category · SteinerCategory¶

41 accepted declarations. Python and Rust forms, contracts, evidence, failure boundaries, and checked usages are kept together below.

Worked example¶

The complete checked program Compose cells, compute polygraphic homology, and inspect the Street model uses this API directly.

# SPDX-License-Identifier: MIT
# Copyright (c) 2026 Benjamin Antieau and Deven Manam
"""Compose cells, compute polygraphic homology, and inspect the Street model."""

from noether.category import AtomId, SteinerCategory


def main() -> None:
    triangle = SteinerCategory.oriental(2)
    first = triangle.atom(AtomId(1, 0))
    second = triangle.atom(AtomId(1, 2))
    long_path = first.compose_at(second, 0)

    assert long_path.top_atoms == [0, 2]
    assert triangle.exact_cell_counts() == [3, 4, 1]
    assert triangle.polygraphic_homology_invariants()[0].free_rank == 1

    categorical = triangle.categorical_homology()
    nerve = categorical.skeleton(2)
    assert nerve.verify()
    assert categorical.component_poset_slice(2).poset.order == 6
    print(
        "2-oriental:",
        f"cells={triangle.exact_cell_counts()}",
        f"Street simplices={nerve.simplex_counts}",
        "stable hPos_0 weight 2 has 6 elements",
    )


if __name__ == "__main__":
    main()

Run it with .venv/bin/python examples/28_steiner_categories.py.

`category.SteinerCategory`¶

Validated finite strong Steiner category.

PythonRust

SteinerCategory(atom_counts: Sequence[int], boundaries: Sequence[Sequence[SignedBoundary]])

noether::category::SteinerCategory · Direct Rust declaration

pub struct SteinerCategory { /* private fields */ }

Contract	Value
Guarantee	checked deterministic result
Determinism	deterministic
Availability	default CPU build
Domain implementation crates	noether
GIL	attached / bounded
Semantic ID	noether/category/SteinerCategory@1

Executable usage: examples/28_steiner_categories.py:5 · examples/28_steiner_categories.py:9 · validation/unit/category/test_acceptance_workload.py:20 · validation/unit/category/test_acceptance_workload.py:24 · validation/unit/category/test_acceptance_workload.py:5 · validation/unit/category/test_acceptance_workload.py:9

`category.SteinerCategory.eq`¶

Test mathematical equality with another value.

PythonRust

(self, other: object) -> bool

noether::category::SteinerCategory · Python projection

Python projection convenience over this Rust mathematical type

Contract	Value
Guarantee	checked deterministic result
Determinism	deterministic
Availability	default CPU build
Domain implementation crates	noether
GIL	attached / bounded
Semantic ID	noether/category/SteinerCategory@1

`category.SteinerCategory.atom`¶

Compute atom for this SteinerCategory.

Detailed contract

Return one atomic cell.

PythonRust

(self, atom: AtomId) -> SteinerCell

noether::category::SteinerCategory::atom · Direct Rust declaration

pub fn atom(&self, atom: AtomId) -> Result<SteinerCell, CategoryError>

Contract	Value
Guarantee	checked deterministic result
Determinism	deterministic
Availability	default CPU build
Domain implementation crates	noether
GIL	attached / bounded
Semantic ID	noether/category/SteinerCategory@1

Executable usage: examples/28_steiner_categories.py:10 · examples/28_steiner_categories.py:11 · validation/unit/category/test_acceptance_workload.py:10 · validation/unit/category/test_acceptance_workload.py:11 · validation/unit/category/test_acceptance_workload.py:25 · validation/unit/category/test_acceptance_workload.py:26

`category.SteinerCategory.atom_count`¶

Return the atom count recorded by this SteinerCategory.

Detailed contract

Total number of atoms.

PythonRust

(self) -> int

noether::category::SteinerCategory::atom_count · Direct Rust declaration

pub fn atom_count(&self) -> usize

Contract	Value
Guarantee	checked deterministic result
Determinism	deterministic
Availability	default CPU build
Domain implementation crates	noether
GIL	attached / bounded
Semantic ID	noether/category/SteinerCategory@1

`category.SteinerCategory.atom_count_in_degree`¶

Compute atom count in degree for this SteinerCategory.

Detailed contract

Number of atoms in one degree; zero beyond the stored complex.

PythonRust

(self, degree: int) -> int

noether::category::SteinerCategory::atom_count_in_degree · Direct Rust declaration

pub fn atom_count_in_degree(&self, degree: usize) -> usize

Contract	Value
Guarantee	checked deterministic result
Determinism	deterministic
Availability	default CPU build
Domain implementation crates	noether
GIL	attached / bounded
Semantic ID	noether/category/SteinerCategory@1

`category.SteinerCategory.atom_counts`¶

Return the atom counts recorded by this SteinerCategory.

