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commalg · SchemeCompletion

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

Worked example

The complete checked program Build and verify an isomorphism between two presentations of F_25 uses this API directly.

# SPDX-License-Identifier: MIT
# Copyright (c) 2026 Benjamin Antieau and Deven Manam
"""Build and verify an isomorphism between two presentations of F_25."""

from noether.numthy import FiniteField


def main() -> None:
    source = FiniteField.extension(5, 2)
    target = FiniteField.with_modulus(5, [2, 0, 1])
    isomorphism = source.isomorphism_to(target)
    isomorphism.verify()
    value = source.element_from_coefficients([3, 2])
    assert isomorphism.apply_inverse(isomorphism(value)) == value
    print(f"a maps to {isomorphism.generator_image}")


if __name__ == "__main__":
    main()

Run it with .venv/bin/python examples/09_finite_field_isomorphisms.py.

commalg.SchemeCompletion

Fixed-order completed local quotient presentation.

class SchemeCompletion

noether::commalg::SchemeCompletion · Direct Rust declaration

pub struct SchemeCompletion<R: Field> { /* private fields */ }
Contract Value
Guarantee checked deterministic result
Determinism deterministic
Availability default CPU build
Domain implementation crates noether, flint-core, groebner-kernel, module-core, poly-core, resolution-kernel
GIL attached / bounded
Semantic ID noether/commalg/SchemeCompletion@1

Executable usage: validation/unit/commalg/test_acceptance_workload.py:122 · validation/unit/commalg/test_acceptance_workload.py:18 · validation/unit/commalg/test_scheme_facade.py:18 · validation/unit/commalg/test_scheme_facade.py:268

commalg.SchemeCompletion.ambient_ring

Return the ambient ring recorded by this SchemeCompletion.

Detailed contract

Fixed-truncation ambient power-series context.

(self) -> PowerSeriesRing[C]

noether::commalg::SchemeCompletion::ambient_ring · Direct Rust declaration

pub fn ambient_ring(&self) -> &PowerSeriesRing<R>
Contract Value
Guarantee checked deterministic result
Determinism deterministic
Availability default CPU build
Domain implementation crates noether, flint-core, groebner-kernel, module-core, poly-core, resolution-kernel
GIL attached / bounded
Semantic ID noether/commalg/SchemeCompletion@1

Executable usage: validation/unit/commalg/test_acceptance_workload.py:120 · validation/unit/commalg/test_construction_algebras.py:36 · validation/unit/commalg/test_construction_algebras.py:37 · validation/unit/commalg/test_construction_algebras.py:44 · validation/unit/commalg/test_scheme_facade.py:262

commalg.SchemeCompletion.center

Return the center recorded by this SchemeCompletion.

Detailed contract

Rational center of the completion.

(self) -> SchemePoint

noether::commalg::SchemeCompletion::center · Direct Rust declaration

pub fn center(&self) -> &SchemePoint<R>
Contract Value
Guarantee checked deterministic result
Determinism deterministic
Availability default CPU build
Domain implementation crates noether, flint-core, groebner-kernel, module-core, poly-core, resolution-kernel
GIL attached / bounded
Semantic ID noether/commalg/SchemeCompletion@1

Executable usage: validation/unit/combin/test_graph_facade.py:102 · validation/unit/commalg/test_acceptance_workload.py:104 · validation/unit/commalg/test_acceptance_workload.py:112 · validation/unit/commalg/test_acceptance_workload.py:96 · validation/unit/commalg/test_acceptance_workload.py:98 · validation/unit/commalg/test_scheme_facade.py:223

commalg.SchemeCompletion.equations

Return the equations recorded by this SchemeCompletion.

Detailed contract

Truncated equations generating the completed local quotient presentation.

