": {
          "additionalProperties": false,
          "properties": {},
          "type": "object"
        },
        "parameters": [],
        "result_schema": {
          "items": {
            "x-python-type": "PadicLocalFieldElement"
          },
          "type": "array",
          "x-python-type": "list[PadicLocalFieldElement]"
        },
        "signature": "(self: Vector[_PadicLocalIntegralContext]) -> list[PadicLocalFieldElement]"
      },
      {
        "input_schema": {
          "additionalProperties": false,
          "properties": {},
          "type": "object"
        },
        "parameters": [],
        "result_schema": {
          "items": {
            "x-python-type": "RealBall"
          },
          "type": "array",
          "x-python-type": "list[RealBall]"
        },
        "signature": "(self: Vector[RealBallField]) -> list[RealBall]"
      },
      {
        "input_schema": {
          "additionalProperties": false,
          "properties": {},
          "type": "object"
        },
        "parameters": [],
        "result_schema": {
          "items": {
            "x-python-type": "ComplexBall"
          },
          "type": "array",
          "x-python-type": "list[ComplexBall]"
        },
        "signature": "(self: Vector[ComplexBallField]) -> list[ComplexBall]"
      },
      {
        "input_schema": {
          "additionalProperties": false,
          "properties": {},
          "type": "object"
        },
        "parameters": [],
        "result_schema": {
          "items": {
            "x-python-type": "PowerSeries[C]"
          },
          "type": "array",
          "x-python-type": "list[PowerSeries[C]]"
        },
        "signature": "(self: Vector[PowerSeriesRing[C]]) -> list[PowerSeries[C]]"
      },
      {
        "input_schema": {
          "additionalProperties": false,
          "properties": {},
          "type": "object"
        },
        "parameters": [],
        "result_schema": {
          "items": {
            "x-python-type": "Polynomial[_PolynomialCoefficientContext]"
          },
          "type": "array",
          "x-python-type": "list[Polynomial[_PolynomialCoefficientContext]]"
        },
        "signature": "(self: Vector[PolynomialRing[_PolynomialCoefficientContext]]) -> list[Polynomial[_PolynomialCoefficientContext]]"
      },
      {
        "input_schema": {
          "additionalProperties": false,
          "properties": {},
          "type": "object"
        },
        "parameters": [],
        "result_schema": {
          "items": {
            "x-python-type": "QuotientElement[C]"
          },
          "type": "array",
          "x-python-type": "list[QuotientElement[C]]"
        },
        "signature": "(self: Vector[QuotientRing[C]]) -> list[QuotientElement[C]]"
      }
    ],
    "qualified_name": "Vector.to_values",
    "releases_gil": true,
    "result_schema": {
      "items": {
        "type": "integer"
      },
      "type": "array",
      "x-python-type": "list[int]"
    },
    "signature": "(self: Vector[IntegerRing]) -> list[int]",
    "signatures": [
      "(self: Vector[IntegerRing]) -> list[int]",
      "(self: Vector[RationalField]) -> list[_Fraction]",
      "(self: Vector[ResidueRing]) -> list[ResidueClass]",
      "(self: Vector[FiniteField]) -> list[FiniteFieldElement]",
      "(self: Vector[RationalFunctionField[RationalField]]) -> list[RationalFunction[RationalField]]",
      "(self: Vector[_NumberFieldContext]) -> list[NumberFieldElement]",
      "(self: Vector[_PadicContext]) -> list[PadicElement]",
      "(self: Vector[_QadicContext]) -> list[QadicElement]",
      "(self: Vector[_PadicLocalIntegralContext]) -> list[PadicLocalFieldElement]",
      "(self: Vector[RealBallField]) -> list[RealBall]",
      "(self: Vector[ComplexBallField]) -> list[ComplexBall]",
      "(self: Vector[PowerSeriesRing[C]]) -> list[PowerSeries[C]]",
      "(self: Vector[PolynomialRing[_PolynomialCoefficientContext]]) -> list[Polynomial[_PolynomialCoefficientContext]]",
      "(self: Vector[QuotientRing[C]]) -> list[QuotientElement[C]]"
    ],
    "summary": "Convert this `Vector` to values.",
    "summary_source": "derived projection vocabulary",
    "type_id": "noether/linalg/Vector@1"
  },
  {
    "canonical_name": "linalg.Vector.zeros",
    "determinism": "deterministic",
    "doc": "",
    "domain": "linalg",
    "file": "noether/linalg/__init__.pyi",
    "input_schema": {
      "additionalProperties": false,
      "properties": {
        "dimension": {
          "type": "integer"
        },
        "ring": {
          "x-python-type": "R"
        }
      },
      "required": [
        "ring",
        "dimension"
      ],
      "type": "object"
    },
    "kind": "classmethod",
    "line": 447,
    "module": "noether.linalg",
    "name": "zeros",
    "overloads": [
      {
        "input_schema": {
          "additionalProperties": false,
          "properties": {
            "dimension": {
              "type": "integer"
            },
            "ring": {
              "x-python-type": "R"
            }
          },
          "required": [
            "ring",
            "dimension"
          ],
          "type": "object"
        },
        "parameters": [
          {
            "kind": "positional_or_keyword",
            "name": "ring",
            "required": true,
            "schema": {
              "x-python-type": "R"
            }
          },
          {
            "kind": "positional_or_keyword",
            "name": "dimension",
            "required": true,
            "schema": {
              "type": "integer"
            }
          }
        ],
        "result_schema": {
          "x-python-type": "Vector[R]"
        },
        "signature": "(cls, ring: R, dimension: int) -> Vector[R]"
      }
    ],
    "qualified_name": "Vector.zeros",
    "releases_gil": false,
    "result_schema": {
      "x-python-type": "Vector[R]"
    },
    "signature": "(cls, ring: R, dimension: int) -> Vector[R]",
    "signatures": [
      "(cls, ring: R, dimension: int) -> Vector[R]"
    ],
    "summary": "Construct a `Vector` using zeros.",
    "summary_source": "derived projection vocabulary",
    "type_id": "noether/linalg/Vector@1"
  },
  {
    "canonical_name": "linalg.World",
    "determinism": "deterministic",
    "doc": "Explicit owner of one initialized MPI world.",
    "domain": "linalg",
    "file": "noether/linalg/__init__.pyi",
    "input_schema": {},
    "kind": "class",
    "line": 1618,
    "module": "noether.linalg",
    "name": "World",
    "overloads": [],
    "qualified_name": "World",
    "releases_gil": false,
    "result_schema": {},
    "signature": "",
    "signatures": [],
    "summary": "Explicit owner of one initialized MPI world.",
    "summary_source": "public docstring"
  },
  {
    "algorithms": [
      "system_mpi_world"
    ],
    "backends": [
      "m4rst-distributed",
      "m4rst"
    ],
    "canonical_name": "linalg.World.initialize",
    "determinism": "deterministic",
    "doc": "Initialize MPI exactly once for this process.",
    "domain": "linalg",
    "example": "from noether import capabilities\ninstalled = capabilities()\nif \"distributed\" in installed.compiled_features:\n    from noether.linalg import DistributedMatrix, World\n    from noether.numthy import FiniteField\n    world = World.initialize()\n    matrix = DistributedMatrix.identity(world, FiniteField.prime(2), 3)\n    assert matrix.rank() == 3\nelse:\n    assert installed.mpi_runtime is None",
    "file": "noether/linalg/__init__.pyi",
    "input_schema": {
      "additionalProperties": false,
      "properties": {},
      "type": "object"
    },
    "kind": "classmethod",
    "line": 1620,
    "module": "noether.linalg",
    "name": "initialize",
    "overloads": [
      {
        "input_schema": {
          "additionalProperties": false,
          "properties": {},
          "type": "object"
        },
        "parameters": [],
        "result_schema": {
          "x-python-type": "World"
        },
        "signature": "(cls) -> World"
      }
    ],
    "qualified_name": "World.initialize",
    "releases_gil": true,
    "result_schema": {
      "x-python-type": "World"
    },
    "signature": "(cls) -> World",
    "signatures": [
      "(cls) -> World"
    ],
    "summary": "Initialize MPI exactly once for this process.",
    "summary_source": "public docstring"
  },
  {
    "canonical_name": "linalg.World.is_root",
    "determinism": "deterministic",
    "doc": "Whether this process is rank zero.",
    "domain": "linalg",
    "file": "noether/linalg/__init__.pyi",
    "input_schema": {
      "additionalProperties": false,
      "properties": {},
      "type": "object"
    },
    "kind": "property",
    "line": 1626,
    "module": "noether.linalg",
    "name": "is_root",
    "overloads": [
      {
        "input_schema": {
          "additionalProperties": false,
          "properties": {},
          "type": "object"
        },
        "parameters": [],
        "result_schema": {
          "type": "boolean"
        },
        "signature": "(self) -> bool"
      }
    ],
    "qualified_name": "World.is_root",
    "releases_gil": false,
    "result_schema": {
      "type": "boolean"
    },
    "signature": "(self) -> bool",
    "signatures": [
      "(self) -> bool"
    ],
    "summary": "Whether this process is rank zero.",
    "summary_source": "public docstring"
  },
  {
    "canonical_name": "linalg.World.library_version",
    "determinism": "deterministic",
    "doc": "MPI implementation/version string.",
    "domain": "linalg",
    "file": "noether/linalg/__init__.pyi",
    "input_schema": {
      "additionalProperties": false,
      "properties": {},
      "type": "object"
    },
    "kind": "property",
    "line": 1628,
    "module": "noether.linalg",
    "name": "library_version",
    "overloads": [
      {
        "input_schema": {
          "additionalProperties": false,
          "properties": {},
          "type": "object"
        },
        "parameters": [],
        "result_schema": {
          "type": "string"
        },
        "signature": "(self) -> str"
      }
    ],
    "qualified_name": "World.library_version",
    "releases_gil": false,
    "result_schema": {
      "type": "string"
    },
    "signature": "(self) -> str",
    "signatures": [
      "(self) -> str"
    ],
    "summary": "MPI implementation/version string.",
    "summary_source": "public docstring"
  },
  {
    "canonical_name": "linalg.World.mpi_rank",
    "determinism": "deterministic",
    "doc": "This process's MPI rank.",
    "domain": "linalg",
    "file": "noether/linalg/__init__.pyi",
    "input_schema": {
      "additionalProperties": false,
      "properties": {},
      "type": "object"
    },
    "kind": "property",
    "line": 1624,
    "module": "noether.linalg",
    "name": "mpi_rank",
    "overloads": [
      {
        "input_schema": {
          "additionalProperties": false,
          "properties": {},
          "type": "object"
        },
        "parameters": [],
        "result_schema": {
          "type": "integer"
        },
        "signature": "(self) -> int"
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    ],
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    "domain": "numthy",
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    "summary": "Return the underlying curve recorded by this `EllipticCurve`.",
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    "type_id": "noether/numthy/EllipticCurve@1"
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    "canonical_name": "numthy.EllipticCurve.verify",
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    "domain": "numthy",
    "file": "noether/numthy/__init__.pyi",
    "input_schema": {
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    "qualified_name": "EllipticCurvePoint.scalar_mul",
    "releases_gil": false,
    "result_schema": {
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    },
    "signature": "(self, scalar: int) -> EllipticCurvePoint[E]",
    "signatures": [
      "(self, scalar: int) -> EllipticCurvePoint[E]"
    ],
    "summary": "Compute scalar mul for this `EllipticCurvePoint`.",
    "summary_source": "derived projection vocabulary",
    "type_id": "noether/numthy/EllipticCurvePoint@1"
  },
  {
    "canonical_name": "numthy.EllipticCurvePoint.to_json",
    "determinism": "deterministic",
    "doc": "",
    "domain": "numthy",
    "file": "noether/numthy/__init__.pyi",
    "input_schema": {
      "additionalProperties": false,
      "properties": {},
      "type": "object"
    },
    "kind": "method",
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    "module": "noether.numthy",
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          "properties": {},
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        "signature": "(self) -> str"
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    ],
    "qualified_name": "EllipticCurvePoint.to_json",
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    "signatures": [
      "(self) -> str"
    ],
    "summary": "Serialize this `EllipticCurvePoint` to its canonical typed JSON representation.",
    "summary_source": "derived projection vocabulary",
    "type_id": "noether/numthy/EllipticCurvePoint@1"
  },
  {
    "canonical_name": "numthy.EllipticCurvePoint.underlying_curve_point",
    "determinism": "deterministic",
    "doc": "",
    "domain": "numthy",
    "file": "noether/numthy/__init__.pyi",
    "input_schema": {
      "additionalProperties": false,
      "properties": {},
      "type": "object"
    },
    "kind": "property",
    "line": 364,
    "module": "noether.numthy",
    "name": "underlying_curve_point",
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      {
        "input_schema": {
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          "properties": {},
          "type": "object"
        },
        "parameters": [],
        "result_schema": {
          "x-python-type": "CurvePoint[E]"
        },
        "signature": "(self) -> CurvePoint[E]"
      }
    ],
    "qualified_name": "EllipticCurvePoint.underlying_curve_point",
    "releases_gil": false,
    "result_schema": {
      "x-python-type": "CurvePoint[E]"
    },
    "signature": "(self) -> CurvePoint[E]",
    "signatures": [
      "(self) -> CurvePoint[E]"
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    "summary": "Return the underlying curve point recorded by this `EllipticCurvePoint`.",
    "summary_source": "derived projection vocabulary",
    "type_id": "noether/numthy/EllipticCurvePoint@1"
  },
  {
    "canonical_name": "numthy.EllipticCurvePoint.verify",
    "determinism": "deterministic",
    "doc": "",
    "domain": "numthy",
    "file": "noether/numthy/__init__.pyi",
    "input_schema": {
      "additionalProperties": false,
      "properties": {},
      "type": "object"
    },
    "kind": "method",
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    "module": "noether.numthy",
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        "signature": "(self) -> None"
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    ],
    "qualified_name": "EllipticCurvePoint.verify",
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    "result_schema": {
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    "signature": "(self) -> None",
    "signatures": [
      "(self) -> None"
    ],
    "summary": "Replay and verify the mathematical evidence carried by this `EllipticCurvePoint`.",
    "summary_source": "derived projection vocabulary",
    "type_id": "noether/numthy/EllipticCurvePoint@1"
  },
  {
    "canonical_name": "numthy.EllipticCurvePoint.verify_exact_order",
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    "doc": "",
    "domain": "numthy",
    "file": "noether/numthy/__init__.pyi",
    "input_schema": {
      "additionalProperties": false,
      "properties": {
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      },
      "required": [
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      "type": "object"
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    "kind": "method",
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    "module": "noether.numthy",
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    ],
    "qualified_name": "EllipticCurvePoint.verify_exact_order",
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    "signature": "(self, candidate: int) -> None",
    "signatures": [
      "(self, candidate: int) -> None"
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    "summary": "Compute verify exact order for this `EllipticCurvePoint`.",
    "summary_source": "derived projection vocabulary",
    "type_id": "noether/numthy/EllipticCurvePoint@1"
  },
  {
    "canonical_name": "numthy.FiniteField",
    "determinism": "deterministic",
    "doc": "A finite field with one explicit, verified presentation.",
    "domain": "numthy",
    "file": "noether/numthy/__init__.pyi",
    "input_schema": {},
    "kind": "class",
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    "module": "noether.numthy",
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    "qualified_name": "FiniteField",
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    "result_schema": {},
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    "signatures": [],
    "summary": "A finite field with one explicit, verified presentation.",
    "summary_source": "public docstring",
    "type_id": "noether/numthy/FiniteField@1"
  },
  {
    "canonical_name": "numthy.FiniteField.__call__",
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    "doc": "",
    "domain": "numthy",
    "file": "noether/numthy/__init__.pyi",
    "input_schema": {
      "additionalProperties": false,
      "properties": {
        "value": {
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      },
      "required": [
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      "type": "object"
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    "kind": "method",
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    "module": "noether.numthy",
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        "result_schema": {
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        "signature": "(self, value: int) -> FiniteFieldElement"
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    ],
    "qualified_name": "FiniteField.__call__",
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    "result_schema": {
      "x-python-type": "FiniteFieldElement"
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    "signature": "(self, value: int) -> FiniteFieldElement",
    "signatures": [
      "(self, value: int) -> FiniteFieldElement"
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    "summary": "Coerce the supplied input through this context or mathematical map.",
    "summary_source": "derived projection vocabulary",
    "type_id": "noether/numthy/FiniteField@1"
  },
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    "canonical_name": "numthy.FiniteField.cardinality",
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    "doc": "",
    "domain": "numthy",
    "file": "noether/numthy/__init__.pyi",
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      "additionalProperties": false,
      "properties": {},
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    "module": "noether.numthy",
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    "qualified_name": "FiniteField.cardinality",
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      "(self) -> int"
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    "summary": "Compute cardinality for this `FiniteField`.",
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    "type_id": "noether/numthy/FiniteField@1"
  },
  {
    "canonical_name": "numthy.FiniteField.characteristic",
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    "doc": "",
    "domain": "numthy",
    "file": "noether/numthy/__init__.pyi",
    "input_schema": {
      "additionalProperties": false,
      "properties": {},
      "type": "object"
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    "kind": "property",
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    "module": "noether.numthy",
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    ],
    "qualified_name": "FiniteField.characteristic",
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    "signature": "(self) -> int",
    "signatures": [
      "(self) -> int"
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    "summary": "Return the characteristic recorded by this `FiniteField`.",
    "summary_source": "derived projection vocabulary",
    "type_id": "noether/numthy/FiniteField@1"
