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linalg · IntegerSolutionModule

7 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.

linalg.IntegerSolutionModule

A nonempty affine lattice of integer solutions.

class IntegerSolutionModule

noether::linalg::IntegerSolutionModule · Direct Rust declaration

pub struct IntegerSolutionModule { /* private fields */ }
Contract Value
Guarantee checked deterministic result
Determinism deterministic
Availability default CPU build
Domain implementation crates noether, flint-core, m4rst, m4rst-distributed, m4rste, mzs, pwvla, spar-core, sparla, sparb, lattice-enum
GIL attached / bounded
Semantic ID noether/linalg/IntegerSolutionModule@1

Executable usage: validation/unit/linalg/test_exact_solution_modules.py:21 · validation/unit/linalg/test_exact_solution_modules.py:34

linalg.IntegerSolutionModule.homogeneous_generators

Return the homogeneous generators recorded by this IntegerSolutionModule.

Detailed contract

Canonical columns generating every solution of A*x=0.

(self) -> Matrix[IntegerRing]

noether::linalg::IntegerSolutionModule::homogeneous_generators · Direct Rust declaration

pub fn homogeneous_generators(&self) -> &Matrix<IntegerRing>
Contract Value
Guarantee checked deterministic result
Determinism deterministic
Availability default CPU build
Domain implementation crates noether, flint-core, m4rst, m4rst-distributed, m4rste, mzs, pwvla, spar-core, sparla, sparb, lattice-enum
GIL attached / bounded
Semantic ID noether/linalg/IntegerSolutionModule@1

Executable usage: examples/38_exact_linear_systems.py:19 · validation/unit/linalg/test_exact_solution_modules.py:39 · validation/unit/linalg/test_exact_solution_modules.py:91 · validation/parity/linalg/test_exact_solution_modules_parity.py:164 · validation/parity/linalg/test_exact_solution_modules_parity.py:202

linalg.IntegerSolutionModule.homogeneous_rank

Return the homogeneous rank recorded by this IntegerSolutionModule.

Detailed contract

Rank of the free homogeneous solution lattice.

(self) -> int

noether::linalg::IntegerSolutionModule::homogeneous_rank · Direct Rust declaration

pub fn homogeneous_rank(&self) -> usize
Contract Value
Guarantee checked deterministic result
Determinism deterministic
Availability default CPU build
Domain implementation crates noether, flint-core, m4rst, m4rst-distributed, m4rste, mzs, pwvla, spar-core, sparla, sparb, lattice-enum
GIL attached / bounded
Semantic ID noether/linalg/IntegerSolutionModule@1

Executable usage: validation/unit/linalg/test_exact_solution_modules.py:40 · validation/bench/sage_exact_systems.py:68

linalg.IntegerSolutionModule.operator

Return the operator recorded by this IntegerSolutionModule.

Detailed contract

Coefficient matrix A in A*x=b.

(self) -> Matrix[IntegerRing]

noether::linalg::IntegerSolutionModule::operator · Direct Rust declaration

pub fn operator(&self) -> &Matrix<IntegerRing>
Contract Value
Guarantee checked deterministic result
Determinism deterministic
Availability default CPU build
Domain implementation crates noether, flint-core, m4rst, m4rst-distributed, m4rste, mzs, pwvla, spar-core, sparla, sparb, lattice-enum
GIL attached / bounded
Semantic ID noether/linalg/IntegerSolutionModule@1

Executable usage: examples/05_sparse_exact_operators.py:10 · examples/05_sparse_exact_operators.py:16 · examples/05_sparse_exact_operators.py:18 · examples/05_sparse_exact_operators.py:19 · examples/38_exact_linear_systems.py:12 · examples/38_exact_linear_systems.py:14

linalg.IntegerSolutionModule.particular_solution

Return the particular solution recorded by this IntegerSolutionModule.

Detailed contract

One exact integer solution.

(self) -> Vector[IntegerRing]

noether::linalg::IntegerSolutionModule::particular_solution · Direct Rust declaration

pub fn particular_solution(&self) -> &Vector<IntegerRing>
Contract Value
Guarantee checked deterministic result
Determinism deterministic
Availability default CPU build
Domain implementation crates noether, flint-core, m4rst, m4rst-distributed, m4rste, mzs, pwvla, spar-core, sparla, sparb, lattice-enum
GIL attached / bounded
Semantic ID noether/linalg/IntegerSolutionModule@1

Executable usage: examples/38_exact_linear_systems.py:18 · examples/38_exact_linear_systems.py:38 · validation/unit/linalg/test_exact_solution_modules.py:38 · validation/unit/linalg/test_exact_solution_modules.py:45 · validation/unit/linalg/test_exact_solution_modules.py:90 · validation/parity/linalg/test_exact_solution_modules_parity.py:176

linalg.IntegerSolutionModule.rhs

Return the rhs recorded by this IntegerSolutionModule.

Detailed contract

Right-hand side b.

(self) -> Vector[IntegerRing]

noether::linalg::IntegerSolutionModule::rhs · Direct Rust declaration

pub fn rhs(&self) -> &Vector<IntegerRing>
Contract Value
Guarantee checked deterministic result
Determinism deterministic
Availability default CPU build
Domain implementation crates noether, flint-core, m4rst, m4rst-distributed, m4rste, mzs, pwvla, spar-core, sparla, sparb, lattice-enum
GIL attached / bounded
Semantic ID noether/linalg/IntegerSolutionModule@1

Executable usage: validation/unit/geometry/test_acceptance_workload.py:60 · validation/unit/geometry/test_acceptance_workload.py:61 · validation/unit/geometry/test_acceptance_workload.py:75 · validation/unit/geometry/test_acceptance_workload.py:78 · validation/unit/linalg/test_exact_linear_algebra_facade.py:236 · validation/unit/linalg/test_exact_linear_algebra_facade.py:237

linalg.IntegerSolutionModule.verify

Replay and verify the mathematical evidence carried by this IntegerSolutionModule.

Detailed contract

Recheck the affine equation and replay the complete Smith calculation.

(self) -> None

noether::linalg::IntegerSolutionModule::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, m4rst, m4rst-distributed, m4rste, mzs, pwvla, spar-core, sparla, sparb, lattice-enum
GIL released
Semantic ID noether/linalg/IntegerSolutionModule@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