Detailed contract

Numbers of atoms in each stored degree.

PythonRust

(self) -> list[int]

noether::category::SteinerCategory::atom_counts · Direct Rust declaration

pub fn atom_counts(&self) -> &[usize]

Contract	Value
Guarantee	checked deterministic result
Determinism	deterministic
Availability	default CPU build
Domain implementation crates	noether
GIL	attached / bounded
Semantic ID	noether/category/SteinerCategory@1

Executable usage: validation/unit/category/test_acceptance_workload.py:22 · validation/unit/category/test_category.py:19

`category.SteinerCategory.automorphism_group`¶

Compute automorphism group for this SteinerCategory.

Detailed contract

Complete automorphism group acting on flattened atomic ids.

PythonRust

(self) -> PermutationGroup

noether::category::SteinerCategory::automorphism_group · Direct Rust declaration

pub fn automorphism_group(&self) -> Result<PermutationGroup, CategoryError>

Contract	Value
Guarantee	checked deterministic result
Determinism	deterministic
Availability	default CPU build
Domain implementation crates	noether
GIL	released
Semantic ID	noether/category/SteinerCategory@1

Executable usage: examples/21_graph_symmetry.py:16 · examples/21_graph_symmetry.py:18 · examples/29_integral_line_configurations.py:10 · validation/unit/category/test_category.py:55 · validation/unit/combin/test_acceptance_workload.py:31 · validation/unit/combin/test_graph_facade.py:91

`category.SteinerCategory.boundary`¶

Compute boundary for this SteinerCategory.

Detailed contract

Signed boundary of one atom.

PythonRust

(self, atom: AtomId) -> SignedBoundary

noether::category::SteinerCategory::boundary · Direct Rust declaration

pub fn boundary(&self, atom: AtomId) -> Result<&SignedBoundary, CategoryError>

Contract	Value
Guarantee	checked deterministic result
Determinism	deterministic
Availability	default CPU build
Domain implementation crates	noether
GIL	attached / bounded
Semantic ID	noether/category/SteinerCategory@1

Executable usage: validation/unit/commalg/test_kahler_differentials.py:3 · validation/unit/conftest.py:5 · validation/unit/homalg/test_chain_complex.py:116 · validation/unit/homalg/test_chain_complex.py:117 · validation/unit/homalg/test_chain_complex.py:185 · validation/unit/homalg/test_chain_complex.py:186

`category.SteinerCategory.canonical_form`¶

Compute canonical form for this SteinerCategory.

Detailed contract

Canonical representative under degree-preserving atom permutations.

PythonRust

(self) -> SteinerCategory

noether::category::SteinerCategory::canonical_form · Direct Rust declaration

pub fn canonical_form(&self) -> Result<Self, CategoryError>

Contract	Value
Guarantee	checked deterministic result
Determinism	deterministic
Availability	default CPU build
Domain implementation crates	noether
GIL	released
Semantic ID	noether/category/SteinerCategory@1

Executable usage: validation/unit/category/test_category.py:38 · validation/unit/combin/test_acceptance_workload.py:27 · validation/unit/combin/test_acceptance_workload.py:29 · validation/unit/infra/test_discovery.py:122 · validation/unit/infra/test_discovery.py:125 · validation/unit/infra/test_discovery.py:56

`category.SteinerCategory.canonical_label`¶

Compute canonical label for this SteinerCategory.

Detailed contract

Exact canonical labeling and complete atomic automorphism group.

PythonRust

(self) -> SteinerCanonicalLabel

noether::category::SteinerCategory::canonical_label · Direct Rust declaration

pub fn canonical_label(&self) -> Result<SteinerCanonicalLabel, CategoryError>

Contract	Value
Guarantee	checked deterministic result
Determinism	deterministic
Availability	default CPU build
Domain implementation crates	noether
GIL	released
Semantic ID	noether/category/SteinerCategory@1

Executable usage: examples/21_graph_symmetry.py:14 · validation/unit/category/test_category.py:16 · validation/unit/category/test_category.py:17 · validation/unit/combin/test_graph_facade.py:84 · validation/unit/combin/test_graph_facade.py:85 · validation/unit/combinatorics/test_poset.py:106

`category.SteinerCategory.categorical_homology`¶

Compute categorical homology for this SteinerCategory.

Detailed contract

Exact lazy presentation of categorical homology with coefficients in N.

PythonRust

(self) -> CategoricalHomologyPresentation

noether::category::SteinerCategory::categorical_homology · Direct Rust declaration

pub fn categorical_homology(&self) -> CategoricalHomologyPresentation

Contract	Value
Guarantee	checked deterministic result
Determinism	deterministic
Availability	default CPU build
Domain implementation crates	noether
GIL	released
Semantic ID	noether/category/SteinerCategory@1

Executable usage: examples/28_steiner_categories.py:18 · validation/unit/category/test_category.py:70 · validation/unit/category/test_category.py:76

`category.SteinerCategory.cell`¶

Compute cell for this SteinerCategory.