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  {
    "canonical_name": "numthy.FiniteField.defining_modulus",
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    "domain": "numthy",
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    "input_schema": {
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      "properties": {},
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    "kind": "property",
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    "module": "noether.numthy",
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              "type": "array",
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          "x-python-type": "list[int] | None"
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        "signature": "(self) -> list[int] | None"
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      "x-python-type": "list[int] | None"
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    "signature": "(self) -> list[int] | None",
    "signatures": [
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    "summary": "Return the defining modulus recorded by this `FiniteField`.",
    "summary_source": "derived projection vocabulary",
    "type_id": "noether/numthy/FiniteField@1"
  },
  {
    "canonical_name": "numthy.FiniteField.degree",
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    "doc": "",
    "domain": "numthy",
    "file": "noether/numthy/__init__.pyi",
    "input_schema": {
      "additionalProperties": false,
      "properties": {},
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    "kind": "property",
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    "module": "noether.numthy",
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    "summary": "Return the degree recorded by this `FiniteField`.",
    "summary_source": "derived projection vocabulary",
    "type_id": "noether/numthy/FiniteField@1"
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  {
    "canonical_name": "numthy.FiniteField.element_from_coefficients",
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    "doc": "",
    "domain": "numthy",
    "file": "noether/numthy/__init__.pyi",
    "input_schema": {
      "additionalProperties": false,
      "properties": {
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          "type": "array",
          "x-python-type": "list[int]"
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      "required": [
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          },
          "required": [
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          }
        ],
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        "signature": "(self, coefficients: list[int]) -> FiniteFieldElement"
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    "qualified_name": "FiniteField.element_from_coefficients",
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    "signature": "(self, coefficients: list[int]) -> FiniteFieldElement",
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      "(self, coefficients: list[int]) -> FiniteFieldElement"
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    "summary": "Compute element from coefficients for this `FiniteField`.",
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    "canonical_name": "numthy.FiniteField.embedding_to",
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    "doc": "Build a checked embedding by specifying the image of this field's\ndistinguished generator.",
    "domain": "numthy",
    "file": "noether/numthy/__init__.pyi",
    "input_schema": {
      "additionalProperties": false,
      "properties": {
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          "x-python-type": "FiniteField"
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          "required": [
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          {
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        ],
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    ],
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    "canonical_name": "numthy.FiniteField.extension",
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        "signature": "(cls, value: str) -> FiniteField"
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    "summary": "Deserialize and revalidate a finite-field presentation.",
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    "summary": "Compute generator for this `FiniteField`.",
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    "canonical_name": "numthy.FiniteField.inverse",
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    "module": "noether.numthy",
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        "parameters": [
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    "summary": "Raise this value to the supplied power.",
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    "canonical_name": "numthy.PadicElement.field",
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    "summary": "Compute zero for this `PadicField`.",
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    "summary": "Compute element from qadic for this `PadicLocalField`.",
    "summary_source": "derived projection vocabulary",
    "type_id": "noether/numthy/PadicLocalField@1"
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    "canonical_name": "numthy.PadicLocalField.equation_order",
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    "doc": "",
    "domain": "numthy",
    "file": "noether/numthy/__init__.pyi",
    "input_schema": {
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      "properties": {},
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    "module": "noether.numthy",
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    "qualified_name": "PadicLocalField.equation_order",
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    "summary": "Compute equation order for this `PadicLocalField`.",
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    "type_id": "noether/numthy/PadicLocalField@1"
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    "doc": "",
    "domain": "numthy",
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    "summary": "Deserialize and revalidate a `PadicLocalField` from typed JSON.",
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    "canonical_name": "numthy.PadicLocalField.integral_element",
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    "doc": "",
    "domain": "numthy",
    "file": "noether/numthy/__init__.pyi",
    "input_schema": {
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    "name": "integral_element",
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    "qualified_name": "PadicLocalField.integral_element",
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    "signature": "(self, coefficients: list[QadicElement]) -> PadicLocalFieldElement",
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    "summary": "Compute integral element for this `PadicLocalField`.",
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    "type_id": "noether/numthy/PadicLocalField@1"
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    "canonical_name": "numthy.PadicLocalField.local_precision",
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    "doc": "",
    "domain": "numthy",
    "file": "noether/numthy/__init__.pyi",
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    "summary": "Return the local precision recorded by this `PadicLocalField`.",
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    "canonical_name": "numthy.PadicLocalField.maximal_order",
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    "doc": "Compute and certify the maximal order of this finite p-adic local field.",
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    "file": "noether/numthy/__init__.pyi",
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    "summary": "Compute and certify the maximal order of this finite p-adic local field.",
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    "canonical_name": "numthy.PadicLocalField.one",
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    "domain": "numthy",
    "file": "noether/numthy/__init__.pyi",
    "input_schema": {
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    "summary": "Compute one for this `PadicLocalField`.",
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    "canonical_name": "numthy.PadicLocalField.order",
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    "doc": "",
    "domain": "numthy",
    "file": "noether/numthy/__init__.pyi",
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      "properties": {
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      "required": [
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    "summary": "Compute order for this `PadicLocalField`.",
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    "canonical_name": "numthy.PadicLocalField.precision_restriction",
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    "domain": "numthy",
    "file": "noether/numthy/__init__.pyi",
    "input_schema": {
      "additionalProperties": false,
      "properties": {
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        "result_schema": {
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    "summary": "Compute precision restriction for this `PadicLocalField`.",
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    "canonical_name": "numthy.PadicLocalField.prime",
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    "doc": "",
    "domain": "numthy",
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      "properties": {},
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    "module": "noether.numthy",
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    "qualified_name": "PadicLocalField.prime",
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    "summary": "Return the prime recorded by this `PadicLocalField`.",
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    "canonical_name": "numthy.PadicLocalField.ramification_index",
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    "doc": "",
    "domain": "numthy",
    "file": "noether/numthy/__init__.pyi",
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    "qualified_name": "PadicLocalField.ramification_index",
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    "summary": "Return the ramification index recorded by this `PadicLocalField`.",
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    "canonical_name": "numthy.PadicLocalField.residue",
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    "domain": "numthy",
    "file": "noether/numthy/__init__.pyi",
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        "result_schema": {
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          ],
          "x-python-type": "FiniteFieldElement | None"
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    "qualified_name": "PadicLocalField.residue",
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    "signature": "(self, value: PadicLocalFieldElement) -> FiniteFieldElement | None",
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    "summary": "Compute residue for this `PadicLocalField`.",
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    "qualified_name": "PadicLocalField.residue_degree",
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    "summary": "Return the residue degree recorded by this `PadicLocalField`.",
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    "canonical_name": "numthy.PadicLocalField.residue_field",
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    "domain": "numthy",
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    "summary": "Return the residue field recorded by this `PadicLocalField`.",
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    "canonical_name": "numthy.PadicLocalField.restrict",
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    "domain": "numthy",
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      "properties": {
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      "required": [
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          },
          {
            "kind": "positional_or_keyword",
            "name": "target_precision",
            "required": true,
            "schema": {
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          }
        ],
        "result_schema": {
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        "signature": "(self, value: PadicLocalFieldElement, target_precision: PadicLocalPrecision) -> PadicLocalFieldElement"
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    ],
    "qualified_name": "PadicLocalField.restrict",
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    "result_schema": {
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    "signature": "(self, value: PadicLocalFieldElement, target_precision: PadicLocalPrecision) -> PadicLocalFieldElement",
    "signatures": [
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    "summary": "Compute restrict for this `PadicLocalField`.",
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    "domain": "numthy",
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    "canonical_name": "numthy.PadicLocalFractionalIdeal.maximal_ideal_exponent",
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    "summary": "Compute maximal ideal exponent for this `PadicLocalFractionalIdeal`.",
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    "summary": "Compute mul for this `PadicLocalFractionalIdeal`.",
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    "summary": "Compute norm valuation for this `PadicLocalFractionalIdeal`.",
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    "summary": "Return the order recorded by this `PadicLocalFractionalIdeal`.",
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    "doc": "",
    "domain": "numthy",
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    "doc": "",
    "domain": "numthy",
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    "domain": "numthy",
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    "qualified_name": "QadicRing.coefficient_frobenius",
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    "signature": "(self, exponent: int) -> QadicRingFrobenius",
    "signatures": [
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    "summary": "Compute coefficient frobenius for this `QadicRing`.",
    "summary_source": "derived projection vocabulary",
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    "summary": "Return the coefficient ring recorded by this `QadicRing`.",
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    "domain": "numthy",
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    "input_schema": {
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    "domain": "numthy",
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    "summary": "Compute generator for this `QadicRing`.",
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    "domain": "numthy",
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    "input_schema": {
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      "properties": {},
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    "module": "noether.numthy",
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    "qualified_name": "QadicRing.one",
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    "summary": "Compute one for this `QadicRing`.",
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    "domain": "numthy",
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    "input_schema": {
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    "module": "noether.numthy",
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    "qualified_name": "QadicRing.precision_restriction",
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    "summary": "Compute precision restriction for this `QadicRing`.",
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    "domain": "numthy",
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    "summary": "Return the prime recorded by this `QadicRing`.",
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    "signature": "(self, value: QadicElement) -> FiniteFieldElement",
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    "summary": "Compute residue for this `QadicRing`.",
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    "domain": "numthy",
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    "summary": "Return the residue degree recorded by this `QadicRing`.",
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    "input_schema": {
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    "qualified_name": "QadicRing.residue_field",
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    "summary": "Return the residue field recorded by this `QadicRing`.",
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    "canonical_name": "numthy.QadicRing.restrict",
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    "domain": "numthy",
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    "input_schema": {
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      "properties": {
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    "signature": "(self, value: Polynomial[QadicRing]) -> Polynomial[QadicRing]",
    "signatures": [
      "(self, value: Polynomial[QadicRing]) -> Polynomial[QadicRing]"
    ],
    "summary": "Compute apply polynomial for this `QadicRingRestriction`.",
    "summary_source": "derived projection vocabulary",
    "type_id": "noether/numthy/QadicRingRestriction@1"
  },
  {
    "canonical_name": "numthy.QadicRingRestriction.apply_power_series",
    "determinism": "deterministic",
    "doc": "",
    "domain": "numthy",
    "file": "noether/numthy/__init__.pyi",
    "input_schema": {
      "additionalProperties": false,
      "properties": {
        "value": {
          "x-python-type": "PowerSeries[QadicRing]"
        }
      },
      "required": [
        "value"
      ],
      "type": "object"
    },
    "kind": "method",
    "line": 897,
    "module": "noether.numthy",
    "name": "apply_power_series",
    "overloads": [
      {
        "input_schema": {
          "additionalProperties": false,
          "properties": {
            "value": {
              "x-python-type": "PowerSeries[QadicRing]"
            }
          },
          "required": [
            "value"
          ],
          "type": "object"
        },
        "parameters": [
          {
            "kind": "positional_or_keyword",
            "name": "value",
            "required": true,
            "schema": {
              "x-python-type": "PowerSeries[QadicRing]"
            }
          }
        ],
        "result_schema": {
          "x-python-type": "PowerSeries[QadicRing]"
        },
        "signature": "(self, value: PowerSeries[QadicRing]) -> PowerSeries[QadicRing]"
      }
    ],
    "qualified_name": "QadicRingRestriction.apply_power_series",
    "releases_gil": false,
    "result_schema": {
      "x-python-type": "PowerSeries[QadicRing]"
    },
    "signature": "(self, value: PowerSeries[QadicRing]) -> PowerSeries[QadicRing]",
    "signatures": [
      "(self, value: PowerSeries[QadicRing]) -> PowerSeries[QadicRing]"
    ],
    "summary": "Compute apply power series for this `QadicRingRestriction`.",
    "summary_source": "derived projection vocabulary",
    "type_id": "noether/numthy/QadicRingRestriction@1"
  },
  {
    "canonical_name": "numthy.QadicRingRestriction.apply_vector",
    "determinism": "deterministic",
    "doc": "",
    "domain": "numthy",
    "file": "noether/numthy/__init__.pyi",
    "input_schema": {
      "additionalProperties": false,
      "properties": {
        "value": {
          "x-python-type": "Vector[QadicRing]"
        }
      },
      "required": [
        "value"
      ],
      "type": "object"
    },
    "kind": "method",
    "line": 898,
    "module": "noether.numthy",
    "name": "apply_vector",
    "overloads": [
      {
        "input_schema": {
          "additionalProperties": false,
          "properties": {
            "value": {
              "x-python-type": "Vector[QadicRing]"
            }
          },
          "required": [
            "value"
          ],
          "type": "object"
        },
        "parameters": [
          {
            "kind": "positional_or_keyword",
            "name": "value",
            "required": true,
            "schema": {
              "x-python-type": "Vector[QadicRing]"
            }
          }
        ],
        "result_schema": {
          "x-python-type": "Vector[QadicRing]"
        },
