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linalg · MinimumNonrealizabilityCertificate

5 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.MinimumNonrealizabilityCertificate

Exact rational evidence that a line configuration is not a minimum shell.

class MinimumNonrealizabilityCertificate

noether::linalg::MinimumNonrealizabilityCertificate · Direct Rust declaration

pub struct MinimumNonrealizabilityCertificate { /* 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/MinimumNonrealizabilityCertificate@1

Executable usage: validation/unit/linalg/test_lattice_facade.py:19 · validation/unit/linalg/test_lattice_facade.py:210

linalg.MinimumNonrealizabilityCertificate.configuration

Return the configuration recorded by this MinimumNonrealizabilityCertificate.

Detailed contract

Candidate configuration proved nonrealizable.

(self) -> IntegralLineConfiguration

noether::linalg::MinimumNonrealizabilityCertificate::configuration · Direct Rust declaration

pub fn configuration(&self) -> &IntegralLineConfiguration
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/MinimumNonrealizabilityCertificate@1

Executable usage: examples/29_integral_line_configurations.py:29 · validation/unit/linalg/test_lattice_facade.py:100 · validation/unit/linalg/test_lattice_facade.py:101 · validation/unit/linalg/test_lattice_facade.py:102 · validation/unit/linalg/test_lattice_facade.py:103 · validation/unit/linalg/test_lattice_facade.py:106

linalg.MinimumNonrealizabilityCertificate.kind

Return the kind recorded by this MinimumNonrealizabilityCertificate.

Detailed contract

Stable kind of rational certificate.

(self) -> str

noether::linalg::MinimumNonrealizabilityCertificate::kind · Direct Rust declaration

pub fn kind(&self) -> &'static str
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/MinimumNonrealizabilityCertificate@1

Executable usage: validation/unit/combinatorics/test_poset.py:95 · validation/unit/commalg/test_curve_facade.py:137 · validation/unit/commalg/test_curve_facade.py:43 · validation/unit/commalg/test_curve_facade.py:44 · validation/unit/commalg/test_curve_facade.py:45 · validation/unit/commalg/test_curve_facade.py:80

linalg.MinimumNonrealizabilityCertificate.obstacle_count

Return the obstacle count recorded by this MinimumNonrealizabilityCertificate.

Detailed contract

Number of finite separating vectors used by the certificate.

(self) -> int

noether::linalg::MinimumNonrealizabilityCertificate::obstacle_count · Direct Rust declaration

pub fn obstacle_count(&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/MinimumNonrealizabilityCertificate@1

linalg.MinimumNonrealizabilityCertificate.verify

Replay and verify the mathematical evidence carried by this MinimumNonrealizabilityCertificate.

Detailed contract

Recheck the rational Farkas or dual-bound identity exactly.

(self) -> None

noether::linalg::MinimumNonrealizabilityCertificate::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/MinimumNonrealizabilityCertificate@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