Detailed contract

Construct the unwhiskered cell generated by a nonempty top atomic chain.

PythonRust

(self, dimension: int, top_atoms: Sequence[int]) -> SteinerCell

noether::category::SteinerCategory::cell · Direct Rust declaration

pub fn cell( &self, dimension: usize, top_atoms: &[usize], ) -> Result<SteinerCell, CategoryError>

Contract	Value
Guarantee	checked deterministic result
Determinism	deterministic
Availability	default CPU build
Domain implementation crates	noether
GIL	attached / bounded
Semantic ID	noether/category/SteinerCategory@1

Executable usage: validation/unit/category/test_category.py:24 · validation/unit/category/test_category.py:26 · validation/unit/category/test_category.py:27 · validation/unit/category/test_category.py:29 · validation/parity/combinatorics/test_tree_core_parity.py:13 · validation/parity/combinatorics/test_tree_pyo3.py:16

`category.SteinerCategory.cell_from_supports`¶

Compute cell from supports for this SteinerCategory.

Detailed contract

Reconstruct a cell from its complete Steiner double sequence. Unlike Self::cell, this retains common lower-dimensional support and therefore represents whiskered cells faithfully. Each support must be a sorted degree-local subset. The two sequences are checked against all boundary, augmentation, and top-chain equations before the cell is returned.

PythonRust

(self, negative_supports: Sequence[Sequence[int]], positive_supports: Sequence[Sequence[int]]) -> SteinerCell

noether::category::SteinerCategory::cell_from_supports · Direct Rust declaration

pub fn cell_from_supports( &self, negative_supports: Vec<Vec<usize>>, positive_supports: Vec<Vec<usize>>, ) -> Result<SteinerCell, CategoryError>

Contract	Value
Guarantee	checked deterministic result
Determinism	deterministic
Availability	default CPU build
Domain implementation crates	noether
GIL	attached / bounded
Semantic ID	noether/category/SteinerCategory@1

Executable usage: validation/unit/category/test_category.py:25

`category.SteinerCategory.cells`¶

Compute cells for this SteinerCategory.

Detailed contract

Enumerate all cells at one dimension, including iterated identities.

PythonRust

(self, dimension: int) -> list[SteinerCell]

noether::category::SteinerCategory::cells · Direct Rust declaration

pub fn cells(&self, dimension: usize) -> Result<Vec<SteinerCell>, CategoryError>

Contract	Value
Guarantee	checked deterministic result
Determinism	deterministic
Availability	default CPU build
Domain implementation crates	noether
GIL	released
Semantic ID	noether/category/SteinerCategory@1

Executable usage: examples/28_steiner_categories.py:24 · examples/28_steiner_categories.py:3 · examples/32_poset_core_and_morse.py:22 · validation/unit/geometry/test_geometry_facade.py:60 · validation/bench/macaulay2_affine_normalization.py:636 · validation/bench/macaulay2_affine_normalization.py:637

`category.SteinerCategory.component_poset`¶

Compute component poset for this SteinerCategory.

Detailed contract

Gepner–Heine Poset of components. Its elements are the objects of this strong Steiner category, ordered by the existence of a 1-cell.

PythonRust

(self) -> Poset

noether::category::SteinerCategory::component_poset · Direct Rust declaration

pub fn component_poset(&self) -> Result<Poset, CategoryError>

Contract	Value
Guarantee	checked deterministic result
Determinism	deterministic
Availability	default CPU build
Domain implementation crates	noether
GIL	released
Semantic ID	noether/category/SteinerCategory@1

Executable usage: validation/unit/category/test_acceptance_workload.py:28 · validation/unit/infra/test_discovery.py:187

`category.SteinerCategory.cube`¶

Construct a SteinerCategory using cube.

Detailed contract

The directed dimension-cube, formed by repeated Gray tensor product.

PythonRust

(dimension: int) -> SteinerCategory

noether::category::SteinerCategory::cube · Direct Rust declaration

pub fn cube(dimension: usize) -> Result<Self, CategoryError>

Contract	Value
Guarantee	checked deterministic result
Determinism	deterministic
Availability	default CPU build
Domain implementation crates	noether
GIL	attached / bounded
Semantic ID	noether/category/SteinerCategory@1

Executable usage: validation/unit/homalg/test_module_facade.py:336 · validation/unit/homalg/test_module_facade.py:338 · validation/unit/homalg/test_module_facade.py:343 · validation/unit/homalg/test_module_facade.py:345 · validation/unit/homalg/test_module_facade.py:348 · validation/parity/polyhedral/test_convex_hull_parity.py:14

`category.SteinerCategory.dimension`¶

Return the dimension recorded by this SteinerCategory.