        "signature": "(self, value: Vector[QadicRing]) -> Vector[QadicRing]"
      }
    ],
    "qualified_name": "QadicRingRestriction.apply_vector",
    "releases_gil": false,
    "result_schema": {
      "x-python-type": "Vector[QadicRing]"
    },
    "signature": "(self, value: Vector[QadicRing]) -> Vector[QadicRing]",
    "signatures": [
      "(self, value: Vector[QadicRing]) -> Vector[QadicRing]"
    ],
    "summary": "Compute apply vector for this `QadicRingRestriction`.",
    "summary_source": "derived projection vocabulary",
    "type_id": "noether/numthy/QadicRingRestriction@1"
  },
  {
    "canonical_name": "numthy.QadicRingRestriction.from_json",
    "determinism": "deterministic",
    "doc": "",
    "domain": "numthy",
    "file": "noether/numthy/__init__.pyi",
    "input_schema": {
      "additionalProperties": false,
      "properties": {
        "value": {
          "type": "string"
        }
      },
      "required": [
        "value"
      ],
      "type": "object"
    },
    "kind": "classmethod",
    "line": 903,
    "module": "noether.numthy",
    "name": "from_json",
    "overloads": [
      {
        "input_schema": {
          "additionalProperties": false,
          "properties": {
            "value": {
              "type": "string"
            }
          },
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            "value"
          ],
          "type": "object"
        },
        "parameters": [
          {
            "kind": "positional_or_keyword",
            "name": "value",
            "required": true,
            "schema": {
              "type": "string"
            }
          }
        ],
        "result_schema": {
          "x-python-type": "QadicRingRestriction"
        },
        "signature": "(cls, value: str) -> QadicRingRestriction"
      }
    ],
    "qualified_name": "QadicRingRestriction.from_json",
    "releases_gil": false,
    "result_schema": {
      "x-python-type": "QadicRingRestriction"
    },
    "signature": "(cls, value: str) -> QadicRingRestriction",
    "signatures": [
      "(cls, value: str) -> QadicRingRestriction"
    ],
    "summary": "Deserialize and revalidate a `QadicRingRestriction` from typed JSON.",
    "summary_source": "derived projection vocabulary",
    "type_id": "noether/numthy/QadicRingRestriction@1"
  },
  {
    "canonical_name": "numthy.QadicRingRestriction.source",
    "determinism": "deterministic",
    "doc": "",
    "domain": "numthy",
    "file": "noether/numthy/__init__.pyi",
    "input_schema": {
      "additionalProperties": false,
      "properties": {},
      "type": "object"
    },
    "kind": "property",
    "line": 892,
    "module": "noether.numthy",
    "name": "source",
    "overloads": [
      {
        "input_schema": {
          "additionalProperties": false,
          "properties": {},
          "type": "object"
        },
        "parameters": [],
        "result_schema": {
          "x-python-type": "QadicRing"
        },
        "signature": "(self) -> QadicRing"
      }
    ],
    "qualified_name": "QadicRingRestriction.source",
    "releases_gil": false,
    "result_schema": {
      "x-python-type": "QadicRing"
    },
    "signature": "(self) -> QadicRing",
    "signatures": [
      "(self) -> QadicRing"
    ],
    "summary": "Return the source recorded by this `QadicRingRestriction`.",
    "summary_source": "derived projection vocabulary",
    "type_id": "noether/numthy/QadicRingRestriction@1"
  },
  {
    "canonical_name": "numthy.QadicRingRestriction.target",
    "determinism": "deterministic",
    "doc": "",
    "domain": "numthy",
    "file": "noether/numthy/__init__.pyi",
    "input_schema": {
      "additionalProperties": false,
      "properties": {},
      "type": "object"
    },
    "kind": "property",
    "line": 894,
    "module": "noether.numthy",
    "name": "target",
    "overloads": [
      {
        "input_schema": {
          "additionalProperties": false,
          "properties": {},
          "type": "object"
        },
        "parameters": [],
        "result_schema": {
          "x-python-type": "QadicRing"
        },
        "signature": "(self) -> QadicRing"
      }
    ],
    "qualified_name": "QadicRingRestriction.target",
    "releases_gil": false,
    "result_schema": {
      "x-python-type": "QadicRing"
    },
    "signature": "(self) -> QadicRing",
    "signatures": [
      "(self) -> QadicRing"
    ],
    "summary": "Return the target recorded by this `QadicRingRestriction`.",
    "summary_source": "derived projection vocabulary",
    "type_id": "noether/numthy/QadicRingRestriction@1"
  },
  {
    "canonical_name": "numthy.QadicRingRestriction.to_json",
    "determinism": "deterministic",
    "doc": "",
    "domain": "numthy",
    "file": "noether/numthy/__init__.pyi",
    "input_schema": {
      "additionalProperties": false,
      "properties": {},
      "type": "object"
    },
    "kind": "method",
    "line": 901,
    "module": "noether.numthy",
    "name": "to_json",
    "overloads": [
      {
        "input_schema": {
          "additionalProperties": false,
          "properties": {},
          "type": "object"
        },
        "parameters": [],
        "result_schema": {
          "type": "string"
        },
        "signature": "(self) -> str"
      }
    ],
    "qualified_name": "QadicRingRestriction.to_json",
    "releases_gil": false,
    "result_schema": {
      "type": "string"
    },
    "signature": "(self) -> str",
    "signatures": [
      "(self) -> str"
    ],
    "summary": "Serialize this `QadicRingRestriction` to its canonical typed JSON representation.",
    "summary_source": "derived projection vocabulary",
    "type_id": "noether/numthy/QadicRingRestriction@1"
  },
  {
    "canonical_name": "numthy.QadicRingRestriction.verify",
    "determinism": "deterministic",
    "doc": "",
    "domain": "numthy",
    "file": "noether/numthy/__init__.pyi",
    "input_schema": {
      "additionalProperties": false,
      "properties": {},
      "type": "object"
    },
    "kind": "method",
    "line": 900,
    "module": "noether.numthy",
    "name": "verify",
    "overloads": [
      {
        "input_schema": {
          "additionalProperties": false,
          "properties": {},
          "type": "object"
        },
        "parameters": [],
        "result_schema": {
          "type": "null"
        },
        "signature": "(self) -> None"
      }
    ],
    "qualified_name": "QadicRingRestriction.verify",
    "releases_gil": false,
    "result_schema": {
      "type": "null"
    },
    "signature": "(self) -> None",
    "signatures": [
      "(self) -> None"
    ],
    "summary": "Replay and verify the mathematical evidence carried by this `QadicRingRestriction`.",
    "summary_source": "derived projection vocabulary",
    "type_id": "noether/numthy/QadicRingRestriction@1"
  },
  {
    "canonical_name": "numthy.cyclotomic_polynomial",
    "determinism": "deterministic",
    "doc": "Return the positive-index cyclotomic polynomial in `Z[x]`.",
    "domain": "numthy",
    "file": "noether/numthy/__init__.pyi",
    "input_schema": {
      "additionalProperties": false,
      "properties": {
        "index": {
          "type": "integer"
        }
      },
      "required": [
        "index"
      ],
      "type": "object"
    },
    "kind": "function",
    "line": 88,
    "module": "noether.numthy",
    "name": "cyclotomic_polynomial",
    "overloads": [
      {
        "input_schema": {
          "additionalProperties": false,
          "properties": {
            "index": {
              "type": "integer"
            }
          },
          "required": [
            "index"
          ],
          "type": "object"
        },
        "parameters": [
          {
            "kind": "positional_or_keyword",
            "name": "index",
            "required": true,
            "schema": {
              "type": "integer"
            }
          }
        ],
        "result_schema": {
          "x-python-type": "Polynomial[IntegerRing]"
        },
        "signature": "(index: int) -> Polynomial[IntegerRing]"
      }
    ],
    "qualified_name": "cyclotomic_polynomial",
    "releases_gil": true,
    "result_schema": {
      "x-python-type": "Polynomial[IntegerRing]"
    },
    "signature": "(index: int) -> Polynomial[IntegerRing]",
    "signatures": [
      "(index: int) -> Polynomial[IntegerRing]"
    ],
    "summary": "Return the positive-index cyclotomic polynomial in `Z[x]`.",
    "summary_source": "public docstring"
  },
  {
    "algorithms": [
      "proved_integer_factorization"
    ],
    "backends": [
      "flint"
    ],
    "canonical_name": "numthy.factor",
    "determinism": "deterministic",
    "doc": "Return the complete proved factorization of a nonzero Python integer.",
    "domain": "numthy",
    "evidence": [
      "IntegerFactorization.verify"
    ],
    "example": "from noether.numthy import factor\nresult = factor(-360)\nassert result.factors == [(2, 3), (3, 2), (5, 1)]\nassert result.verify(-360)",
    "file": "noether/numthy/__init__.pyi",
    "input_schema": {
      "additionalProperties": false,
      "properties": {
        "value": {
          "type": "integer"
        }
      },
      "required": [
        "value"
      ],
      "type": "object"
    },
    "kind": "function",
    "line": 83,
    "module": "noether.numthy",
    "name": "factor",
    "overloads": [
      {
        "input_schema": {
          "additionalProperties": false,
          "properties": {
            "value": {
              "type": "integer"
            }
          },
          "required": [
            "value"
          ],
          "type": "object"
        },
        "parameters": [
          {
            "kind": "positional_or_keyword",
            "name": "value",
            "required": true,
            "schema": {
              "type": "integer"
            }
          }
        ],
        "result_schema": {
          "x-python-type": "IntegerFactorization"
        },
        "signature": "(value: int) -> IntegerFactorization"
      }
    ],
    "qualified_name": "factor",
    "releases_gil": true,
    "result_schema": {
      "x-python-type": "IntegerFactorization"
    },
    "signature": "(value: int) -> IntegerFactorization",
    "signatures": [
      "(value: int) -> IntegerFactorization"
    ],
    "summary": "Return the complete proved factorization of a nonzero integer.",
    "summary_source": "editorial metadata"
  },
  {
    "canonical_name": "numthy.is_prime",
    "determinism": "deterministic",
    "doc": "Test primality with FLINT's deterministic primality prover.",
    "domain": "numthy",
    "file": "noether/numthy/__init__.pyi",
    "input_schema": {
      "additionalProperties": false,
      "properties": {
        "value": {
          "type": "integer"
        }
      },
      "required": [
        "value"
      ],
      "type": "object"
    },
    "kind": "function",
    "line": 87,
    "module": "noether.numthy",
    "name": "is_prime",
    "overloads": [
      {
        "input_schema": {
          "additionalProperties": false,
          "properties": {
            "value": {
              "type": "integer"
            }
          },
          "required": [
            "value"
          ],
          "type": "object"
        },
        "parameters": [
          {
            "kind": "positional_or_keyword",
            "name": "value",
            "required": true,
            "schema": {
              "type": "integer"
            }
          }
        ],
        "result_schema": {
          "type": "boolean"
        },
        "signature": "(value: int) -> bool"
      }
    ],
    "qualified_name": "is_prime",
    "releases_gil": true,
    "result_schema": {
      "type": "boolean"
    },
    "signature": "(value: int) -> bool",
    "signatures": [
      "(value: int) -> bool"
    ],
    "summary": "Test primality with FLINT's deterministic primality prover.",
    "summary_source": "public docstring"
  },
  {
    "canonical_name": "numthy.partial_factor",
    "determinism": "deterministic",
    "doc": "Return deterministic partial factorization within explicit work bounds.",
    "domain": "numthy",
    "file": "noether/numthy/__init__.pyi",
    "input_schema": {
      "additionalProperties": false,
      "properties": {
        "primality_proof_bits": {
          "type": "integer"
        },
        "trial_bound": {
          "type": "integer"
        },
        "value": {
          "type": "integer"
        }
      },
      "required": [
        "value"
      ],
      "type": "object"
    },
    "kind": "function",
    "line": 84,
    "module": "noether.numthy",
    "name": "partial_factor",
    "overloads": [
      {
        "input_schema": {
          "additionalProperties": false,
          "properties": {
            "primality_proof_bits": {
              "type": "integer"
            },
            "trial_bound": {
              "type": "integer"
            },
            "value": {
              "type": "integer"
            }
          },
          "required": [
            "value"
          ],
          "type": "object"
        },
        "parameters": [
          {
            "kind": "positional_or_keyword",
            "name": "value",
            "required": true,
            "schema": {
              "type": "integer"
            }
          },
          {
            "default": "10000",
            "kind": "keyword_only",
            "name": "trial_bound",
            "required": false,
            "schema": {
              "type": "integer"
            }
          },
          {
            "default": "256",
            "kind": "keyword_only",
            "name": "primality_proof_bits",
            "required": false,
            "schema": {
              "type": "integer"
            }
          }
        ],
        "result_schema": {
          "x-python-type": "PartialIntegerFactorization"
        },
        "signature": "(value: int, *, trial_bound: int=10000, primality_proof_bits: int=256) -> PartialIntegerFactorization"
      }
    ],
    "qualified_name": "partial_factor",
    "releases_gil": true,
    "result_schema": {
      "x-python-type": "PartialIntegerFactorization"
    },
    "signature": "(value: int, *, trial_bound: int=10000, primality_proof_bits: int=256) -> PartialIntegerFactorization",
    "signatures": [
      "(value: int, *, trial_bound: int=10000, primality_proof_bits: int=256) -> PartialIntegerFactorization"
    ],
    "summary": "Return a deterministic partial factorization within explicit work bounds.",
    "summary_source": "editorial metadata"
  },
  {
    "canonical_name": "topology.AdmissibleCohomologyOperation",
    "determinism": "deterministic",
    "doc": "A checked admissible word in the mod-p Steenrod algebra.",
    "domain": "topology",
    "file": "noether/topology/__init__.pyi",
    "input_schema": {},
    "kind": "class",
    "line": 27,
    "module": "noether.topology",
    "name": "AdmissibleCohomologyOperation",
    "overloads": [],
    "qualified_name": "AdmissibleCohomologyOperation",
    "releases_gil": false,
    "result_schema": {},
    "signature": "",
    "signatures": [],
    "summary": "A checked admissible word in the mod-p Steenrod algebra.",
    "summary_source": "public docstring"
  },
  {
    "canonical_name": "topology.AdmissibleCohomologyOperation.bockstein_reduced_power",
    "determinism": "deterministic",
    "doc": "",
    "domain": "topology",
    "file": "noether/topology/__init__.pyi",
    "input_schema": {
      "additionalProperties": false,
      "properties": {
        "power": {
          "type": "integer"
        },
        "prime": {
          "type": "integer"
        }
      },
      "required": [
        "prime",
        "power"
      ],
      "type": "object"
    },
    "kind": "staticmethod",
    "line": 41,
    "module": "noether.topology",
    "name": "bockstein_reduced_power",
    "overloads": [
      {
        "input_schema": {
          "additionalProperties": false,
          "properties": {
            "power": {
              "type": "integer"
            },
            "prime": {
              "type": "integer"
            }
          },
          "required": [
            "prime",
            "power"
          ],
          "type": "object"
        },
        "parameters": [
          {
            "kind": "positional_or_keyword",
            "name": "prime",
            "required": true,
            "schema": {
              "type": "integer"
            }
          },
          {
            "kind": "positional_or_keyword",
            "name": "power",
            "required": true,
            "schema": {
              "type": "integer"
            }
          }
        ],
        "result_schema": {
          "x-python-type": "AdmissibleCohomologyOperation"
        },
        "signature": "(prime: int, power: int) -> AdmissibleCohomologyOperation"
      }
    ],
    "qualified_name": "AdmissibleCohomologyOperation.bockstein_reduced_power",
    "releases_gil": false,
    "result_schema": {
      "x-python-type": "AdmissibleCohomologyOperation"
    },
    "signature": "(prime: int, power: int) -> AdmissibleCohomologyOperation",
    "signatures": [
      "(prime: int, power: int) -> AdmissibleCohomologyOperation"
    ],
    "summary": "Construct a `AdmissibleCohomologyOperation` using bockstein reduced power.",
    "summary_source": "derived projection vocabulary"
  },
  {
    "canonical_name": "topology.AdmissibleCohomologyOperation.bocksteins",
    "determinism": "deterministic",
    "doc": "",
    "domain": "topology",
    "file": "noether/topology/__init__.pyi",
    "input_schema": {
      "additionalProperties": false,
      "properties": {},
      "type": "object"
    },
    "kind": "property",
    "line": 47,
    "module": "noether.topology",
    "name": "bocksteins",
    "overloads": [
      {
        "input_schema": {
          "additionalProperties": false,
          "properties": {},
          "type": "object"
        },
        "parameters": [],
        "result_schema": {
          "anyOf": [
            {
              "items": {
                "type": "boolean"
              },
              "type": "array",
              "x-python-type": "list[bool]"
            },
            {
              "type": "null"
            }
          ],
          "x-python-type": "list[bool] | None"
        },
        "signature": "(self) -> list[bool] | None"
      }
    ],
    "qualified_name": "AdmissibleCohomologyOperation.bocksteins",
    "releases_gil": false,
    "result_schema": {
      "anyOf": [
        {
          "items": {
            "type": "boolean"
          },
          "type": "array",
          "x-python-type": "list[bool]"
        },
        {
          "type": "null"
        }
      ],
      "x-python-type": "list[bool] | None"
    },
    "signature": "(self) -> list[bool] | None",
    "signatures": [
      "(self) -> list[bool] | None"
    ],
    "summary": "Return the bocksteins recorded by this `AdmissibleCohomologyOperation`.",
    "summary_source": "derived projection vocabulary"
  },
  {
    "canonical_name": "topology.AdmissibleCohomologyOperation.degree_shift",
    "determinism": "deterministic",
    "doc": "",
    "domain": "topology",
    "file": "noether/topology/__init__.pyi",
    "input_schema": {
      "additionalProperties": false,
      "properties": {},
      "type": "object"
    },
    "kind": "property",
    "line": 52,
    "module": "noether.topology",
    "name": "degree_shift",
    "overloads": [
      {
        "input_schema": {
          "additionalProperties": false,
          "properties": {},
          "type": "object"
        },
        "parameters": [],
        "result_schema": {
          "type": "integer"
        },
        "signature": "(self) -> int"
      }
    ],
    "qualified_name": "AdmissibleCohomologyOperation.degree_shift",
    "releases_gil": false,
    "result_schema": {
      "type": "integer"
    },
    "signature": "(self) -> int",
    "signatures": [
      "(self) -> int"
    ],
    "summary": "Return the degree shift recorded by this `AdmissibleCohomologyOperation`.",
    "summary_source": "derived projection vocabulary"
  },
  {
    "canonical_name": "topology.AdmissibleCohomologyOperation.display_name",
    "determinism": "deterministic",
    "doc": "",
    "domain": "topology",
    "file": "noether/topology/__init__.pyi",
    "input_schema": {
      "additionalProperties": false,
      "properties": {},
      "type": "object"
    },
    "kind": "property",
    "line": 56,
    "module": "noether.topology",
    "name": "display_name",
    "overloads": [
      {
        "input_schema": {
          "additionalProperties": false,
          "properties": {},
