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linalg · IntegralLineOrbitAction

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

Worked example

The complete checked program Certify symmetries, realizations, and completions of integral lines uses this API directly.

# SPDX-License-Identifier: MIT
# Copyright (c) 2026 Benjamin Antieau and Deven Manam
"""Certify symmetries, realizations, and completions of integral lines."""

from noether.linalg import IntegralLineConfiguration


def main() -> None:
    axes = IntegralLineConfiguration(2, [[1, 0], [0, 1]])
    symmetries = axes.automorphism_group()
    symmetries.verify()
    assert symmetries.order == 8

    a2_lines = IntegralLineConfiguration(2, [[1, 0], [0, 1], [1, -1]])
    realization = a2_lines.realize_minimum(max_iterations=20)
    realization.verify()
    assert realization.status == "realized"
    assert realization.realization is not None

    completion = axes.complete_minimum(max_iterations=20)
    completion.verify()
    assert completion.status == "completed"
    assert completion.realization is not None

    print(
        "integral lines:",
        f"axis symmetry order={symmetries.order}",
        f"A2 Gram matrix={realization.realization.gram_matrix}",
        f"axis completion={completion.realization.configuration.vectors}",
    )


if __name__ == "__main__":
    main()

Run it with .venv/bin/python examples/29_integral_line_configurations.py.

linalg.IntegralLineOrbitAction

Exact finite action of a line-configuration stabilizer on external lines.

class IntegralLineOrbitAction

noether::linalg::IntegralLineOrbitAction · Direct Rust declaration

pub struct IntegralLineOrbitAction { /* 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/IntegralLineOrbitAction@1

linalg.IntegralLineOrbitAction.action

Simultaneous finite permutation action.

Detailed contract

Simultaneous permutation action induced by the matrix generators.

(self) -> PermutationGroup

noether::linalg::IntegralLineOrbitAction::action · Direct Rust declaration

pub fn action(&self) -> &PermutationGroup
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/IntegralLineOrbitAction@1

Executable usage: validation/unit/infra/test_repository_hygiene.py:144 · validation/unit/infra/test_repository_hygiene.py:181 · validation/unit/infra/test_repository_hygiene.py:182 · validation/unit/infra/test_repository_hygiene.py:218 · validation/unit/linalg/test_lattice_facade.py:154 · validation/unit/linalg/test_lattice_facade.py:155

linalg.IntegralLineOrbitAction.automorphisms

Stabilizer inducing the action.

Detailed contract

Stabilizer whose matrix generators induce this action.

(self) -> IntegralLineAutomorphismGroup

noether::linalg::IntegralLineOrbitAction::automorphisms · Direct Rust declaration

pub fn automorphisms(&self) -> &IntegralLineAutomorphismGroup
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/IntegralLineOrbitAction@1

Executable usage: validation/unit/groups/test_acceptance_workload.py:113 · validation/unit/groups/test_acceptance_workload.py:115 · validation/unit/groups/test_acceptance_workload.py:116 · validation/unit/groups/test_acceptance_workload.py:117 · validation/unit/groups/test_acceptance_workload.py:118 · validation/unit/linalg/test_lattice_facade.py:193

linalg.IntegralLineOrbitAction.line

Canonical primitive line at one action point.

Detailed contract

Canonical line at one action point.

(self, point: int) -> tuple[int, ...]

noether::linalg::IntegralLineOrbitAction::line · Direct Rust declaration

pub fn line(&self, point: Point) -> Result<&[Integer], LineConfigurationError>
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/IntegralLineOrbitAction@1

Executable usage: validation/unit/commalg/test_acceptance_workload.py:148 · validation/unit/commalg/test_acceptance_workload.py:149 · validation/unit/commalg/test_polynomial_homomorphism.py:18 · validation/unit/commalg/test_polynomial_homomorphism.py:21 · validation/unit/commalg/test_polynomial_homomorphism.py:27 · validation/unit/commalg/test_polynomial_homomorphism.py:30

linalg.IntegralLineOrbitAction.lines

Flattened line vector at every action point.

Detailed contract

Flattened canonical primitive lines indexed by the action domain.

(self) -> tuple[tuple[int, ...], ...]

noether::linalg::IntegralLineOrbitAction::lines · Direct Rust declaration

pub fn lines(&self) -> &[Vec<Integer>]
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/IntegralLineOrbitAction@1

Executable usage: examples/29_integral_line_configurations.py:26 · examples/29_integral_line_configurations.py:3 · validation/unit/infra/test_macaulay2_benchmark_protocol.py:421 · validation/unit/infra/test_macaulay2_benchmark_protocol.py:436 · validation/unit/infra/test_macaulay2_benchmark_protocol.py:439 · validation/unit/infra/test_macaulay2_benchmark_protocol.py:440

linalg.IntegralLineOrbitAction.orbit_ranges

Half-open point range belonging to each tagged orbit.

Detailed contract

Half-open flattened point range for each tagged seed orbit.

(self) -> list[tuple[int, int]]

noether::linalg::IntegralLineOrbitAction::orbit_ranges · Direct Rust declaration

pub fn orbit_ranges(&self) -> &[(usize, 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/IntegralLineOrbitAction@1

linalg.IntegralLineOrbitAction.seed_points

Point occupied by each tagged seed.

Detailed contract

Point occupied by each seed in the flattened action domain.

(self) -> list[int]

noether::linalg::IntegralLineOrbitAction::seed_points · Direct Rust declaration

pub fn seed_points(&self) -> &[Point] ⓘ
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/IntegralLineOrbitAction@1

linalg.IntegralLineOrbitAction.seeds

Canonical tagged seed lines.

Detailed contract

Canonical primitive line used to seed each tagged orbit.

(self) -> tuple[tuple[int, ...], ...]

noether::linalg::IntegralLineOrbitAction::seeds · Direct Rust declaration

pub fn seeds(&self) -> &[Vec<Integer>]
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/IntegralLineOrbitAction@1

Executable usage: validation/unit/infra/test_mpi_scale_report.py:262 · validation/parity/m1_bsgs.py:273 · validation/parity/m1_bsgs.py:301 · validation/parity/m1_bsgs.py:59

linalg.IntegralLineOrbitAction.subgroup_action

Induced permutation action of a verified subgroup.

Detailed contract

Induced action of a verified subgroup on the same external-line domain.

(self, subgroup: IntegralLineSubgroup) -> PermutationGroup

noether::linalg::IntegralLineOrbitAction::subgroup_action · Direct Rust declaration

pub fn subgroup_action( &self, subgroup: &IntegralLineSubgroup, ) -> Result<PermutationGroup, LineConfigurationError>
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/IntegralLineOrbitAction@1

linalg.IntegralLineOrbitAction.verify

Rebuild every orbit and induced action exactly.

Detailed contract

Rebuild every tagged orbit and induced generator action exactly.

(self) -> None

noether::linalg::IntegralLineOrbitAction::verify · Direct Rust declaration

pub fn verify(&self) -> Result<(), LineConfigurationError>
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/IntegralLineOrbitAction@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