Detailed contract

Largest degree containing an atom, or -1 for the empty category.

PythonRust

(self) -> int

noether::category::SteinerCategory::dimension · Direct Rust declaration

pub fn dimension(&self) -> i64

Contract	Value
Guarantee	checked deterministic result
Determinism	deterministic
Availability	default CPU build
Domain implementation crates	noether
GIL	attached / bounded
Semantic ID	noether/category/SteinerCategory@1

Executable usage: examples/03_exact_linear_systems.py:15 · examples/03_exact_linear_systems.py:16 · examples/15_groebner_twisted_cubic.py:14 · examples/15_groebner_twisted_cubic.py:28 · examples/27_weight_characters.py:25 · examples/34_finite_group_cohomology.py:13

`category.SteinerCategory.disjoint_union`¶

Compute disjoint union for this SteinerCategory.

Detailed contract

Coproduct (disjoint union) of two finite strong Steiner categories.

PythonRust

(self, other: SteinerCategory) -> SteinerCategory

noether::category::SteinerCategory::disjoint_union · Direct Rust declaration

pub fn disjoint_union(&self, other: &Self) -> Result<Self, CategoryError>

Contract	Value
Guarantee	checked deterministic result
Determinism	deterministic
Availability	default CPU build
Domain implementation crates	noether
GIL	attached / bounded
Semantic ID	noether/category/SteinerCategory@1

Executable usage: validation/unit/category/test_category.py:51 · validation/unit/category/test_category.py:53

`category.SteinerCategory.dual`¶

Compute dual for this SteinerCategory.

Detailed contract

Reverse the selected globular dimensions. A selected positive degree negates that differential, equivalently interchanging the negative and positive atomic boundary supports. The entries must be distinct positive dimensions present in the category. The result is returned only when this same atomic basis remains strong and loop-free; a strict categorical dual can exist outside that presentation class.

PythonRust

(self, dimensions: Sequence[int]) -> SteinerCategory

noether::category::SteinerCategory::dual · Direct Rust declaration

pub fn dual(&self, dimensions: &[usize]) -> Result<Self, CategoryError>

Contract	Value
Guarantee	checked deterministic result
Determinism	deterministic
Availability	default CPU build
Domain implementation crates	noether
GIL	released
Semantic ID	noether/category/SteinerCategory@1

Executable usage: examples/40_exterior_algebra.py:28 · validation/unit/category/test_category.py:18 · validation/unit/groups/test_weight_characters.py:47 · validation/unit/homalg/test_exterior_algebra.py:123 · validation/unit/homalg/test_exterior_algebra.py:124 · validation/unit/homalg/test_exterior_algebra.py:125

`category.SteinerCategory.empty`¶

Construct a SteinerCategory using empty.

Detailed contract

The empty strict omega-category.

PythonRust

() -> SteinerCategory

noether::category::SteinerCategory::empty · Direct Rust declaration

pub fn empty() -> Self

Contract	Value
Guarantee	checked deterministic result
Determinism	deterministic
Availability	default CPU build
Domain implementation crates	noether
GIL	attached / bounded
Semantic ID	noether/category/SteinerCategory@1

Executable usage: validation/unit/category/test_category.py:49 · validation/unit/category/test_category.py:52 · validation/unit/category/test_category.py:53 · validation/unit/combinatorics/test_matroid.py:74 · validation/unit/combinatorics/test_matroid.py:75 · validation/unit/combinatorics/test_matroid.py:76

`category.SteinerCategory.exact_cell_counts`¶

Compute exact cell counts for this SteinerCategory.

Detailed contract

Exact-dimensional nonidentity cell counts.

PythonRust

(self) -> list[int]

noether::category::SteinerCategory::exact_cell_counts · Direct Rust declaration

pub fn exact_cell_counts(&self) -> Result<Vec<usize>, CategoryError>

Contract	Value
Guarantee	checked deterministic result
Determinism	deterministic
Availability	default CPU build
Domain implementation crates	noether
GIL	released
Semantic ID	noether/category/SteinerCategory@1

Executable usage: examples/28_steiner_categories.py:15 · examples/28_steiner_categories.py:24 · validation/unit/category/test_acceptance_workload.py:14 · validation/unit/category/test_category.py:54 · validation/unit/infra/test_discovery.py:189

`category.SteinerCategory.exact_cells`¶

Compute exact cells for this SteinerCategory.

Detailed contract

Enumerate exact-dimensional nonidentity cells in one degree.