          "type": "object"
        },
        "parameters": [],
        "result_schema": {
          "type": "string"
        },
        "signature": "(self) -> str"
      }
    ],
    "qualified_name": "AdmissibleCohomologyOperation.display_name",
    "releases_gil": false,
    "result_schema": {
      "type": "string"
    },
    "signature": "(self) -> str",
    "signatures": [
      "(self) -> str"
    ],
    "summary": "Return the display name recorded by this `AdmissibleCohomologyOperation`.",
    "summary_source": "derived projection vocabulary"
  },
  {
    "canonical_name": "topology.AdmissibleCohomologyOperation.excess",
    "determinism": "deterministic",
    "doc": "",
    "domain": "topology",
    "file": "noether/topology/__init__.pyi",
    "input_schema": {
      "additionalProperties": false,
      "properties": {},
      "type": "object"
    },
    "kind": "property",
    "line": 54,
    "module": "noether.topology",
    "name": "excess",
    "overloads": [
      {
        "input_schema": {
          "additionalProperties": false,
          "properties": {},
          "type": "object"
        },
        "parameters": [],
        "result_schema": {
          "type": "integer"
        },
        "signature": "(self) -> int"
      }
    ],
    "qualified_name": "AdmissibleCohomologyOperation.excess",
    "releases_gil": false,
    "result_schema": {
      "type": "integer"
    },
    "signature": "(self) -> int",
    "signatures": [
      "(self) -> int"
    ],
    "summary": "Return the excess recorded by this `AdmissibleCohomologyOperation`.",
    "summary_source": "derived projection vocabulary"
  },
  {
    "canonical_name": "topology.AdmissibleCohomologyOperation.from_json",
    "determinism": "deterministic",
    "doc": "",
    "domain": "topology",
    "file": "noether/topology/__init__.pyi",
    "input_schema": {
      "additionalProperties": false,
      "properties": {
        "value": {
          "type": "string"
        }
      },
      "required": [
        "value"
      ],
      "type": "object"
    },
    "kind": "classmethod",
    "line": 59,
    "module": "noether.topology",
    "name": "from_json",
    "overloads": [
      {
        "input_schema": {
          "additionalProperties": false,
          "properties": {
            "value": {
              "type": "string"
            }
          },
          "required": [
            "value"
          ],
          "type": "object"
        },
        "parameters": [
          {
            "kind": "positional_or_keyword",
            "name": "value",
            "required": true,
            "schema": {
              "type": "string"
            }
          }
        ],
        "result_schema": {
          "x-python-type": "AdmissibleCohomologyOperation"
        },
        "signature": "(cls, value: str) -> AdmissibleCohomologyOperation"
      }
    ],
    "qualified_name": "AdmissibleCohomologyOperation.from_json",
    "releases_gil": false,
    "result_schema": {
      "x-python-type": "AdmissibleCohomologyOperation"
    },
    "signature": "(cls, value: str) -> AdmissibleCohomologyOperation",
    "signatures": [
      "(cls, value: str) -> AdmissibleCohomologyOperation"
    ],
    "summary": "Deserialize and revalidate a `AdmissibleCohomologyOperation` from typed JSON.",
    "summary_source": "derived projection vocabulary"
  },
  {
    "canonical_name": "topology.AdmissibleCohomologyOperation.identity",
    "determinism": "deterministic",
    "doc": "",
    "domain": "topology",
    "file": "noether/topology/__init__.pyi",
    "input_schema": {
      "additionalProperties": false,
      "properties": {
        "prime": {
          "type": "integer"
        }
      },
      "required": [
        "prime"
      ],
      "type": "object"
    },
    "kind": "staticmethod",
    "line": 35,
    "module": "noether.topology",
    "name": "identity",
    "overloads": [
      {
        "input_schema": {
          "additionalProperties": false,
          "properties": {
            "prime": {
              "type": "integer"
            }
          },
          "required": [
            "prime"
          ],
          "type": "object"
        },
        "parameters": [
          {
            "kind": "positional_or_keyword",
            "name": "prime",
            "required": true,
            "schema": {
              "type": "integer"
            }
          }
        ],
        "result_schema": {
          "x-python-type": "AdmissibleCohomologyOperation"
        },
        "signature": "(prime: int) -> AdmissibleCohomologyOperation"
      }
    ],
    "qualified_name": "AdmissibleCohomologyOperation.identity",
    "releases_gil": false,
    "result_schema": {
      "x-python-type": "AdmissibleCohomologyOperation"
    },
    "signature": "(prime: int) -> AdmissibleCohomologyOperation",
    "signatures": [
      "(prime: int) -> AdmissibleCohomologyOperation"
    ],
    "summary": "Construct a `AdmissibleCohomologyOperation` using identity.",
    "summary_source": "derived projection vocabulary"
  },
  {
    "canonical_name": "topology.AdmissibleCohomologyOperation.is_identity",
    "determinism": "deterministic",
    "doc": "",
    "domain": "topology",
    "file": "noether/topology/__init__.pyi",
    "input_schema": {
      "additionalProperties": false,
      "properties": {},
      "type": "object"
    },
    "kind": "method",
    "line": 50,
    "module": "noether.topology",
    "name": "is_identity",
    "overloads": [
      {
        "input_schema": {
          "additionalProperties": false,
          "properties": {},
          "type": "object"
        },
        "parameters": [],
        "result_schema": {
          "type": "boolean"
        },
        "signature": "(self) -> bool"
      }
    ],
    "qualified_name": "AdmissibleCohomologyOperation.is_identity",
    "releases_gil": false,
    "result_schema": {
      "type": "boolean"
    },
    "signature": "(self) -> bool",
    "signatures": [
      "(self) -> bool"
    ],
    "summary": "Test whether this `AdmissibleCohomologyOperation` is identity.",
    "summary_source": "derived projection vocabulary"
  },
  {
    "canonical_name": "topology.AdmissibleCohomologyOperation.mod_two",
    "determinism": "deterministic",
    "doc": "",
    "domain": "topology",
    "file": "noether/topology/__init__.pyi",
    "input_schema": {
      "additionalProperties": false,
      "properties": {
        "sequence": {
          "items": {
            "type": "integer"
          },
          "type": "array",
          "x-python-type": "list[int]"
        }
      },
      "required": [
        "sequence"
      ],
      "type": "object"
    },
    "kind": "staticmethod",
    "line": 29,
    "module": "noether.topology",
    "name": "mod_two",
    "overloads": [
      {
        "input_schema": {
          "additionalProperties": false,
          "properties": {
            "sequence": {
              "items": {
                "type": "integer"
              },
              "type": "array",
              "x-python-type": "list[int]"
            }
          },
          "required": [
            "sequence"
          ],
          "type": "object"
        },
        "parameters": [
          {
            "kind": "positional_or_keyword",
            "name": "sequence",
            "required": true,
            "schema": {
              "items": {
                "type": "integer"
              },
              "type": "array",
              "x-python-type": "list[int]"
            }
          }
        ],
        "result_schema": {
          "x-python-type": "AdmissibleCohomologyOperation"
        },
        "signature": "(sequence: list[int]) -> AdmissibleCohomologyOperation"
      }
    ],
    "qualified_name": "AdmissibleCohomologyOperation.mod_two",
    "releases_gil": false,
    "result_schema": {
      "x-python-type": "AdmissibleCohomologyOperation"
    },
    "signature": "(sequence: list[int]) -> AdmissibleCohomologyOperation",
    "signatures": [
      "(sequence: list[int]) -> AdmissibleCohomologyOperation"
    ],
    "summary": "Construct a `AdmissibleCohomologyOperation` using mod two.",
    "summary_source": "derived projection vocabulary"
  },
  {
    "canonical_name": "topology.AdmissibleCohomologyOperation.odd_primary",
    "determinism": "deterministic",
    "doc": "",
    "domain": "topology",
    "file": "noether/topology/__init__.pyi",
    "input_schema": {
      "additionalProperties": false,
      "properties": {
        "bocksteins": {
          "items": {
            "type": "boolean"
          },
          "type": "array",
          "x-python-type": "list[bool]"
        },
        "prime": {
          "type": "integer"
        },
        "reduced_powers": {
          "items": {
            "type": "integer"
          },
          "type": "array",
          "x-python-type": "list[int]"
        }
      },
      "required": [
        "prime",
        "bocksteins",
        "reduced_powers"
      ],
      "type": "object"
    },
    "kind": "staticmethod",
    "line": 31,
    "module": "noether.topology",
    "name": "odd_primary",
    "overloads": [
      {
        "input_schema": {
          "additionalProperties": false,
          "properties": {
            "bocksteins": {
              "items": {
                "type": "boolean"
              },
              "type": "array",
              "x-python-type": "list[bool]"
            },
            "prime": {
              "type": "integer"
            },
            "reduced_powers": {
              "items": {
                "type": "integer"
              },
              "type": "array",
              "x-python-type": "list[int]"
            }
          },
          "required": [
            "prime",
            "bocksteins",
            "reduced_powers"
          ],
          "type": "object"
        },
        "parameters": [
          {
            "kind": "positional_or_keyword",
            "name": "prime",
            "required": true,
            "schema": {
              "type": "integer"
            }
          },
          {
            "kind": "positional_or_keyword",
            "name": "bocksteins",
            "required": true,
            "schema": {
              "items": {
                "type": "boolean"
              },
              "type": "array",
              "x-python-type": "list[bool]"
            }
          },
          {
            "kind": "positional_or_keyword",
            "name": "reduced_powers",
            "required": true,
            "schema": {
              "items": {
                "type": "integer"
              },
              "type": "array",
              "x-python-type": "list[int]"
            }
          }
        ],
        "result_schema": {
          "x-python-type": "AdmissibleCohomologyOperation"
        },
        "signature": "(prime: int, bocksteins: list[bool], reduced_powers: list[int]) -> AdmissibleCohomologyOperation"
      }
    ],
    "qualified_name": "AdmissibleCohomologyOperation.odd_primary",
    "releases_gil": false,
    "result_schema": {
      "x-python-type": "AdmissibleCohomologyOperation"
    },
    "signature": "(prime: int, bocksteins: list[bool], reduced_powers: list[int]) -> AdmissibleCohomologyOperation",
    "signatures": [
      "(prime: int, bocksteins: list[bool], reduced_powers: list[int]) -> AdmissibleCohomologyOperation"
    ],
    "summary": "Construct a `AdmissibleCohomologyOperation` using odd primary.",
    "summary_source": "derived projection vocabulary"
  },
  {
    "canonical_name": "topology.AdmissibleCohomologyOperation.prime",
    "determinism": "deterministic",
    "doc": "",
    "domain": "topology",
    "file": "noether/topology/__init__.pyi",
    "input_schema": {
      "additionalProperties": false,
      "properties": {},
      "type": "object"
    },
    "kind": "property",
    "line": 43,
    "module": "noether.topology",
    "name": "prime",
    "overloads": [
      {
        "input_schema": {
          "additionalProperties": false,
          "properties": {},
          "type": "object"
        },
        "parameters": [],
        "result_schema": {
          "type": "integer"
        },
        "signature": "(self) -> int"
      }
    ],
    "qualified_name": "AdmissibleCohomologyOperation.prime",
    "releases_gil": false,
    "result_schema": {
      "type": "integer"
    },
    "signature": "(self) -> int",
    "signatures": [
      "(self) -> int"
    ],
    "summary": "Return the prime recorded by this `AdmissibleCohomologyOperation`.",
    "summary_source": "derived projection vocabulary"
  },
  {
    "canonical_name": "topology.AdmissibleCohomologyOperation.reduced_power",
    "determinism": "deterministic",
    "doc": "",
    "domain": "topology",
    "file": "noether/topology/__init__.pyi",
    "input_schema": {
      "additionalProperties": false,
      "properties": {
        "power": {
          "type": "integer"
        },
        "prime": {
          "type": "integer"
        }
      },
      "required": [
        "prime",
        "power"
      ],
      "type": "object"
    },
    "kind": "staticmethod",
    "line": 39,
    "module": "noether.topology",
    "name": "reduced_power",
    "overloads": [
      {
        "input_schema": {
          "additionalProperties": false,
          "properties": {
            "power": {
              "type": "integer"
            },
            "prime": {
              "type": "integer"
            }
          },
          "required": [
            "prime",
            "power"
          ],
          "type": "object"
        },
        "parameters": [
          {
            "kind": "positional_or_keyword",
            "name": "prime",
            "required": true,
            "schema": {
              "type": "integer"
            }
          },
          {
            "kind": "positional_or_keyword",
            "name": "power",
            "required": true,
            "schema": {
              "type": "integer"
            }
          }
        ],
        "result_schema": {
          "x-python-type": "AdmissibleCohomologyOperation"
        },
        "signature": "(prime: int, power: int) -> AdmissibleCohomologyOperation"
      }
    ],
    "qualified_name": "AdmissibleCohomologyOperation.reduced_power",
    "releases_gil": false,
    "result_schema": {
      "x-python-type": "AdmissibleCohomologyOperation"
    },
    "signature": "(prime: int, power: int) -> AdmissibleCohomologyOperation",
    "signatures": [
      "(prime: int, power: int) -> AdmissibleCohomologyOperation"
    ],
    "summary": "Construct a `AdmissibleCohomologyOperation` using reduced power.",
    "summary_source": "derived projection vocabulary"
  },
  {
    "canonical_name": "topology.AdmissibleCohomologyOperation.reduced_powers",
    "determinism": "deterministic",
    "doc": "",
    "domain": "topology",
    "file": "noether/topology/__init__.pyi",
    "input_schema": {
      "additionalProperties": false,
      "properties": {},
      "type": "object"
    },
    "kind": "property",
    "line": 49,
    "module": "noether.topology",
    "name": "reduced_powers",
    "overloads": [
      {
        "input_schema": {
          "additionalProperties": false,
          "properties": {},
          "type": "object"
        },
        "parameters": [],
        "result_schema": {
          "anyOf": [
            {
              "items": {
                "type": "integer"
              },
              "type": "array",
              "x-python-type": "list[int]"
            },
            {
              "type": "null"
            }
          ],
          "x-python-type": "list[int] | None"
        },
        "signature": "(self) -> list[int] | None"
      }
    ],
    "qualified_name": "AdmissibleCohomologyOperation.reduced_powers",
    "releases_gil": false,
    "result_schema": {
      "anyOf": [
        {
          "items": {
            "type": "integer"
          },
          "type": "array",
          "x-python-type": "list[int]"
        },
        {
          "type": "null"
        }
      ],
      "x-python-type": "list[int] | None"
    },
    "signature": "(self) -> list[int] | None",
    "signatures": [
      "(self) -> list[int] | None"
    ],
    "summary": "Return the reduced powers recorded by this `AdmissibleCohomologyOperation`.",
    "summary_source": "derived projection vocabulary"
  },
  {
    "canonical_name": "topology.AdmissibleCohomologyOperation.square_sequence",
    "determinism": "deterministic",
    "doc": "",
    "domain": "topology",
    "file": "noether/topology/__init__.pyi",
    "input_schema": {
      "additionalProperties": false,
      "properties": {},
      "type": "object"
    },
    "kind": "property",
    "line": 45,
    "module": "noether.topology",
    "name": "square_sequence",
    "overloads": [
      {
        "input_schema": {
          "additionalProperties": false,
          "properties": {},
          "type": "object"
        },
        "parameters": [],
        "result_schema": {
          "anyOf": [
            {
              "items": {
                "type": "integer"
              },
              "type": "array",
              "x-python-type": "list[int]"
            },
            {
              "type": "null"
            }
          ],
          "x-python-type": "list[int] | None"
        },
        "signature": "(self) -> list[int] | None"
      }
    ],
    "qualified_name": "AdmissibleCohomologyOperation.square_sequence",
    "releases_gil": false,
    "result_schema": {
      "anyOf": [
        {
          "items": {
            "type": "integer"
          },
          "type": "array",
          "x-python-type": "list[int]"
        },
        {
          "type": "null"
        }
      ],
      "x-python-type": "list[int] | None"
    },
    "signature": "(self) -> list[int] | None",
    "signatures": [
      "(self) -> list[int] | None"
    ],
    "summary": "Return the square sequence recorded by this `AdmissibleCohomologyOperation`.",
    "summary_source": "derived projection vocabulary"
  },
  {
    "canonical_name": "topology.AdmissibleCohomologyOperation.steenrod_square",
    "determinism": "deterministic",
    "doc": "",
    "domain": "topology",
    "file": "noether/topology/__init__.pyi",
    "input_schema": {
      "additionalProperties": false,
      "properties": {
        "square": {
          "type": "integer"
        }
      },
      "required": [
        "square"
      ],
      "type": "object"
    },
    "kind": "staticmethod",
    "line": 37,
    "module": "noether.topology",
    "name": "steenrod_square",
    "overloads": [
      {
        "input_schema": {
          "additionalProperties": false,
          "properties": {
            "square": {
              "type": "integer"
            }
          },
          "required": [
            "square"
          ],
          "type": "object"
        },
        "parameters": [
          {
            "kind": "positional_or_keyword",
            "name": "square",
            "required": true,
            "schema": {
              "type": "integer"
            }
          }
        ],
        "result_schema": {
          "x-python-type": "AdmissibleCohomologyOperation"
        },
        "signature": "(square: int) -> AdmissibleCohomologyOperation"
      }
    ],
    "qualified_name": "AdmissibleCohomologyOperation.steenrod_square",
    "releases_gil": false,
    "result_schema": {
      "x-python-type": "AdmissibleCohomologyOperation"
    },
    "signature": "(square: int) -> AdmissibleCohomologyOperation",
    "signatures": [
      "(square: int) -> AdmissibleCohomologyOperation"
    ],
    "summary": "Construct a `AdmissibleCohomologyOperation` using steenrod square.",
    "summary_source": "derived projection vocabulary"
  },
  {
    "canonical_name": "topology.AdmissibleCohomologyOperation.to_json",
    "determinism": "deterministic",
    "doc": "",
    "domain": "topology",
    "file": "noether/topology/__init__.pyi",
    "input_schema": {
      "additionalProperties": false,
      "properties": {},
      "type": "object"
    },
    "kind": "method",
    "line": 57,
    "module": "noether.topology",
    "name": "to_json",
    "overloads": [
      {
        "input_schema": {
          "additionalProperties": false,
          "properties": {},
          "type": "object"
        },
        "parameters": [],
        "result_schema": {
          "type": "string"
        },
        "signature": "(self) -> str"
      }
    ],
    "qualified_name": "AdmissibleCohomologyOperation.to_json",
    "releases_gil": false,
    "result_schema": {
      "type": "string"
    },
    "signature": "(self) -> str",
    "signatures": [
      "(self) -> str"
    ],
