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homalg · SubmoduleMembership

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

Worked example

The complete checked program Reproduce Fisher's order-five elliptic-function normalization uses this API directly.

# SPDX-License-Identifier: MIT
# Copyright (c) 2026 Benjamin Antieau and Deven Manam
"""Reproduce Fisher's order-five elliptic-function normalization."""

from noether import Q
from noether.commalg import RationalFunctionField, WeilDivisor
from noether.numthy import EllipticCurve, NthPowerClassGroup


def main() -> None:
    coefficients = RationalFunctionField(Q, "lambda")
    parameter = coefficients.generator()
    curve = EllipticCurve(
        coefficients,
        [1 - parameter, -parameter, -parameter, 0, 0],
    )
    point = curve.point(0, 0)
    point.verify_exact_order(5)

    miller = curve.miller_function(point, 5)
    divisor = WeilDivisor(
        curve.underlying_curve,
        [
            (curve.origin().underlying_curve_point.as_closed_point(), 5),
            (point.underlying_curve_point.as_closed_point(), -5),
        ],
    )
    function = divisor.riemann_roch_space().basis[0]
    assert (function / miller.function).as_constant() is not None

    evaluation_at_two = function.evaluate_at((2 * point).underlying_curve_point)
    evaluation_at_four = function.evaluate_at((4 * point).underlying_curve_point)
    assert evaluation_at_two.value is not None
    assert evaluation_at_four.value is not None
    classes = NthPowerClassGroup(coefficients, 5)
    normalization = classes.class_of(evaluation_at_two.value) ** -2 * classes.class_of(
        evaluation_at_four.value
    )
    assert normalization.is_identity()
    print("order 5 verified; Riemann--Roch and Miller agree; normalization is trivial")


if __name__ == "__main__":
    main()

Run it with .venv/bin/python examples/26_elliptic_descent_functions.py.

homalg.SubmoduleMembership

Replayable coefficients proving exact submodule membership.

class SubmoduleMembership

noether::homalg::SubmoduleMembership · Direct Rust declaration

pub struct SubmoduleMembership<R: Ring> { /* private fields */ }
Contract Value
Guarantee checked deterministic result
Determinism deterministic
Availability default CPU build
Domain implementation crates noether, flint-core, groebner-kernel, m4rst, m4rste, module-core, resolution-kernel, sparla
GIL attached / bounded
Semantic ID noether/homalg/SubmoduleMembership@1

Executable usage: validation/unit/homalg/test_exact_submodules.py:26 · validation/unit/homalg/test_exact_submodules.py:48 · validation/unit/homalg/test_exact_submodules.py:53 · validation/unit/homalg/test_exact_submodules.py:58

homalg.SubmoduleMembership.coefficients

Return the coefficients recorded by this SubmoduleMembership.

Detailed contract

Coefficients in the retained canonical generators.

(self) -> Vector[_ExactModuleRing]

noether::homalg::SubmoduleMembership::coefficients · Direct Rust declaration

pub fn coefficients(&self) -> &Vector<R>
Contract Value
Guarantee checked deterministic result
Determinism deterministic
Availability default CPU build
Domain implementation crates noether, flint-core, groebner-kernel, m4rst, m4rste, module-core, resolution-kernel, sparla
GIL attached / bounded
Semantic ID noether/homalg/SubmoduleMembership@1

Executable usage: examples/26_elliptic_descent_functions.py:11 · examples/26_elliptic_descent_functions.py:12 · examples/26_elliptic_descent_functions.py:14 · examples/26_elliptic_descent_functions.py:35 · examples/31_local_fields_and_precision.py:11 · examples/31_local_fields_and_precision.py:12

homalg.SubmoduleMembership.element

Return the element recorded by this SubmoduleMembership.

Detailed contract

Ambient element certified to lie in the submodule.

(self) -> ModuleElement[_ExactModuleRing]

noether::homalg::SubmoduleMembership::element · Direct Rust declaration

pub fn element(&self) -> &ModuleElement<R>
Contract Value
Guarantee checked deterministic result
Determinism deterministic
Availability default CPU build
Domain implementation crates noether, flint-core, groebner-kernel, m4rst, m4rste, module-core, resolution-kernel, sparla
GIL attached / bounded
Semantic ID noether/homalg/SubmoduleMembership@1

Executable usage: examples/19_permutation_groups.py:17 · examples/19_permutation_groups.py:18 · examples/30_modules_and_resolutions.py:3 · examples/38_exact_linear_systems.py:52 · examples/38_exact_linear_systems.py:56 · examples/38_exact_linear_systems.py:58

homalg.SubmoduleMembership.from_json

Deserialize and revalidate a SubmoduleMembership from typed JSON.

Detailed contract

Deserialize and replay a membership proof.

(cls, value: str) -> SubmoduleMembership[_ExactModuleRing]

noether::homalg::SubmoduleMembership::from_json · Direct Rust declaration

pub fn from_json(value: &str) -> Result<Self, NoetherError>where R: for<'de> Deserialize<'de>, R::Element: for<'de> Deserialize<'de>,
Contract Value
Guarantee checked deterministic result
Determinism deterministic
Availability default CPU build
Domain implementation crates noether, flint-core, groebner-kernel, m4rst, m4rste, module-core, resolution-kernel, sparla
GIL released
Semantic ID noether/homalg/SubmoduleMembership@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

homalg.SubmoduleMembership.submodule

Return the submodule recorded by this SubmoduleMembership.

Detailed contract

Submodule whose membership is certified.

(self) -> Submodule[_ExactModuleRing]

noether::homalg::SubmoduleMembership::submodule · Direct Rust declaration

pub fn submodule(&self) -> &Submodule<R>
Contract Value
Guarantee checked deterministic result
Determinism deterministic
Availability default CPU build
Domain implementation crates noether, flint-core, groebner-kernel, m4rst, m4rste, module-core, resolution-kernel, sparla
GIL attached / bounded
Semantic ID noether/homalg/SubmoduleMembership@1

Executable usage: examples/30_modules_and_resolutions.py:26 · examples/30_modules_and_resolutions.py:27 · examples/38_exact_linear_systems.py:70 · examples/38_exact_linear_systems.py:71 · validation/unit/homalg/test_exact_submodules.py:101 · validation/unit/homalg/test_exact_submodules.py:106

homalg.SubmoduleMembership.to_json

Serialize this SubmoduleMembership to its canonical typed JSON representation.

Detailed contract

Serialize this proof with exact context identity and replay data.

(self) -> str

noether::homalg::SubmoduleMembership::to_json · Direct Rust declaration

pub fn to_json(&self) -> Result<String, NoetherError>where R: Serialize, R::Element: Serialize,
Contract Value
Guarantee checked deterministic result
Determinism deterministic
Availability default CPU build
Domain implementation crates noether, flint-core, groebner-kernel, m4rst, m4rste, module-core, resolution-kernel, sparla
GIL released
Semantic ID noether/homalg/SubmoduleMembership@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

homalg.SubmoduleMembership.verify

Replay and verify the mathematical evidence carried by this SubmoduleMembership.

Detailed contract

Reconstruct the represented element from the canonical generator map.

(self) -> None

noether::homalg::SubmoduleMembership::verify · Direct Rust declaration

pub fn verify(&self) -> Result<(), VerificationError>
Contract Value
Guarantee checked deterministic result
Determinism deterministic
Availability default CPU build
Domain implementation crates noether, flint-core, groebner-kernel, m4rst, m4rste, module-core, resolution-kernel, sparla
GIL released
Semantic ID noether/homalg/SubmoduleMembership@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