(self) -> list[PowerSeries[C]]

noether::commalg::SchemeCompletion::equations · Direct Rust declaration

pub fn equations(&self) -> &[PowerSeries<R>]
Contract Value
Guarantee checked deterministic result
Determinism deterministic
Availability default CPU build
Domain implementation crates noether, flint-core, groebner-kernel, module-core, poly-core, resolution-kernel
GIL attached / bounded
Semantic ID noether/commalg/SchemeCompletion@1

Executable usage: validation/unit/commalg/test_acceptance_workload.py:121 · validation/unit/commalg/test_kahler_differentials.py:121 · validation/unit/commalg/test_scheme_facade.py:263 · validation/parity/polyhedral/test_convex_hull_parity.py:205 · validation/bench/tritangent_stress.py:83 · validation/corpus/combin/nowhere-zero-modular-flows.py:3

commalg.SchemeCompletion.from_json

Deserialize and revalidate a SchemeCompletion from typed JSON.

Detailed contract

Deserialize and recompute the translated truncated equations.

(cls, value: str) -> SchemeCompletion[FiniteField]

noether::commalg::SchemeCompletion::from_json · Direct Rust declaration

pub fn from_json(value: &str) -> Result<Self, NoetherError>
Contract Value
Guarantee checked deterministic result
Determinism deterministic
Availability default CPU build
Domain implementation crates noether, flint-core, groebner-kernel, module-core, poly-core, resolution-kernel
GIL attached / bounded
Semantic ID noether/commalg/SchemeCompletion@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

commalg.SchemeCompletion.source

Return the source recorded by this SchemeCompletion.

Detailed contract

Scheme whose completed local presentation is recorded.

(self) -> Scheme[C]

noether::commalg::SchemeCompletion::source · Direct Rust declaration

pub fn source(&self) -> &Scheme<R>
Contract Value
Guarantee checked deterministic result
Determinism deterministic
Availability default CPU build
Domain implementation crates noether, flint-core, groebner-kernel, module-core, poly-core, resolution-kernel
GIL attached / bounded
Semantic ID noether/commalg/SchemeCompletion@1

Executable usage: examples/09_finite_field_isomorphisms.py:11 · examples/09_finite_field_isomorphisms.py:13 · examples/09_finite_field_isomorphisms.py:9 · examples/30_modules_and_resolutions.py:34 · examples/30_modules_and_resolutions.py:37 · validation/unit/category/test_acceptance_workload.py:25

commalg.SchemeCompletion.to_json

Serialize this SchemeCompletion to its canonical typed JSON representation.

Detailed contract

Serialize the local quotient presentation and fixed truncation.

(self) -> str

noether::commalg::SchemeCompletion::to_json · Direct Rust declaration

pub fn to_json(&self) -> Result<String, NoetherError>
Contract Value
Guarantee checked deterministic result
Determinism deterministic
Availability default CPU build
Domain implementation crates noether, flint-core, groebner-kernel, module-core, poly-core, resolution-kernel
GIL attached / bounded
Semantic ID noether/commalg/SchemeCompletion@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

commalg.SchemeCompletion.truncation

Return the truncation recorded by this SchemeCompletion.

Detailed contract

Immutable absolute monomial truncation.

(self) -> SeriesTruncation

noether::commalg::SchemeCompletion::truncation · Direct Rust declaration

pub fn truncation(&self) -> &SeriesTruncation
Contract Value
Guarantee checked deterministic result
Determinism deterministic
Availability default CPU build
Domain implementation crates noether, flint-core, groebner-kernel, module-core, poly-core, resolution-kernel
GIL attached / bounded
Semantic ID noether/commalg/SchemeCompletion@1

Executable usage: examples/34_finite_group_cohomology.py:11 · examples/34_finite_group_cohomology.py:12 · examples/34_finite_group_cohomology.py:13 · examples/34_finite_group_cohomology.py:15 · examples/34_finite_group_cohomology.py:20 · validation/unit/commalg/test_acceptance_workload.py:28

commalg.SchemeCompletion.verify

Replay and verify the mathematical evidence carried by this SchemeCompletion.

Detailed contract

Reconstruct and compare the translated fixed-order local equations.

(self) -> None

noether::commalg::SchemeCompletion::verify · Direct Rust declaration

pub fn verify(&self) -> Result<(), GeometryError>
Contract Value
Guarantee checked deterministic result
Determinism deterministic
Availability default CPU build
Domain implementation crates noether, flint-core, groebner-kernel, module-core, poly-core, resolution-kernel
GIL attached / bounded
Semantic ID noether/commalg/SchemeCompletion@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