    "summary": "Serialize this `AdmissibleCohomologyOperation` to its canonical typed JSON representation.",
    "summary_source": "derived projection vocabulary"
  },
  {
    "canonical_name": "topology.Barcode",
    "determinism": "deterministic",
    "doc": "Persistent-homology intervals carrying their exact coefficient field.",
    "domain": "topology",
    "file": "noether/topology/__init__.pyi",
    "input_schema": {
      "additionalProperties": false,
      "properties": {
        "bars": {
          "items": {
            "maxItems": 3,
            "minItems": 3,
            "prefixItems": [
              {
                "type": "integer"
              },
              {
                "type": "number"
              },
              {
                "anyOf": [
                  {
                    "type": "number"
                  },
                  {
                    "type": "null"
                  }
                ],
                "x-python-type": "float | None"
              }
            ],
            "type": "array",
            "x-python-type": "tuple[int, float, float | None]"
          },
          "type": "array",
          "x-python-type": "list[tuple[int, float, float | None]]"
        },
        "ring": {
          "anyOf": [
            {
              "x-python-type": "RationalField"
            },
            {
              "x-python-type": "FiniteField"
            }
          ],
          "x-python-type": "RationalField | FiniteField"
        }
      },
      "required": [
        "bars",
        "ring"
      ],
      "type": "object"
    },
    "kind": "class",
    "line": 743,
    "module": "noether.topology",
    "name": "Barcode",
    "overloads": [
      {
        "input_schema": {
          "additionalProperties": false,
          "properties": {
            "bars": {
              "items": {
                "maxItems": 3,
                "minItems": 3,
                "prefixItems": [
                  {
                    "type": "integer"
                  },
                  {
                    "type": "number"
                  },
                  {
                    "anyOf": [
                      {
                        "type": "number"
                      },
                      {
                        "type": "null"
                      }
                    ],
                    "x-python-type": "float | None"
                  }
                ],
                "type": "array",
                "x-python-type": "tuple[int, float, float | None]"
              },
              "type": "array",
              "x-python-type": "list[tuple[int, float, float | None]]"
            },
            "ring": {
              "anyOf": [
                {
                  "x-python-type": "RationalField"
                },
                {
                  "x-python-type": "FiniteField"
                }
              ],
              "x-python-type": "RationalField | FiniteField"
            }
          },
          "required": [
            "bars",
            "ring"
          ],
          "type": "object"
        },
        "parameters": [
          {
            "kind": "positional_or_keyword",
            "name": "bars",
            "required": true,
            "schema": {
              "items": {
                "maxItems": 3,
                "minItems": 3,
                "prefixItems": [
                  {
                    "type": "integer"
                  },
                  {
                    "type": "number"
                  },
                  {
                    "anyOf": [
                      {
                        "type": "number"
                      },
                      {
                        "type": "null"
                      }
                    ],
                    "x-python-type": "float | None"
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    "releases_gil": false,
    "result_schema": {
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    "signature": "(self, other: Barcode, *, power: float=2.0) -> float",
    "signatures": [
      "(self, other: Barcode, *, power: float=2.0) -> float"
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    "summary": "Wasserstein distance, with dimensions and coefficient context checked.",
    "summary_source": "public docstring",
    "type_id": "noether/topology/Barcode@1"
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  {
    "canonical_name": "topology.DominatedVertex",
    "determinism": "deterministic",
    "doc": "One replayable dominated-vertex deletion.",
    "domain": "topology",
    "file": "noether/topology/__init__.pyi",
    "input_schema": {
      "additionalProperties": false,
      "properties": {
        "dominator": {
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        "vertex": {
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      "required": [
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        "dominator"
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      "type": "object"
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    "kind": "class",
    "line": 323,
    "module": "noether.topology",
    "name": "DominatedVertex",
    "overloads": [
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        "input_schema": {
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          "properties": {
            "dominator": {
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            "vertex": {
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          },
          "required": [
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            "dominator"
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          "type": "object"
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        "parameters": [
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            "kind": "positional_or_keyword",
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            "required": true,
            "schema": {
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          },
          {
            "kind": "positional_or_keyword",
            "name": "dominator",
            "required": true,
            "schema": {
              "type": "integer"
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          }
        ],
        "result_schema": {
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        "signature": "(self, vertex: int, dominator: int) -> None"
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    "qualified_name": "DominatedVertex",
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    "result_schema": {
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    "signature": "(self, vertex: int, dominator: int) -> None",
    "signatures": [
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    "summary": "One replayable dominated-vertex deletion.",
    "summary_source": "public docstring"
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    "canonical_name": "topology.DominatedVertex.dominator",
    "determinism": "deterministic",
    "doc": "",
    "domain": "topology",
    "file": "noether/topology/__init__.pyi",
    "input_schema": {
      "additionalProperties": false,
      "properties": {},
      "type": "object"
    },
    "kind": "property",
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    "module": "noether.topology",
    "name": "dominator",
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          "properties": {},
          "type": "object"
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    "qualified_name": "DominatedVertex.dominator",
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    "result_schema": {
      "type": "integer"
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    "signature": "(self) -> int",
    "signatures": [
      "(self) -> int"
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    "summary": "Return the dominator recorded by this `DominatedVertex`.",
    "summary_source": "derived projection vocabulary"
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  {
    "canonical_name": "topology.DominatedVertex.vertex",
    "determinism": "deterministic",
    "doc": "",
    "domain": "topology",
    "file": "noether/topology/__init__.pyi",
    "input_schema": {
      "additionalProperties": false,
      "properties": {},
      "type": "object"
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    "kind": "property",
    "line": 326,
    "module": "noether.topology",
    "name": "vertex",
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          "properties": {},
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        "signature": "(self) -> int"
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    "qualified_name": "DominatedVertex.vertex",
    "releases_gil": false,
    "result_schema": {
      "type": "integer"
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    "signature": "(self) -> int",
    "signatures": [
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    "summary": "Return the vertex recorded by this `DominatedVertex`.",
    "summary_source": "derived projection vocabulary"
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  {
    "canonical_name": "topology.EilenbergMacLaneCohomologyGenerator",
    "determinism": "deterministic",
    "doc": "One named generator in a complete modular cohomology presentation.",
    "domain": "topology",
    "file": "noether/topology/__init__.pyi",
    "input_schema": {},
    "kind": "class",
    "line": 255,
    "module": "noether.topology",
    "name": "EilenbergMacLaneCohomologyGenerator",
    "overloads": [],
    "qualified_name": "EilenbergMacLaneCohomologyGenerator",
    "releases_gil": false,
    "result_schema": {},
    "signature": "",
    "signatures": [],
    "summary": "One named generator in a complete modular cohomology presentation.",
    "summary_source": "public docstring"
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  {
    "canonical_name": "topology.EilenbergMacLaneCohomologyGenerator.degree",
    "determinism": "deterministic",
    "doc": "",
    "domain": "topology",
    "file": "noether/topology/__init__.pyi",
    "input_schema": {
      "additionalProperties": false,
      "properties": {},
      "type": "object"
    },
    "kind": "property",
    "line": 259,
    "module": "noether.topology",
    "name": "degree",
    "overloads": [
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          "properties": {},
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        "parameters": [],
        "result_schema": {
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        "signature": "(self) -> int"
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    "qualified_name": "EilenbergMacLaneCohomologyGenerator.degree",
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    "result_schema": {
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    "signature": "(self) -> int",
    "signatures": [
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    "summary": "Return the degree recorded by this `EilenbergMacLaneCohomologyGenerator`.",
    "summary_source": "derived projection vocabulary"
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  {
    "canonical_name": "topology.EilenbergMacLaneCohomologyGenerator.display_name",
    "determinism": "deterministic",
    "doc": "",
    "domain": "topology",
    "file": "noether/topology/__init__.pyi",
    "input_schema": {
      "additionalProperties": false,
      "properties": {},
      "type": "object"
    },
    "kind": "property",
    "line": 261,
    "module": "noether.topology",
    "name": "display_name",
    "overloads": [
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        "input_schema": {
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          "properties": {},
          "type": "object"
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        "parameters": [],
        "result_schema": {
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        "signature": "(self) -> str"
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    "qualified_name": "EilenbergMacLaneCohomologyGenerator.display_name",
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    "result_schema": {
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    "signature": "(self) -> str",
    "signatures": [
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    "summary": "Return the display name recorded by this `EilenbergMacLaneCohomologyGenerator`.",
    "summary_source": "derived projection vocabulary"
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  {
    "canonical_name": "topology.EilenbergMacLaneCohomologyGenerator.id",
    "determinism": "deterministic",
    "doc": "",
    "domain": "topology",
    "file": "noether/topology/__init__.pyi",
    "input_schema": {
      "additionalProperties": false,
      "properties": {},
      "type": "object"
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    "kind": "property",
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    "module": "noether.topology",
    "name": "id",
    "overloads": [
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        "input_schema": {
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          "properties": {},
          "type": "object"
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        "parameters": [],
        "result_schema": {
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        "signature": "(self) -> str"
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    "qualified_name": "EilenbergMacLaneCohomologyGenerator.id",
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    "result_schema": {
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    "signature": "(self) -> str",
    "signatures": [
      "(self) -> str"
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    "summary": "Return the id recorded by this `EilenbergMacLaneCohomologyGenerator`.",
    "summary_source": "derived projection vocabulary"
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  {
    "canonical_name": "topology.EilenbergMacLaneCohomologyGenerator.origin",
    "determinism": "deterministic",
    "doc": "",
    "domain": "topology",
    "file": "noether/topology/__init__.pyi",
    "input_schema": {
      "additionalProperties": false,
      "properties": {},
      "type": "object"
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    "kind": "property",
    "line": 263,
    "module": "noether.topology",
    "name": "origin",
    "overloads": [
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          "properties": {},
          "type": "object"
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        "parameters": [],
        "result_schema": {
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        "signature": "(self) -> _EilenbergMacLaneGeneratorOrigin"
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    "qualified_name": "EilenbergMacLaneCohomologyGenerator.origin",
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    "result_schema": {
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    "signature": "(self) -> _EilenbergMacLaneGeneratorOrigin",
    "signatures": [
      "(self) -> _EilenbergMacLaneGeneratorOrigin"
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    "summary": "Return the origin recorded by this `EilenbergMacLaneCohomologyGenerator`.",
    "summary_source": "derived projection vocabulary"
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    "canonical_name": "topology.EilenbergMacLaneCohomologyOperation",
    "determinism": "deterministic",
    "doc": "One named primary operation on a modular cohomology generator.",
    "domain": "topology",
    "file": "noether/topology/__init__.pyi",
    "input_schema": {},
    "kind": "class",
    "line": 265,
    "module": "noether.topology",
    "name": "EilenbergMacLaneCohomologyOperation",
    "overloads": [],
    "qualified_name": "EilenbergMacLaneCohomologyOperation",
    "releases_gil": false,
    "result_schema": {},
    "signature": "",
    "signatures": [],
    "summary": "One named primary operation on a modular cohomology generator.",
    "summary_source": "public docstring"
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    "canonical_name": "topology.EilenbergMacLaneCohomologyOperation.operation",
    "determinism": "deterministic",
    "doc": "",
    "domain": "topology",
    "file": "noether/topology/__init__.pyi",
    "input_schema": {
      "additionalProperties": false,
      "properties": {},
      "type": "object"
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    "kind": "property",
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    "module": "noether.topology",
    "name": "operation",
    "overloads": [
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          "properties": {},
          "type": "object"
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        "parameters": [],
        "result_schema": {
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        "signature": "(self) -> AdmissibleCohomologyOperation"
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    ],
    "qualified_name": "EilenbergMacLaneCohomologyOperation.operation",
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    "result_schema": {
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    "signature": "(self) -> AdmissibleCohomologyOperation",
    "signatures": [
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    "summary": "Return the operation recorded by this `EilenbergMacLaneCohomologyOperation`.",
    "summary_source": "derived projection vocabulary"
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  {
    "canonical_name": "topology.EilenbergMacLaneCohomologyOperation.source_generator",
    "determinism": "deterministic",
    "doc": "",
    "domain": "topology",
    "file": "noether/topology/__init__.pyi",
    "input_schema": {
      "additionalProperties": false,
      "properties": {},
      "type": "object"
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    "kind": "property",
    "line": 269,
    "module": "noether.topology",
    "name": "source_generator",
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        "input_schema": {
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          "properties": {},
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        "parameters": [],
        "result_schema": {
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    "qualified_name": "EilenbergMacLaneCohomologyOperation.source_generator",
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    "result_schema": {
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    "signature": "(self) -> int",
    "signatures": [
      "(self) -> int"
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    "summary": "Return the source generator recorded by this `EilenbergMacLaneCohomologyOperation`.",
    "summary_source": "derived projection vocabulary"
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    "canonical_name": "topology.EilenbergMacLaneCohomologyOperation.target",
    "determinism": "deterministic",
    "doc": "",
    "domain": "topology",
    "file": "noether/topology/__init__.pyi",
    "input_schema": {
      "additionalProperties": false,
      "properties": {},
      "type": "object"
    },
    "kind": "property",
    "line": 271,
    "module": "noether.topology",
    "name": "target",
    "overloads": [
      {
        "input_schema": {
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          "properties": {},
          "type": "object"
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        "parameters": [],
        "result_schema": {
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        "signature": "(self) -> GradedCommutativePolynomial"
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    ],
    "qualified_name": "EilenbergMacLaneCohomologyOperation.target",
    "releases_gil": false,
    "result_schema": {
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    "signature": "(self) -> GradedCommutativePolynomial",
    "signatures": [
      "(self) -> GradedCommutativePolynomial"
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    "summary": "Return the target recorded by this `EilenbergMacLaneCohomologyOperation`.",
    "summary_source": "derived projection vocabulary"
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    "canonical_name": "topology.EilenbergMacLaneIntegralCohomology",
    "determinism": "deterministic",
    "doc": "Exact bounded additive integral cohomology of a symbolic K(A,n).",
    "domain": "topology",
    "file": "noether/topology/__init__.pyi",
    "input_schema": {},