PythonRust

(self, dimension: int) -> list[SteinerCell]

noether::category::SteinerCategory::exact_cells · Direct Rust declaration

pub fn exact_cells( &self, dimension: usize, ) -> Result<Vec<SteinerCell>, CategoryError>

Contract	Value
Guarantee	checked deterministic result
Determinism	deterministic
Availability	default CPU build
Domain implementation crates	noether
GIL	released
Semantic ID	noether/category/SteinerCategory@1

Executable usage: validation/unit/infra/test_discovery.py:188

`category.SteinerCategory.from_json`¶

Deserialize and revalidate a SteinerCategory from typed JSON.

PythonRust

(value: str) -> SteinerCategory

noether::category::SteinerCategory · Python projection

Python projection convenience over this Rust mathematical type

Contract	Value
Guarantee	checked deterministic result
Determinism	deterministic
Availability	default CPU build
Domain implementation crates	noether
GIL	attached / bounded
Semantic ID	noether/category/SteinerCategory@1

Executable usage: validation/unit/category/test_category.py:14 · validation/unit/combin/test_acceptance_workload.py:65 · validation/unit/combin/test_acceptance_workload.py:86 · validation/unit/combin/test_graph_facade.py:106 · validation/unit/combin/test_graph_facade.py:25 · validation/unit/combinatorics/test_kashiwara_crystal.py:32

`category.SteinerCategory.fundamental_poset`¶

Compute fundamental poset for this SteinerCategory.

Detailed contract

Gepner–Heine fundamental Poset at an ordered pair of objects. This uses the strong-Steiner description by directed atomic paths and atomic 2-cell relations, avoiding the generic iterated-boundary filter.

PythonRust

(self, source: SteinerCell, target: SteinerCell) -> Poset

noether::category::SteinerCategory::fundamental_poset · Direct Rust declaration

pub fn fundamental_poset( &self, source: &SteinerCell, target: &SteinerCell, ) -> Result<Poset, CategoryError>

Contract	Value
Guarantee	checked deterministic result
Determinism	deterministic
Availability	default CPU build
Domain implementation crates	noether
GIL	released
Semantic ID	noether/category/SteinerCategory@1

Executable usage: validation/unit/category/test_category.py:33 · validation/unit/category/test_category.py:76 · validation/unit/infra/test_discovery.py:183

`category.SteinerCategory.globe`¶

Construct a SteinerCategory using globe.

Detailed contract

The walking dimension-cell.

PythonRust

(dimension: int) -> SteinerCategory

noether::category::SteinerCategory::globe · Direct Rust declaration

pub fn globe(dimension: usize) -> Result<Self, CategoryError>

Contract	Value
Guarantee	checked deterministic result
Determinism	deterministic
Availability	default CPU build
Domain implementation crates	noether
GIL	attached / bounded
Semantic ID	noether/category/SteinerCategory@1

Executable usage: validation/unit/category/test_acceptance_workload.py:20 · validation/unit/category/test_category.py:38 · validation/unit/category/test_category.py:65 · validation/unit/category/test_category.py:69

`category.SteinerCategory.gray_tensor`¶

Compute gray tensor for this SteinerCategory.

Detailed contract

Non-symmetric Gray tensor product.

PythonRust

(self, other: SteinerCategory) -> SteinerCategory

noether::category::SteinerCategory::gray_tensor · Direct Rust declaration

pub fn gray_tensor(&self, other: &Self) -> Result<Self, CategoryError>

Contract	Value
Guarantee	checked deterministic result
Determinism	deterministic
Availability	default CPU build
Domain implementation crates	noether
GIL	released
Semantic ID	noether/category/SteinerCategory@1

Executable usage: validation/unit/category/test_acceptance_workload.py:21

`category.SteinerCategory.homotopy_poset`¶

Compute homotopy poset for this SteinerCategory.

Detailed contract

Compute the Gepner–Heine homotopy Poset at a valid basepoint.

PythonRust

(self, basepoint: OrientedBasepoint) -> Poset

noether::category::SteinerCategory::homotopy_poset · Direct Rust declaration

pub fn homotopy_poset( &self, basepoint: &OrientedBasepoint, ) -> Result<Poset, CategoryError>

Contract	Value
Guarantee	checked deterministic result
Determinism	deterministic
Availability	default CPU build
Domain implementation crates	noether
GIL	released
Semantic ID	noether/category/SteinerCategory@1

Executable usage: validation/unit/infra/test_discovery.py:191

`category.SteinerCategory.is_isomorphic`¶

Test whether this SteinerCategory is isomorphic.

Detailed contract

Whether two categories are strictly isomorphic.

PythonRust

(self, other: SteinerCategory) -> bool

noether::category::SteinerCategory::is_isomorphic · Direct Rust declaration

pub fn is_isomorphic(&self, other: &Self) -> Result<bool, CategoryError>

Contract	Value
Guarantee	checked deterministic result
Determinism	deterministic
Availability	default CPU build
Domain implementation crates	noether
GIL	released
Semantic ID	noether/category/SteinerCategory@1

`category.SteinerCategory.mapping_category`¶

Compute mapping category for this SteinerCategory.

