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homalg · KernelSubmodule

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

Worked example

The complete checked program Exact affine systems and canonical submodules over Z and Z/nZ uses this API directly.

# SPDX-License-Identifier: MIT
# Copyright (c) 2026 Benjamin Antieau and Deven Manam
"""Exact affine systems and canonical submodules over Z and Z/nZ."""

from __future__ import annotations

from noether import IntegerRing, Z
from noether.commalg import ResidueRing
from noether.homalg import FreeModule, ModuleElement, ModuleHomomorphism, Submodule
from noether.linalg import Matrix, Vector

operator = Matrix.from_rows(Z, [[2, 4, 0], [0, 0, 3]])
right_hand_side = Vector.from_values(Z, [6, 9])
integer_result = operator.solve_exact(right_hand_side)
assert integer_result.is_consistent
integer_solutions = integer_result.solutions
assert integer_solutions is not None
assert operator @ integer_solutions.particular_solution == right_hand_side
assert operator @ integer_solutions.homogeneous_generators == Matrix.zeros(Z, (2, 1))
integer_result.verify()

# Empty systems retain a replayable divisibility obstruction.
impossible = Matrix.from_rows(Z, [[6]]).solve_exact(Vector.from_values(Z, [9]))
assert not impossible.is_consistent
assert impossible.obstruction is not None
assert impossible.obstruction.left_gcd == 6
assert abs(impossible.obstruction.combined_rhs) == 9
impossible.obstruction.verify()

# Composite moduli are not treated as fields.  The result is an affine module,
# including all homogeneous torsion directions.
z12 = ResidueRing(12)
congruence = Matrix.from_rows(z12, [[8]]).solve_exact(Vector.from_values(z12, [4]))
assert congruence.is_consistent
congruence_solutions = congruence.solutions
assert congruence_solutions is not None
assert {
    (int(congruence_solutions.particular_solution[0]) + 3 * offset) % 12 for offset in range(4)
} == {
    2,
    5,
    8,
    11,
}
congruence.verify()

# Submodules retain a canonical normal form, so equality and lattice operations
# compare mathematics rather than the supplied generating lists.
ambient = FreeModule(Z, 2)


def element(x: int, y: int) -> ModuleElement[IntegerRing]:
    return ModuleElement(ambient, Vector.from_values(Z, [x, y]))


lattice = Submodule.from_generators(ambient, [element(2, 0), element(0, 3)])
assert lattice.finite_index() == 6
witness = lattice.membership_witness(element(6, 9))
assert witness is not None
witness.verify()
assert lattice.saturation().finite_index() == 1

homomorphism = ModuleHomomorphism(
    FreeModule(Z, 3),
    ambient,
    operator,
)
kernel_result = homomorphism.kernel()
image_result = homomorphism.image()
kernel = kernel_result.submodule
image = image_result.submodule
assert kernel.generator_count == 1
assert image.contains(element(2, 0))
kernel_result.verify()
image_result.verify()

print(
    "exact systems: integer affine rank 1; four Z/12Z solutions; "
    "index-6 lattice saturated and verified"
)

Run it with .venv/bin/python examples/38_exact_linear_systems.py.

homalg.KernelSubmodule

A map-bound certificate for a complete exact kernel.

class KernelSubmodule

noether::homalg::KernelSubmodule · Direct Rust declaration

pub struct KernelSubmodule<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/KernelSubmodule@1

Executable usage: validation/unit/homalg/test_exact_submodules.py:103 · validation/unit/homalg/test_exact_submodules.py:22 · validation/unit/homalg/test_exact_submodules.py:80 · validation/unit/homalg/test_exact_submodules.py:84 · validation/unit/infra/test_public_api_index.py:280

homalg.KernelSubmodule.from_json

Deserialize and revalidate a KernelSubmodule from typed JSON.

Detailed contract

Deserialize and fully replay a kernel certificate.

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

noether::homalg::KernelSubmodule::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, m4rst, m4rste, module-core, resolution-kernel, sparla
GIL released
Semantic ID noether/homalg/KernelSubmodule@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.KernelSubmodule.homomorphism

Return the homomorphism recorded by this KernelSubmodule.

Detailed contract

Homomorphism whose kernel is certified.

(self) -> ModuleHomomorphism[_ExactModuleRing]

noether::homalg::KernelSubmodule::homomorphism · Direct Rust declaration

pub fn homomorphism(&self) -> &ModuleHomomorphism<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/KernelSubmodule@1

Executable usage: examples/38_exact_linear_systems.py:63 · examples/38_exact_linear_systems.py:68 · examples/38_exact_linear_systems.py:69 · validation/unit/commalg/test_base_change.py:118 · validation/unit/commalg/test_base_change.py:123 · validation/unit/commalg/test_base_change.py:124

homalg.KernelSubmodule.submodule

Return the submodule recorded by this KernelSubmodule.

Detailed contract

Canonical submodule of the source proved to be the full kernel.

(self) -> Submodule[_ExactModuleRing]

noether::homalg::KernelSubmodule::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/KernelSubmodule@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.KernelSubmodule.to_json

Serialize this KernelSubmodule to its canonical typed JSON representation.

Detailed contract

Serialize the map together with its canonical kernel evidence.

(self) -> str

noether::homalg::KernelSubmodule::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/KernelSubmodule@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.KernelSubmodule.verify

Replay and verify the mathematical evidence carried by this KernelSubmodule.

Detailed contract

Replay zero-containment and canonical completeness of this kernel. Replay zero-containment and canonical completeness over Z/nZ.

(self) -> None

noether::homalg::KernelSubmodule::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/KernelSubmodule@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