    "kind": "class",
    "line": 192,
    "module": "noether.topology",
    "name": "EilenbergMacLaneIntegralCohomology",
    "overloads": [],
    "qualified_name": "EilenbergMacLaneIntegralCohomology",
    "releases_gil": false,
    "result_schema": {},
    "signature": "",
    "signatures": [],
    "summary": "Exact bounded additive integral cohomology of a symbolic K(A,n).",
    "summary_source": "public docstring",
    "type_id": "noether/topology/EilenbergMacLaneIntegralCohomology@1"
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  {
    "canonical_name": "topology.EilenbergMacLaneIntegralCohomology.from_json",
    "determinism": "deterministic",
    "doc": "",
    "domain": "topology",
    "file": "noether/topology/__init__.pyi",
    "input_schema": {
      "additionalProperties": false,
      "properties": {
        "value": {
          "type": "string"
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      },
      "required": [
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      "type": "object"
    },
    "kind": "classmethod",
    "line": 206,
    "module": "noether.topology",
    "name": "from_json",
    "overloads": [
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        "input_schema": {
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          "properties": {
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          "required": [
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          "type": "object"
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        "parameters": [
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            "kind": "positional_or_keyword",
            "name": "value",
            "required": true,
            "schema": {
              "type": "string"
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          }
        ],
        "result_schema": {
          "x-python-type": "EilenbergMacLaneIntegralCohomology"
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        "signature": "(cls, value: str) -> EilenbergMacLaneIntegralCohomology"
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    ],
    "qualified_name": "EilenbergMacLaneIntegralCohomology.from_json",
    "releases_gil": false,
    "result_schema": {
      "x-python-type": "EilenbergMacLaneIntegralCohomology"
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    "signature": "(cls, value: str) -> EilenbergMacLaneIntegralCohomology",
    "signatures": [
      "(cls, value: str) -> EilenbergMacLaneIntegralCohomology"
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    "summary": "Deserialize and revalidate a `EilenbergMacLaneIntegralCohomology` from typed JSON.",
    "summary_source": "derived projection vocabulary",
    "type_id": "noether/topology/EilenbergMacLaneIntegralCohomology@1"
  },
  {
    "canonical_name": "topology.EilenbergMacLaneIntegralCohomology.group",
    "determinism": "deterministic",
    "doc": "",
    "domain": "topology",
    "file": "noether/topology/__init__.pyi",
    "input_schema": {
      "additionalProperties": false,
      "properties": {
        "degree": {
          "type": "integer"
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      },
      "required": [
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      "type": "object"
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    "kind": "method",
    "line": 202,
    "module": "noether.topology",
    "name": "group",
    "overloads": [
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          "properties": {
            "degree": {
              "type": "integer"
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          },
          "required": [
            "degree"
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          "type": "object"
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        "parameters": [
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            "kind": "positional_or_keyword",
            "name": "degree",
            "required": true,
            "schema": {
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          }
        ],
        "result_schema": {
          "anyOf": [
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              "x-python-type": "FinitelyGeneratedAbelianGroup"
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            {
              "type": "null"
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          ],
          "x-python-type": "FinitelyGeneratedAbelianGroup | None"
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        "signature": "(self, degree: int) -> FinitelyGeneratedAbelianGroup | None"
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    ],
    "qualified_name": "EilenbergMacLaneIntegralCohomology.group",
    "releases_gil": false,
    "result_schema": {
      "anyOf": [
        {
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        {
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      ],
      "x-python-type": "FinitelyGeneratedAbelianGroup | None"
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    "signature": "(self, degree: int) -> FinitelyGeneratedAbelianGroup | None",
    "signatures": [
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    "summary": "Compute group for this `EilenbergMacLaneIntegralCohomology`.",
    "summary_source": "derived projection vocabulary",
    "type_id": "noether/topology/EilenbergMacLaneIntegralCohomology@1"
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  {
    "canonical_name": "topology.EilenbergMacLaneIntegralCohomology.groups",
    "determinism": "deterministic",
    "doc": "",
    "domain": "topology",
    "file": "noether/topology/__init__.pyi",
    "input_schema": {
      "additionalProperties": false,
      "properties": {},
      "type": "object"
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    "kind": "property",
    "line": 201,
    "module": "noether.topology",
    "name": "groups",
    "overloads": [
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          "properties": {},
          "type": "object"
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        "parameters": [],
        "result_schema": {
          "items": {
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          "type": "array",
          "x-python-type": "list[FinitelyGeneratedAbelianGroup]"
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        "signature": "(self) -> list[FinitelyGeneratedAbelianGroup]"
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    ],
    "qualified_name": "EilenbergMacLaneIntegralCohomology.groups",
    "releases_gil": false,
    "result_schema": {
      "items": {
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      "type": "array",
      "x-python-type": "list[FinitelyGeneratedAbelianGroup]"
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    "signature": "(self) -> list[FinitelyGeneratedAbelianGroup]",
    "signatures": [
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    "summary": "Return the groups recorded by this `EilenbergMacLaneIntegralCohomology`.",
    "summary_source": "derived projection vocabulary",
    "type_id": "noether/topology/EilenbergMacLaneIntegralCohomology@1"
  },
  {
    "canonical_name": "topology.EilenbergMacLaneIntegralCohomology.homology",
    "determinism": "deterministic",
    "doc": "",
    "domain": "topology",
    "file": "noether/topology/__init__.pyi",
    "input_schema": {
      "additionalProperties": false,
      "properties": {},
      "type": "object"
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    "kind": "property",
    "line": 199,
    "module": "noether.topology",
    "name": "homology",
    "overloads": [
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          "properties": {},
          "type": "object"
        },
        "parameters": [],
        "result_schema": {
          "x-python-type": "EilenbergMacLaneIntegralHomology"
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        "signature": "(self) -> EilenbergMacLaneIntegralHomology"
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    ],
    "qualified_name": "EilenbergMacLaneIntegralCohomology.homology",
    "releases_gil": false,
    "result_schema": {
      "x-python-type": "EilenbergMacLaneIntegralHomology"
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    "signature": "(self) -> EilenbergMacLaneIntegralHomology",
    "signatures": [
      "(self) -> EilenbergMacLaneIntegralHomology"
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    "summary": "Return the homology recorded by this `EilenbergMacLaneIntegralCohomology`.",
    "summary_source": "derived projection vocabulary",
    "type_id": "noether/topology/EilenbergMacLaneIntegralCohomology@1"
  },
  {
    "canonical_name": "topology.EilenbergMacLaneIntegralCohomology.max_degree",
    "determinism": "deterministic",
    "doc": "",
    "domain": "topology",
    "file": "noether/topology/__init__.pyi",
    "input_schema": {
      "additionalProperties": false,
      "properties": {},
      "type": "object"
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    "kind": "property",
    "line": 197,
    "module": "noether.topology",
    "name": "max_degree",
    "overloads": [
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        "input_schema": {
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          "properties": {},
          "type": "object"
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        "parameters": [],
        "result_schema": {
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        "signature": "(self) -> int"
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    ],
    "qualified_name": "EilenbergMacLaneIntegralCohomology.max_degree",
    "releases_gil": false,
    "result_schema": {
      "type": "integer"
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    "signature": "(self) -> int",
    "signatures": [
      "(self) -> int"
    ],
    "summary": "Return the max degree recorded by this `EilenbergMacLaneIntegralCohomology`.",
    "summary_source": "derived projection vocabulary",
    "type_id": "noether/topology/EilenbergMacLaneIntegralCohomology@1"
  },
  {
    "canonical_name": "topology.EilenbergMacLaneIntegralCohomology.space",
    "determinism": "deterministic",
    "doc": "",
    "domain": "topology",
    "file": "noether/topology/__init__.pyi",
    "input_schema": {
      "additionalProperties": false,
      "properties": {},
      "type": "object"
    },
    "kind": "property",
    "line": 195,
    "module": "noether.topology",
    "name": "space",
    "overloads": [
      {
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          "properties": {},
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    "domain": "topology",
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    "canonical_name": "topology.EilenbergMacLaneIntegralCohomologyGenerator.display_name",
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    "domain": "topology",
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    "doc": "Complete integral cohomology-ring presentation of a symbolic K(A,n).",
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    "summary": "Complete integral cohomology-ring presentation of a symbolic K(A,n).",
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    "domain": "topology",
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    "summary": "Compute additive group for this `EilenbergMacLaneIntegralCohomologyPresentation`.",
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    "type_id": "noether/topology/EilenbergMacLaneIntegralCohomologyPresentation@1"
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    "signature": "(self) -> Literal['Z']",
    "signatures": [
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    "summary": "Return the coefficient ring recorded by this `EilenbergMacLaneIntegralCohomologyPresentation`.",
    "summary_source": "derived projection vocabulary",
    "type_id": "noether/topology/EilenbergMacLaneIntegralCohomologyPresentation@1"
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    "domain": "topology",
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    "summary": "Return the generators recorded by this `EilenbergMacLaneIntegralCohomologyPresentation`.",
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    "type_id": "noether/topology/EilenbergMacLaneIntegralCohomologyPresentation@1"
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    "domain": "topology",
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    "summary": "Return the graded commutative recorded by this `EilenbergMacLaneIntegralCohomologyPresentation`.",
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    "type_id": "noether/topology/EilenbergMacLaneIntegralCohomologyPresentation@1"
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    "summary": "Return the relations recorded by this `EilenbergMacLaneIntegralCohomologyPresentation`.",
    "summary_source": "derived projection vocabulary",
    "type_id": "noether/topology/EilenbergMacLaneIntegralCohomologyPresentation@1"
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    "type_id": "noether/topology/EilenbergMacLaneIntegralCohomologyPresentation@1"
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    "summary": "Return the strategy recorded by this `EilenbergMacLaneIntegralCohomologyPresentation`.",
    "summary_source": "derived projection vocabulary",
    "type_id": "noether/topology/EilenbergMacLaneIntegralCohomologyPresentation@1"
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    "canonical_name": "topology.EilenbergMacLaneIntegralCohomologyPresentation.to_json",
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    "file": "noether/topology/__init__.pyi",
    "input_schema": {
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      "properties": {},
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    "name": "to_json",
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    "doc": "One homogeneous relation in an integral cohomology presentation.",
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    "domain": "topology",
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    "summary": "Return the terms recorded by this `EilenbergMacLaneIntegralCohomologyRelation`.",
    "summary_source": "derived projection vocabulary"
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  {
    "canonical_name": "topology.EilenbergMacLaneIntegralCohomologyRelationTerm",
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    "doc": "One sparse term in an integral cohomology relation.",
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    "input_schema": {},
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    "module": "noether.topology",
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    "summary": "One sparse term in an integral cohomology relation.",
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    "domain": "topology",
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    "summary": "Return the coefficient recorded by this `EilenbergMacLaneIntegralCohomologyRelationTerm`.",
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    "canonical_name": "topology.EilenbergMacLaneIntegralCohomologyRelationTerm.exponents",
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    "domain": "topology",
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    "doc": "Exact bounded integral homology of a symbolic K(A,n).",
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    "summary": "Exact bounded integral homology of a symbolic K(A,n).",
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    "canonical_name": "topology.EilenbergMacLaneIntegralHomology.from_json",
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    "doc": "",
    "domain": "topology",
    "file": "noether/topology/__init__.pyi",
    "input_schema": {
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    "result_schema": {
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    "summary": "Deserialize and revalidate a `EilenbergMacLaneIntegralHomology` from typed JSON.",
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    "canonical_name": "topology.EilenbergMacLaneIntegralHomology.group",
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    "doc": "",
    "domain": "topology",
    "file": "noether/topology/__init__.pyi",
    "input_schema": {
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      "properties": {
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        "result_schema": {
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          "x-python-type": "FinitelyGeneratedAbelianGroup | None"
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    "signature": "(self, degree: int) -> FinitelyGeneratedAbelianGroup | None",
    "signatures": [
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    "summary": "Compute group for this `EilenbergMacLaneIntegralHomology`.",
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    "canonical_name": "topology.EilenbergMacLaneIntegralHomology.groups",
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    "doc": "",
    "domain": "topology",
    "file": "noether/topology/__init__.pyi",
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      "properties": {},
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        "signature": "(self) -> list[FinitelyGeneratedAbelianGroup]"
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    "qualified_name": "EilenbergMacLaneIntegralHomology.groups",
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    "signatures": [
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    "summary": "Return the groups recorded by this `EilenbergMacLaneIntegralHomology`.",
    "summary_source": "derived projection vocabulary",
    "type_id": "noether/topology/EilenbergMacLaneIntegralHomology@1"
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    "canonical_name": "topology.EilenbergMacLaneIntegralHomology.max_degree",
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    "domain": "topology",
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      "properties": {},
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    "kind": "property",
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    "module": "noether.topology",
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    "qualified_name": "EilenbergMacLaneIntegralHomology.max_degree",
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    "result_schema": {
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    "summary": "Return the max degree recorded by this `EilenbergMacLaneIntegralHomology`.",
    "summary_source": "derived projection vocabulary",
    "type_id": "noether/topology/EilenbergMacLaneIntegralHomology@1"
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    "canonical_name": "topology.EilenbergMacLaneIntegralHomology.space",
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    "doc": "",
    "domain": "topology",
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    "kind": "property",
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    "module": "noether.topology",
    "name": "space",
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        "parameters": [],
        "result_schema": {
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        "signature": "(self) -> EilenbergMacLaneSpace"
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    "qualified_name": "EilenbergMacLaneIntegralHomology.space",
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    "result_schema": {
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    "summary": "Return the space recorded by this `EilenbergMacLaneIntegralHomology`.",
    "summary_source": "derived projection vocabulary",
    "type_id": "noether/topology/EilenbergMacLaneIntegralHomology@1"
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    "canonical_name": "topology.EilenbergMacLaneIntegralHomology.strategy",
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    "doc": "",
    "domain": "topology",
    "file": "noether/topology/__init__.pyi",
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      "properties": {},
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    "kind": "property",
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    "module": "noether.topology",
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    "qualified_name": "EilenbergMacLaneIntegralHomology.strategy",
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    "signatures": [
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    "summary": "Return the strategy recorded by this `EilenbergMacLaneIntegralHomology`.",