Detailed contract

Mapping omega-category between two objects.

PythonRust

(self, source: SteinerCell, target: SteinerCell) -> FiniteOmegaCategory

noether::category::SteinerCategory::mapping_category · Direct Rust declaration

pub fn mapping_category( &self, source: &SteinerCell, target: &SteinerCell, ) -> Result<FiniteOmegaCategory, CategoryError>

Contract	Value
Guarantee	checked deterministic result
Determinism	deterministic
Availability	default CPU build
Domain implementation crates	noether
GIL	released
Semantic ID	noether/category/SteinerCategory@1

Executable usage: validation/unit/category/test_acceptance_workload.py:27 · validation/unit/category/test_category.py:35 · validation/unit/infra/test_discovery.py:186

`category.SteinerCategory.monomorphic_oriented_basepoints`¶

Compute monomorphic oriented basepoints for this SteinerCategory.

Detailed contract

Enumerate monomorphic oriented basepoints of one dimension.

PythonRust

(self, dimension: int) -> list[OrientedBasepoint]

noether::category::SteinerCategory::monomorphic_oriented_basepoints · Direct Rust declaration

pub fn monomorphic_oriented_basepoints( &self, dimension: usize, ) -> Result<Vec<OrientedBasepoint>, CategoryError>

Contract	Value
Guarantee	checked deterministic result
Determinism	deterministic
Availability	default CPU build
Domain implementation crates	noether
GIL	released
Semantic ID	noether/category/SteinerCategory@1

`category.SteinerCategory.oriental`¶

Construct the Street oriental of the requested dimension.

Detailed contract

Street’s oriented dimension-simplex.

PythonRust

(dimension: int) -> SteinerCategory

noether::category::SteinerCategory::oriental · Direct Rust declaration

pub fn oriental(dimension: usize) -> Result<Self, CategoryError>

Contract	Value
Guarantee	checked deterministic result
Determinism	deterministic
Availability	default CPU build
Dispatch / backing	noether
GIL	attached / bounded
Semantic ID	noether/category/SteinerCategory@1
Evidence	SteinerCategory.verify
Failures	dimension exceeds the bounded construction limit

Executable usage: examples/28_steiner_categories.py:23 · examples/28_steiner_categories.py:9 · validation/unit/category/test_acceptance_workload.py:24 · validation/unit/category/test_acceptance_workload.py:9 · validation/unit/category/test_category.py:13 · validation/unit/category/test_category.py:23

`category.SteinerCategory.point`¶

Construct a SteinerCategory using point.

Detailed contract

The terminal zero-category.

PythonRust

() -> SteinerCategory

noether::category::SteinerCategory::point · Direct Rust declaration

pub fn point() -> Self

Contract	Value
Guarantee	checked deterministic result
Determinism	deterministic
Availability	default CPU build
Domain implementation crates	noether
GIL	attached / bounded
Semantic ID	noether/category/SteinerCategory@1

Executable usage: examples/26_elliptic_descent_functions.py:17 · examples/26_elliptic_descent_functions.py:18 · examples/26_elliptic_descent_functions.py:20 · examples/26_elliptic_descent_functions.py:25 · examples/26_elliptic_descent_functions.py:31 · examples/26_elliptic_descent_functions.py:32

Example

from noether.category import SteinerCategory
point = SteinerCategory.point()
assert point.dimension == 0
assert point.atom_counts == [1]
point.verify()

`category.SteinerCategory.polygraphic_homology`¶

Compute polygraphic homology for this SteinerCategory.

Detailed contract

Integral polygraphic homology with source-retaining witnesses.

PythonRust

(self) -> GradedHomology[IntegerRing]

noether::category::SteinerCategory::polygraphic_homology · Direct Rust declaration

pub fn polygraphic_homology( &self, ) -> Result<GradedHomology<IntegerRing>, CategoryError>

Contract	Value
Guarantee	checked deterministic result
Determinism	deterministic
Availability	default CPU build
Domain implementation crates	noether
GIL	released
Semantic ID	noether/category/SteinerCategory@1

`category.SteinerCategory.polygraphic_homology_invariants`¶

Compute polygraphic homology invariants for this SteinerCategory.

Detailed contract

Lean integral polygraphic homology invariants by degree.

PythonRust

(self) -> dict[int, FinitelyGeneratedAbelianGroup]

noether::category::SteinerCategory::polygraphic_homology_invariants · Direct Rust declaration

pub fn polygraphic_homology_invariants( &self, ) -> Result<BTreeMap<i32, FinitelyGeneratedAbelianGroup>, CategoryError>

Contract	Value
Guarantee	checked deterministic result
Determinism	deterministic
Availability	default CPU build
Domain implementation crates	noether
GIL	released
Semantic ID	noether/category/SteinerCategory@1

Executable usage: examples/28_steiner_categories.py:16 · validation/unit/category/test_acceptance_workload.py:15

`category.SteinerCategory.precedence_poset`¶

Compute precedence poset for this SteinerCategory.

Detailed contract

The strong precedence Poset on the flattened atomic basis.

PythonRust

(self) -> Poset

noether::category::SteinerCategory::precedence_poset · Direct Rust declaration

pub fn precedence_poset(&self) -> Result<Poset, CategoryError>

Contract	Value
Guarantee	checked deterministic result
Determinism	deterministic
Availability	default CPU build
Domain implementation crates	noether
GIL	attached / bounded
Semantic ID	noether/category/SteinerCategory@1

`category.SteinerCategory.reduced_categorical_homology`¶

Compute reduced categorical homology for this SteinerCategory.

Detailed contract

Reduced categorical homology pointed at one object.

PythonRust

(self, object: SteinerCell) -> CategoricalHomologyPresentation

noether::category::SteinerCategory::reduced_categorical_homology · Direct Rust declaration

pub fn reduced_categorical_homology( &self, object: &SteinerCell, ) -> Result<CategoricalHomologyPresentation, CategoryError>

Contract	Value
Guarantee	checked deterministic result
Determinism	deterministic
Availability	default CPU build
Domain implementation crates	noether
GIL	released
Semantic ID	noether/category/SteinerCategory@1

`category.SteinerCategory.street_nerve_skeleton`¶

Compute street nerve skeleton for this SteinerCategory.

Detailed contract

Exact Street-nerve skeleton through max_simplicial_degree.

PythonRust

(self, max_degree: int) -> StreetNerveSkeleton

noether::category::SteinerCategory::street_nerve_skeleton · Direct Rust declaration

pub fn street_nerve_skeleton( &self, max_simplicial_degree: usize, ) -> Result<StreetNerveSkeleton, CategoryError>

Contract	Value
Guarantee	checked deterministic result
Determinism	deterministic
Availability	default CPU build
Domain implementation crates	noether
GIL	released
Semantic ID	noether/category/SteinerCategory@1

Executable usage: validation/unit/category/test_acceptance_workload.py:16

`category.SteinerCategory.to_json`¶

Serialize this SteinerCategory to its canonical typed JSON representation.

PythonRust

(self) -> str

noether::category::SteinerCategory · Python projection

Python projection convenience over this Rust mathematical type

Contract	Value
Guarantee	checked deterministic result
Determinism	deterministic
Availability	default CPU build
Domain implementation crates	noether
GIL	attached / bounded
Semantic ID	noether/category/SteinerCategory@1

Executable usage: validation/unit/category/test_category.py:14 · validation/unit/category/test_category.py:15 · validation/unit/combin/test_acceptance_workload.py:30 · validation/unit/combin/test_acceptance_workload.py:65 · validation/unit/combin/test_acceptance_workload.py:86 · validation/unit/combin/test_graph_facade.py:106

`category.SteinerCategory.truncate`¶

Compute truncate for this SteinerCategory.

Detailed contract

Strict truncation obtained by forgetting all atoms above dimension.

PythonRust

(self, dimension: int) -> SteinerCategory

noether::category::SteinerCategory::truncate · Direct Rust declaration

pub fn truncate(&self, dimension: usize) -> Result<Self, CategoryError>

Contract	Value
Guarantee	checked deterministic result
Determinism	deterministic
Availability	default CPU build
Domain implementation crates	noether
GIL	released
Semantic ID	noether/category/SteinerCategory@1

Executable usage: validation/unit/category/test_category.py:19 · validation/parity/commalg/test_intersection_theory_sage_parity.py:117 · validation/parity/commalg/test_intersection_theory_sage_parity.py:121 · validation/parity/commalg/test_intersection_theory_sage_parity.py:123 · validation/parity/commalg/test_intersection_theory_sage_parity.py:134 · validation/parity/commalg/test_intersection_theory_sage_parity.py:137

`category.SteinerCategory.underlying_chain_complex`¶

Compute underlying chain complex for this SteinerCategory.

Detailed contract

Underlying integral atomic chain complex.

PythonRust

(self) -> ChainComplex[IntegerRing]

noether::category::SteinerCategory::underlying_chain_complex · Direct Rust declaration

pub fn underlying_chain_complex( &self, ) -> Result<ChainComplex<IntegerRing>, CategoryError>

Contract	Value
Guarantee	checked deterministic result
Determinism	deterministic
Availability	default CPU build
Domain implementation crates	noether
GIL	released
Semantic ID	noether/category/SteinerCategory@1

`category.SteinerCategory.verify`¶

Replay and verify the mathematical evidence carried by this SteinerCategory.