    "summary_source": "derived projection vocabulary",
    "type_id": "noether/topology/EilenbergMacLaneIntegralHomology@1"
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    "canonical_name": "topology.EilenbergMacLaneIntegralHomology.to_json",
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    "name": "to_json",
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    "qualified_name": "EilenbergMacLaneIntegralHomology.to_json",
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    "summary": "Serialize this `EilenbergMacLaneIntegralHomology` to its canonical typed JSON representation.",
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    "summary": "Replay and verify the mathematical evidence carried by this `EilenbergMacLaneIntegralHomology`.",
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    "doc": "A complete prime-field cohomology presentation of a symbolic K(A,n).",
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    "summary": "A complete prime-field cohomology presentation of a symbolic K(A,n).",
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    "canonical_name": "topology.EilenbergMacLaneModularCohomology.field",
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    "module": "noether.topology",
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    "qualified_name": "EilenbergMacLaneModularCohomology.field",
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    "result_schema": {
      "x-python-type": "FiniteField"
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    "signature": "(self) -> FiniteField",
    "signatures": [
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    "summary": "Return the field recorded by this `EilenbergMacLaneModularCohomology`.",
    "summary_source": "derived projection vocabulary",
    "type_id": "noether/topology/EilenbergMacLaneModularCohomology@1"
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    "canonical_name": "topology.EilenbergMacLaneModularCohomology.from_json",
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    "input_schema": {
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    "kind": "classmethod",
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    "module": "noether.topology",
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        "parameters": [
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    "summary": "Return the space recorded by this `EilenbergMacLaneModularCohomology`.",
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    "summary": "Return the strategy recorded by this `EilenbergMacLaneModularCohomology`.",
    "summary_source": "derived projection vocabulary",
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    "domain": "topology",
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    "signature": "(self) -> None",
    "signatures": [
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    "summary": "Replay and verify the mathematical evidence carried by this `EilenbergMacLaneModularCohomology`.",
    "summary_source": "derived projection vocabulary",
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    "canonical_name": "topology.EilenbergMacLaneModularCohomologyTruncation",
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    "doc": "A prime-field cohomology presentation exact through one degree.",
    "domain": "topology",
    "file": "noether/topology/__init__.pyi",
    "input_schema": {},
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    "module": "noether.topology",
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    "summary": "A prime-field cohomology presentation exact through one degree.",
    "summary_source": "public docstring",
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    "canonical_name": "topology.EilenbergMacLaneModularCohomologyTruncation.exact_through_degree",
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    "doc": "",
    "domain": "topology",
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    "summary": "Return the exact through degree recorded by this `EilenbergMacLaneModularCohomologyTruncation`.",
    "summary_source": "derived projection vocabulary",
    "type_id": "noether/topology/EilenbergMacLaneModularCohomologyTruncation@1"
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    "signature": "(self) -> FiniteField",
    "signatures": [
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    "summary": "Return the field recorded by this `EilenbergMacLaneModularCohomologyTruncation`.",
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    "type_id": "noether/topology/EilenbergMacLaneModularCohomologyTruncation@1"
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    "domain": "topology",
    "file": "noether/topology/__init__.pyi",
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    "module": "noether.topology",
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    "qualified_name": "EilenbergMacLaneModularCohomologyTruncation.from_json",
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    "result_schema": {
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    "signature": "(cls, value: str) -> EilenbergMacLaneModularCohomologyTruncation",
    "signatures": [
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    "summary": "Deserialize and revalidate a `EilenbergMacLaneModularCohomologyTruncation` from typed JSON.",
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    "domain": "topology",
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    "name": "generators",
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    "type_id": "noether/topology/EilenbergMacLaneModularCohomologyTruncation@1"
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    "canonical_name": "topology.EilenbergMacLaneModularCohomologyTruncation.presentation",
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    "doc": "",
    "domain": "topology",
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    "input_schema": {
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    "module": "noether.topology",
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    "result_schema": {
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    "summary": "Return the presentation recorded by this `EilenbergMacLaneModularCohomologyTruncation`.",
    "summary_source": "derived projection vocabulary",
    "type_id": "noether/topology/EilenbergMacLaneModularCohomologyTruncation@1"
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    "canonical_name": "topology.EilenbergMacLaneModularCohomologyTruncation.space",
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    "domain": "topology",
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    "qualified_name": "EilenbergMacLaneModularCohomologyTruncation.space",
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    "result_schema": {
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    "signature": "(self) -> EilenbergMacLaneSpace",
    "signatures": [
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    "summary": "Return the space recorded by this `EilenbergMacLaneModularCohomologyTruncation`.",
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    "type_id": "noether/topology/EilenbergMacLaneModularCohomologyTruncation@1"
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    "doc": "",
    "domain": "topology",
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    "input_schema": {
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      "properties": {},
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    "line": 311,
    "module": "noether.topology",
    "name": "strategy",
    "overloads": [
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          "properties": {},
          "type": "object"
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        "parameters": [],
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          "x-python-type": "Literal['trivial_space', 'cartan_serre_admissible_squares', 'cartan_serre_admissible_odd_primary_operations']"
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        "signature": "(self) -> Literal['trivial_space', 'cartan_serre_admissible_squares', 'cartan_serre_admissible_odd_primary_operations']"
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    "qualified_name": "EilenbergMacLaneModularCohomologyTruncation.strategy",
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    "result_schema": {
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    "signature": "(self) -> Literal['trivial_space', 'cartan_serre_admissible_squares', 'cartan_serre_admissible_odd_primary_operations']",
    "signatures": [
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    "summary": "Return the strategy recorded by this `EilenbergMacLaneModularCohomologyTruncation`.",
    "summary_source": "derived projection vocabulary",
    "type_id": "noether/topology/EilenbergMacLaneModularCohomologyTruncation@1"
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  {
    "canonical_name": "topology.EilenbergMacLaneModularCohomologyTruncation.to_json",
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    "domain": "topology",
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    "module": "noether.topology",
    "name": "to_json",
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      "(self, steps: list[ElementaryCollapse]) -> SimplicialCollapseSequence"
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    "summary": "Replay and retain a proposed sequence of elementary collapses.",
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    "canonical_name": "topology.SimplicialComplex.cup_i",
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    "domain": "topology",
    "file": "noether/topology/__init__.pyi",
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    "canonical_name": "topology.SimplicialComplex.cup_product",
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    "file": "noether/topology/__init__.pyi",
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    "canonical_name": "topology.SimplicialComplex.dimension",
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    "doc": "Top dimension; the empty complex has dimension -1.",
    "domain": "topology",
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    "summary": "Top dimension; the empty complex has dimension -1.",
    "summary_source": "public docstring",
    "type_id": "noether/topology/SimplicialComplex@1"
  },
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    "canonical_name": "topology.SimplicialComplex.dominated_vertices",
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    "doc": "Dominated vertices with their least-label dominators.",
    "domain": "topology",
    "file": "noether/topology/__init__.pyi",
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      "properties": {},
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    "type_id": "noether/topology/SimplicialComplex@1"
  },
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    "canonical_name": "topology.SimplicialComplex.elementary_collapse",
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    "doc": "Replay one verified elementary free-face collapse.",
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          }
        ],
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    "signature": "(self, step: ElementaryCollapse) -> SimplicialComplex",
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    "summary": "Replay one verified elementary free-face collapse.",
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  },
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    "canonical_name": "topology.SimplicialComplex.elementary_collapses",
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    "doc": "Every elementary free-face collapse available in this complex.",
    "domain": "topology",
    "file": "noether/topology/__init__.pyi",
    "input_schema": {
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      "properties": {},
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    ],
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    "signatures": [
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    "summary": "Every elementary free-face collapse available in this complex.",
    "summary_source": "public docstring",
    "type_id": "noether/topology/SimplicialComplex@1"
  },
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    "canonical_name": "topology.SimplicialComplex.euler_characteristic",
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    "doc": "Euler characteristic computed independently from cell counts.",
    "domain": "topology",
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    "summary": "Euler characteristic computed independently from cell counts.",
    "summary_source": "public docstring",
    "type_id": "noether/topology/SimplicialComplex@1"
  },
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    "canonical_name": "topology.SimplicialComplex.facets",
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    "doc": "Maximal simplices in canonical lexicographic order.",
    "domain": "topology",
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      "additionalProperties": false,
      "properties": {},
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    "signatures": [
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    "summary": "Maximal simplices in canonical lexicographic order.",
    "summary_source": "public docstring",
    "type_id": "noether/topology/SimplicialComplex@1"
  },
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    "canonical_name": "topology.SimplicialComplex.from_json",
    "determinism": "deterministic",
    "doc": "",
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    "file": "noether/topology/__init__.pyi",
    "input_schema": {
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      "properties": {
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    "module": "noether.topology",
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    ],
    "summary": "Deserialize and revalidate a `SimplicialComplex` from typed JSON.",
    "summary_source": "derived projection vocabulary",
    "type_id": "noether/topology/SimplicialComplex@1"
  },
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    "doc": "Compute checked homology in every chain degree, over `Z` by default.",
    "domain": "topology",
    "evidence": [
      "GradedHomology.verify"
    ],
    "example": "from noether.topology import SimplicialComplex\ncircle = SimplicialComplex([[0, 1], [1, 2], [0, 2]])\nresult = circle.graded_homology()\nassert result.get(1).free_rank == 1\nresult.verify()",
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        "signature": "(self) -> GradedHomology[IntegerRing]"
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      {
        "input_schema": {
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          "properties": {
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        "signature": "(self, *, ring: _CoefficientContext) -> GradedHomology[_CoefficientContext]"
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    "qualified_name": "SimplicialComplex.graded_homology",
    "releases_gil": true,
    "result_schema": {
      "x-python-type": "GradedHomology[IntegerRing]"
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    "signature": "(self) -> GradedHomology[IntegerRing]",
    "signatures": [
      "(self) -> GradedHomology[IntegerRing]",
      "(self, *, ring: _CoefficientContext) -> GradedHomology[_CoefficientContext]"
    ],
    "summary": "Compute checked homology in every chain degree, over `Z` by default.",
    "summary_source": "public docstring",
    "supported_rings": [
      "Z",
      "exact fields"
    ],
    "type_id": "noether/topology/SimplicialComplex@1"
  },
  {
    "canonical_name": "topology.SimplicialComplex.graded_homology_invariants",
    "determinism": "deterministic",
    "doc": "Compute lean integral homology invariants in every chain degree.",
    "domain": "topology",
    "file": "noether/topology/__init__.pyi",
    "input_schema": {
      "additionalProperties": false,
      "properties": {},
      "type": "object"
    },
    "kind": "method",
    "line": 396,
    "module": "noether.topology",
    "name": "graded_homology_invariants",
    "overloads": [
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        "input_schema": {
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          "properties": {},
          "type": "object"
        },
        "parameters": [],
        "result_schema": {
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          },
          "type": "array",
          "x-python-type": "list[FinitelyGeneratedAbelianGroup]"
        },
        "signature": "(self) -> list[FinitelyGeneratedAbelianGroup]"
      }
    ],
    "qualified_name": "SimplicialComplex.graded_homology_invariants",
    "releases_gil": true,
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    "canonical_name": "topology.SimplicialSet.steenrod_reduced_power",
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        "source": {
          "type": "integer"
        }
      },
      "required": [
        "operation",
        "source"
      ],
      "type": "object"
    },
    "kind": "method",
    "line": 647,
    "module": "noether.topology",
    "name": "apply_operation",
    "overloads": [
      {
        "input_schema": {
          "additionalProperties": false,
          "properties": {
            "operation": {
              "x-python-type": "AdmissibleCohomologyOperation"
            },
            "source": {
              "type": "integer"
            }
          },
          "required": [
            "operation",
            "source"
          ],
          "type": "object"
        },
        "parameters": [
          {
            "kind": "positional_or_keyword",
            "name": "operation",
            "required": true,
            "schema": {
              "x-python-type": "AdmissibleCohomologyOperation"
            }
          },
          {
            "kind": "positional_or_keyword",
            "name": "source",
            "required": true,
            "schema": {
              "type": "integer"
            }
          }
        ],
        "result_schema": {
          "items": {
            "x-python-type": "SimplicialCohomologyProductTerm"
          },
          "type": "array",
          "x-python-type": "list[SimplicialCohomologyProductTerm]"
        },
        "signature": "(self, operation: AdmissibleCohomologyOperation, source: int) -> list[SimplicialCohomologyProductTerm]"
      }
    ],
    "qualified_name": "SimplicialSetCohomologyRing.apply_operation",
    "releases_gil": false,
    "result_schema": {
      "items": {
        "x-python-type": "SimplicialCohomologyProductTerm"
      },
      "type": "array",
      "x-python-type": "list[SimplicialCohomologyProductTerm]"
    },
    "signature": "(self, operation: AdmissibleCohomologyOperation, source: int) -> list[SimplicialCohomologyProductTerm]",
    "signatures": [
      "(self, operation: AdmissibleCohomologyOperation, source: int) -> list[SimplicialCohomologyProductTerm]"
    ],
    "summary": "Compute apply operation for this `SimplicialSetCohomologyRing`.",
    "summary_source": "derived projection vocabulary",
    "type_id": "noether/topology/SimplicialSetCohomologyRing@1"
  },
  {
    "canonical_name": "topology.SimplicialSetCohomologyRing.basis",
    "determinism": "deterministic",
    "doc": "",
    "domain": "topology",
    "file": "noether/topology/__init__.pyi",
    "input_schema": {
      "additionalProperties": false,
      "properties": {},
      "type": "object"
    },
    "kind": "property",
    "line": 632,
    "module": "noether.topology",
    "name": "basis",
    "overloads": [
      {
        "input_schema": {
          "additionalProperties": false,
          "properties": {},
          "type": "object"
        },
        "parameters": [],
        "result_schema": {
          "items": {
            "x-python-type": "SimplicialCohomologyBasisElement"
          },
          "type": "array",
          "x-python-type": "list[SimplicialCohomologyBasisElement]"
        },
        "signature": "(self) -> list[SimplicialCohomologyBasisElement]"
      }
    ],
    "qualified_name": "SimplicialSetCohomologyRing.basis",
    "releases_gil": false,
    "result_schema": {
      "items": {
        "x-python-type": "SimplicialCohomologyBasisElement"
      },
      "type": "array",
      "x-python-type": "list[SimplicialCohomologyBasisElement]"
    },
    "signature": "(self) -> list[SimplicialCohomologyBasisElement]",
    "signatures": [
      "(self) -> list[SimplicialCohomologyBasisElement]"