Detailed contract

Recheck d^2 = 0, unitality, and strong loop-freeness.

PythonRust

(self) -> None

noether::category::SteinerCategory::verify · Direct Rust declaration

pub fn verify(&self) -> Result<(), CategoryError>

Contract	Value
Guarantee	checked deterministic result
Determinism	deterministic
Availability	default CPU build
Domain implementation crates	noether
GIL	attached / bounded
Semantic ID	noether/category/SteinerCategory@1

Executable usage: examples/01_bezout_certificates.py:10 · examples/01_bezout_certificates.py:3 · examples/03_exact_linear_systems.py:13 · examples/03_exact_linear_systems.py:3 · examples/04_smith_normal_form.py:12 · examples/06_shortest_lattice_vectors.py:12

category · SteinerCategory¶

Worked example¶

category.SteinerCategory¶

category.SteinerCategory.__eq__¶

category.SteinerCategory.atom¶

category.SteinerCategory.atom_count¶

category.SteinerCategory.atom_count_in_degree¶

category.SteinerCategory.atom_counts¶

category.SteinerCategory.automorphism_group¶

category.SteinerCategory.boundary¶

category.SteinerCategory.canonical_form¶

category.SteinerCategory.canonical_label¶

category.SteinerCategory.categorical_homology¶

category.SteinerCategory.cell¶

category.SteinerCategory.cell_from_supports¶

category.SteinerCategory.cells¶

category.SteinerCategory.component_poset¶

category.SteinerCategory.cube¶

category.SteinerCategory.dimension¶

category.SteinerCategory.disjoint_union¶

category.SteinerCategory.dual¶

category.SteinerCategory.empty¶

category.SteinerCategory.exact_cell_counts¶

category.SteinerCategory.exact_cells¶

category.SteinerCategory.from_json¶

category.SteinerCategory.fundamental_poset¶

category.SteinerCategory.globe¶

category.SteinerCategory.gray_tensor¶

category.SteinerCategory.homotopy_poset¶

category.SteinerCategory.is_isomorphic¶

category.SteinerCategory.mapping_category¶

category.SteinerCategory.monomorphic_oriented_basepoints¶

category.SteinerCategory.oriental¶

category.SteinerCategory.point¶

category.SteinerCategory.polygraphic_homology¶

category.SteinerCategory.polygraphic_homology_invariants¶

category.SteinerCategory.precedence_poset¶

category.SteinerCategory.reduced_categorical_homology¶

category.SteinerCategory.street_nerve_skeleton¶

category.SteinerCategory.to_json¶

category.SteinerCategory.truncate¶

category.SteinerCategory.underlying_chain_complex¶

category.SteinerCategory.verify¶

`category.SteinerCategory`¶

`category.SteinerCategory.eq`¶

`category.SteinerCategory.atom`¶

`category.SteinerCategory.atom_count`¶

`category.SteinerCategory.atom_count_in_degree`¶

`category.SteinerCategory.atom_counts`¶

`category.SteinerCategory.automorphism_group`¶

`category.SteinerCategory.boundary`¶

`category.SteinerCategory.canonical_form`¶

`category.SteinerCategory.canonical_label`¶

`category.SteinerCategory.categorical_homology`¶

`category.SteinerCategory.cell`¶

`category.SteinerCategory.cell_from_supports`¶

`category.SteinerCategory.cells`¶

`category.SteinerCategory.component_poset`¶

`category.SteinerCategory.cube`¶

`category.SteinerCategory.dimension`¶

`category.SteinerCategory.disjoint_union`¶

`category.SteinerCategory.dual`¶

`category.SteinerCategory.empty`¶

`category.SteinerCategory.exact_cell_counts`¶

`category.SteinerCategory.exact_cells`¶

`category.SteinerCategory.from_json`¶

`category.SteinerCategory.fundamental_poset`¶

`category.SteinerCategory.globe`¶

`category.SteinerCategory.gray_tensor`¶

`category.SteinerCategory.homotopy_poset`¶

`category.SteinerCategory.is_isomorphic`¶

`category.SteinerCategory.mapping_category`¶

`category.SteinerCategory.monomorphic_oriented_basepoints`¶

`category.SteinerCategory.oriental`¶

`category.SteinerCategory.point`¶

`category.SteinerCategory.polygraphic_homology`¶

`category.SteinerCategory.polygraphic_homology_invariants`¶

`category.SteinerCategory.precedence_poset`¶

`category.SteinerCategory.reduced_categorical_homology`¶

`category.SteinerCategory.street_nerve_skeleton`¶

`category.SteinerCategory.to_json`¶

`category.SteinerCategory.truncate`¶

`category.SteinerCategory.underlying_chain_complex`¶

`category.SteinerCategory.verify`¶