    ],
    "summary": "Return the basis recorded by this `SimplicialSetCohomologyRing`.",
    "summary_source": "derived projection vocabulary",
    "type_id": "noether/topology/SimplicialSetCohomologyRing@1"
  },
  {
    "canonical_name": "topology.SimplicialSetCohomologyRing.bockstein_reduced_power",
    "determinism": "deterministic",
    "doc": "",
    "domain": "topology",
    "file": "noether/topology/__init__.pyi",
    "input_schema": {
      "additionalProperties": false,
      "properties": {
        "power": {
          "type": "integer"
        },
        "source": {
          "type": "integer"
        }
      },
      "required": [
        "power",
        "source"
      ],
      "type": "object"
    },
    "kind": "method",
    "line": 644,
    "module": "noether.topology",
    "name": "bockstein_reduced_power",
    "overloads": [
      {
        "input_schema": {
          "additionalProperties": false,
          "properties": {
            "power": {
              "type": "integer"
            },
            "source": {
              "type": "integer"
            }
          },
          "required": [
            "power",
            "source"
          ],
          "type": "object"
        },
        "parameters": [
          {
            "kind": "positional_or_keyword",
            "name": "power",
            "required": true,
            "schema": {
              "type": "integer"
            }
          },
          {
            "kind": "positional_or_keyword",
            "name": "source",
            "required": true,
            "schema": {
              "type": "integer"
            }
          }
        ],
        "result_schema": {
          "items": {
            "x-python-type": "SimplicialCohomologyProductTerm"
          },
          "type": "array",
          "x-python-type": "list[SimplicialCohomologyProductTerm]"
        },
        "signature": "(self, power: int, source: int) -> list[SimplicialCohomologyProductTerm]"
      }
    ],
    "qualified_name": "SimplicialSetCohomologyRing.bockstein_reduced_power",
    "releases_gil": false,
    "result_schema": {
      "items": {
        "x-python-type": "SimplicialCohomologyProductTerm"
      },
      "type": "array",
      "x-python-type": "list[SimplicialCohomologyProductTerm]"
    },
    "signature": "(self, power: int, source: int) -> list[SimplicialCohomologyProductTerm]",
    "signatures": [
      "(self, power: int, source: int) -> list[SimplicialCohomologyProductTerm]"
    ],
    "summary": "Compute bockstein reduced power for this `SimplicialSetCohomologyRing`.",
    "summary_source": "derived projection vocabulary",
    "type_id": "noether/topology/SimplicialSetCohomologyRing@1"
  },
  {
    "canonical_name": "topology.SimplicialSetCohomologyRing.field",
    "determinism": "deterministic",
    "doc": "",
    "domain": "topology",
    "file": "noether/topology/__init__.pyi",
    "input_schema": {
      "additionalProperties": false,
      "properties": {},
      "type": "object"
    },
    "kind": "property",
    "line": 630,
    "module": "noether.topology",
    "name": "field",
    "overloads": [
      {
        "input_schema": {
          "additionalProperties": false,
          "properties": {},
          "type": "object"
        },
        "parameters": [],
        "result_schema": {
          "x-python-type": "FiniteField"
        },
        "signature": "(self) -> FiniteField"
      }
    ],
    "qualified_name": "SimplicialSetCohomologyRing.field",
    "releases_gil": false,
    "result_schema": {
      "x-python-type": "FiniteField"
    },
    "signature": "(self) -> FiniteField",
    "signatures": [
      "(self) -> FiniteField"
    ],
    "summary": "Return the field recorded by this `SimplicialSetCohomologyRing`.",
    "summary_source": "derived projection vocabulary",
    "type_id": "noether/topology/SimplicialSetCohomologyRing@1"
  },
  {
    "canonical_name": "topology.SimplicialSetCohomologyRing.from_json",
    "determinism": "deterministic",
    "doc": "",
    "domain": "topology",
    "file": "noether/topology/__init__.pyi",
    "input_schema": {
      "additionalProperties": false,
      "properties": {
        "value": {
          "type": "string"
        }
      },
      "required": [
        "value"
      ],
      "type": "object"
    },
    "kind": "classmethod",
    "line": 653,
    "module": "noether.topology",
    "name": "from_json",
    "overloads": [
      {
        "input_schema": {
          "additionalProperties": false,
          "properties": {
            "value": {
              "type": "string"
            }
          },
          "required": [
            "value"
          ],
          "type": "object"
        },
        "parameters": [
          {
            "kind": "positional_or_keyword",
            "name": "value",
            "required": true,
            "schema": {
              "type": "string"
            }
          }
        ],
        "result_schema": {
          "x-python-type": "SimplicialSetCohomologyRing"
        },
        "signature": "(cls, value: str) -> SimplicialSetCohomologyRing"
      }
    ],
    "qualified_name": "SimplicialSetCohomologyRing.from_json",
    "releases_gil": false,
    "result_schema": {
      "x-python-type": "SimplicialSetCohomologyRing"
    },
    "signature": "(cls, value: str) -> SimplicialSetCohomologyRing",
    "signatures": [
      "(cls, value: str) -> SimplicialSetCohomologyRing"
    ],
    "summary": "Deserialize and revalidate a `SimplicialSetCohomologyRing` from typed JSON.",
    "summary_source": "derived projection vocabulary",
    "type_id": "noether/topology/SimplicialSetCohomologyRing@1"
  },
  {
    "canonical_name": "topology.SimplicialSetCohomologyRing.presentation",
    "determinism": "deterministic",
    "doc": "",
    "domain": "topology",
    "file": "noether/topology/__init__.pyi",
    "input_schema": {
      "additionalProperties": false,
      "properties": {},
      "type": "object"
    },
    "kind": "property",
    "line": 636,
    "module": "noether.topology",
    "name": "presentation",
    "overloads": [
      {
        "input_schema": {
          "additionalProperties": false,
          "properties": {},
          "type": "object"
        },
        "parameters": [],
        "result_schema": {
          "anyOf": [
            {
              "x-python-type": "GradedCommutativePresentation"
            },
            {
              "type": "null"
            }
          ],
          "x-python-type": "GradedCommutativePresentation | None"
        },
        "signature": "(self) -> GradedCommutativePresentation | None"
      }
    ],
    "qualified_name": "SimplicialSetCohomologyRing.presentation",
    "releases_gil": false,
    "result_schema": {
      "anyOf": [
        {
          "x-python-type": "GradedCommutativePresentation"
        },
        {
          "type": "null"
        }
      ],
      "x-python-type": "GradedCommutativePresentation | None"
    },
    "signature": "(self) -> GradedCommutativePresentation | None",
    "signatures": [
      "(self) -> GradedCommutativePresentation | None"
    ],
    "summary": "Return the presentation recorded by this `SimplicialSetCohomologyRing`.",
    "summary_source": "derived projection vocabulary",
    "type_id": "noether/topology/SimplicialSetCohomologyRing@1"
  },
  {
    "canonical_name": "topology.SimplicialSetCohomologyRing.product",
    "determinism": "deterministic",
    "doc": "",
    "domain": "topology",
    "file": "noether/topology/__init__.pyi",
    "input_schema": {
      "additionalProperties": false,
      "properties": {
        "left": {
          "type": "integer"
        },
        "right": {
          "type": "integer"
        }
      },
      "required": [
        "left",
        "right"
      ],
      "type": "object"
    },
    "kind": "method",
    "line": 637,
    "module": "noether.topology",
    "name": "product",
    "overloads": [
      {
        "input_schema": {
          "additionalProperties": false,
          "properties": {
            "left": {
              "type": "integer"
            },
            "right": {
              "type": "integer"
            }
          },
          "required": [
            "left",
            "right"
          ],
          "type": "object"
        },
        "parameters": [
          {
            "kind": "positional_or_keyword",
            "name": "left",
            "required": true,
            "schema": {
              "type": "integer"
            }
          },
          {
            "kind": "positional_or_keyword",
            "name": "right",
            "required": true,
            "schema": {
              "type": "integer"
            }
          }
        ],
        "result_schema": {
          "items": {
            "x-python-type": "SimplicialCohomologyProductTerm"
          },
          "type": "array",
          "x-python-type": "list[SimplicialCohomologyProductTerm]"
        },
        "signature": "(self, left: int, right: int) -> list[SimplicialCohomologyProductTerm]"
      }
    ],
    "qualified_name": "SimplicialSetCohomologyRing.product",
    "releases_gil": false,
    "result_schema": {
      "items": {
        "x-python-type": "SimplicialCohomologyProductTerm"
      },
      "type": "array",
      "x-python-type": "list[SimplicialCohomologyProductTerm]"
    },
    "signature": "(self, left: int, right: int) -> list[SimplicialCohomologyProductTerm]",
    "signatures": [
      "(self, left: int, right: int) -> list[SimplicialCohomologyProductTerm]"
    ],
    "summary": "Compute product for this `SimplicialSetCohomologyRing`.",
    "summary_source": "derived projection vocabulary",
    "type_id": "noether/topology/SimplicialSetCohomologyRing@1"
  },
  {
    "canonical_name": "topology.SimplicialSetCohomologyRing.products",
    "determinism": "deterministic",
    "doc": "",
    "domain": "topology",
    "file": "noether/topology/__init__.pyi",
    "input_schema": {
      "additionalProperties": false,
      "properties": {},
      "type": "object"
    },
    "kind": "property",
    "line": 634,
    "module": "noether.topology",
    "name": "products",
    "overloads": [
      {
        "input_schema": {
          "additionalProperties": false,
          "properties": {},
          "type": "object"
        },
        "parameters": [],
        "result_schema": {
          "items": {
            "x-python-type": "SimplicialCohomologyProduct"
          },
          "type": "array",
          "x-python-type": "list[SimplicialCohomologyProduct]"
        },
        "signature": "(self) -> list[SimplicialCohomologyProduct]"
      }
    ],
    "qualified_name": "SimplicialSetCohomologyRing.products",
    "releases_gil": false,
    "result_schema": {
      "items": {
        "x-python-type": "SimplicialCohomologyProduct"
      },
      "type": "array",
      "x-python-type": "list[SimplicialCohomologyProduct]"
    },
    "signature": "(self) -> list[SimplicialCohomologyProduct]",
    "signatures": [
      "(self) -> list[SimplicialCohomologyProduct]"
    ],
    "summary": "Return the products recorded by this `SimplicialSetCohomologyRing`.",
    "summary_source": "derived projection vocabulary",
    "type_id": "noether/topology/SimplicialSetCohomologyRing@1"
  },
  {
    "canonical_name": "topology.SimplicialSetCohomologyRing.simplicial_set",
    "determinism": "deterministic",
    "doc": "",
    "domain": "topology",
    "file": "noether/topology/__init__.pyi",
    "input_schema": {
      "additionalProperties": false,
      "properties": {},
      "type": "object"
    },
    "kind": "property",
    "line": 628,
    "module": "noether.topology",
    "name": "simplicial_set",
    "overloads": [
      {
        "input_schema": {
          "additionalProperties": false,
          "properties": {},
          "type": "object"
        },
        "parameters": [],
        "result_schema": {
          "x-python-type": "SimplicialSet"
        },
        "signature": "(self) -> SimplicialSet"
      }
    ],
    "qualified_name": "SimplicialSetCohomologyRing.simplicial_set",
    "releases_gil": false,
    "result_schema": {
      "x-python-type": "SimplicialSet"
    },
    "signature": "(self) -> SimplicialSet",
    "signatures": [
      "(self) -> SimplicialSet"
    ],
    "summary": "Return the simplicial set recorded by this `SimplicialSetCohomologyRing`.",
    "summary_source": "derived projection vocabulary",
    "type_id": "noether/topology/SimplicialSetCohomologyRing@1"
  },
  {
    "canonical_name": "topology.SimplicialSetCohomologyRing.steenrod_reduced_power",
    "determinism": "deterministic",
    "doc": "",
    "domain": "topology",
    "file": "noether/topology/__init__.pyi",
    "input_schema": {
      "additionalProperties": false,
      "properties": {
        "power": {
          "type": "integer"
        },
        "source": {
          "type": "integer"
        }
      },
      "required": [
        "power",
        "source"
      ],
      "type": "object"
    },
    "kind": "method",
    "line": 641,
    "module": "noether.topology",
    "name": "steenrod_reduced_power",
    "overloads": [
      {
        "input_schema": {
          "additionalProperties": false,
          "properties": {
            "power": {
              "type": "integer"
            },
            "source": {
              "type": "integer"
            }
          },
          "required": [
            "power",
            "source"
          ],
          "type": "object"
        },
        "parameters": [
          {
            "kind": "positional_or_keyword",
            "name": "power",
            "required": true,
            "schema": {
              "type": "integer"
            }
          },
          {
            "kind": "positional_or_keyword",
            "name": "source",
            "required": true,
            "schema": {
              "type": "integer"
            }
          }
        ],
        "result_schema": {
          "items": {
            "x-python-type": "SimplicialCohomologyProductTerm"
          },
          "type": "array",
          "x-python-type": "list[SimplicialCohomologyProductTerm]"
        },
        "signature": "(self, power: int, source: int) -> list[SimplicialCohomologyProductTerm]"
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    ],
    "qualified_name": "SimplicialSetCohomologyRing.steenrod_reduced_power",
    "releases_gil": false,
    "result_schema": {
      "items": {
        "x-python-type": "SimplicialCohomologyProductTerm"
      },
      "type": "array",
      "x-python-type": "list[SimplicialCohomologyProductTerm]"
    },
    "signature": "(self, power: int, source: int) -> list[SimplicialCohomologyProductTerm]",
    "signatures": [
      "(self, power: int, source: int) -> list[SimplicialCohomologyProductTerm]"
    ],
    "summary": "Compute steenrod reduced power for this `SimplicialSetCohomologyRing`.",
    "summary_source": "derived projection vocabulary",
    "type_id": "noether/topology/SimplicialSetCohomologyRing@1"
  },
  {
    "canonical_name": "topology.SimplicialSetCohomologyRing.steenrod_square",
    "determinism": "deterministic",
    "doc": "",
    "domain": "topology",
    "file": "noether/topology/__init__.pyi",
    "input_schema": {
      "additionalProperties": false,
      "properties": {
        "source": {
          "type": "integer"
        },
        "square": {
          "type": "integer"
        }
      },
      "required": [
        "square",
        "source"
      ],
      "type": "object"
    },
    "kind": "method",
    "line": 638,
    "module": "noether.topology",
    "name": "steenrod_square",
    "overloads": [
      {
        "input_schema": {
          "additionalProperties": false,
          "properties": {
            "source": {
              "type": "integer"
            },
            "square": {
              "type": "integer"
            }
          },
          "required": [
            "square",
            "source"
          ],
          "type": "object"
        },
        "parameters": [
          {
            "kind": "positional_or_keyword",
            "name": "square",
            "required": true,
            "schema": {
              "type": "integer"
            }
          },
          {
            "kind": "positional_or_keyword",
            "name": "source",
            "required": true,
            "schema": {
              "type": "integer"
            }
          }
        ],
        "result_schema": {
          "items": {
            "x-python-type": "SimplicialCohomologyProductTerm"
          },
          "type": "array",
          "x-python-type": "list[SimplicialCohomologyProductTerm]"
        },
        "signature": "(self, square: int, source: int) -> list[SimplicialCohomologyProductTerm]"
      }
    ],
    "qualified_name": "SimplicialSetCohomologyRing.steenrod_square",
    "releases_gil": false,
    "result_schema": {
      "items": {
        "x-python-type": "SimplicialCohomologyProductTerm"
      },
      "type": "array",
      "x-python-type": "list[SimplicialCohomologyProductTerm]"
    },
    "signature": "(self, square: int, source: int) -> list[SimplicialCohomologyProductTerm]",
    "signatures": [
      "(self, square: int, source: int) -> list[SimplicialCohomologyProductTerm]"
    ],
    "summary": "Compute steenrod square for this `SimplicialSetCohomologyRing`.",
    "summary_source": "derived projection vocabulary",
    "type_id": "noether/topology/SimplicialSetCohomologyRing@1"
  },
  {
    "canonical_name": "topology.SimplicialSetCohomologyRing.to_json",
    "determinism": "deterministic",
    "doc": "",
    "domain": "topology",
    "file": "noether/topology/__init__.pyi",
    "input_schema": {
      "additionalProperties": false,
      "properties": {},
      "type": "object"
    },
    "kind": "method",
    "line": 651,
    "module": "noether.topology",
    "name": "to_json",
    "overloads": [
      {
        "input_schema": {
          "additionalProperties": false,
          "properties": {},
          "type": "object"
        },
        "parameters": [],
        "result_schema": {
          "type": "string"
        },
        "signature": "(self) -> str"
      }
    ],
    "qualified_name": "SimplicialSetCohomologyRing.to_json",
    "releases_gil": false,
    "result_schema": {
      "type": "string"
    },
    "signature": "(self) -> str",
    "signatures": [
      "(self) -> str"
    ],
    "summary": "Serialize this `SimplicialSetCohomologyRing` to its canonical typed JSON representation.",
    "summary_source": "derived projection vocabulary",
    "type_id": "noether/topology/SimplicialSetCohomologyRing@1"
  },
  {
    "canonical_name": "topology.SimplicialSetCohomologyRing.verify",
    "determinism": "deterministic",
    "doc": "",
    "domain": "topology",
    "file": "noether/topology/__init__.pyi",
    "input_schema": {
      "additionalProperties": false,
      "properties": {},
      "type": "object"
    },
    "kind": "method",
    "line": 650,
    "module": "noether.topology",
    "name": "verify",
    "overloads": [
      {
        "input_schema": {
          "additionalProperties": false,
          "properties": {},
          "type": "object"
        },
        "parameters": [],
        "result_schema": {
          "type": "null"
        },
        "signature": "(self) -> None"
      }
    ],
    "qualified_name": "SimplicialSetCohomologyRing.verify",
    "releases_gil": false,
    "result_schema": {
      "type": "null"
    },
    "signature": "(self) -> None",
    "signatures": [
      "(self) -> None"
    ],
    "summary": "Replay and verify the mathematical evidence carried by this `SimplicialSetCohomologyRing`.",
    "summary_source": "derived projection vocabulary",
    "type_id": "noether/topology/SimplicialSetCohomologyRing@1"
  },
  {
    "canonical_name": "topology.SimplicialSetElementaryHornCollapse",
    "determinism": "deterministic",
    "doc": "One elementary simplicial-set collapse inverse to a horn attachment.",
    "domain": "topology",
    "file": "noether/topology/__init__.pyi",
    "input_schema": {
      "additionalProperties": false,
      "properties": {
        "face": {
          "type": "integer"
        },
        "free_face": {
          "type": "integer"
        },
        "simplex": {
          "type": "integer"
        },
        "simplex_dimension": {
          "type": "integer"
        }
      },
      "required": [
        "simplex_dimension",
        "simplex",
        "face",
        "free_face"
      ],
      "type": "object"
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    "kind": "class",
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