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numthy · PartialIntegerFactorization

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

Worked example

The complete checked program Compute a fast Smith form without transformation matrices uses this API directly.

# SPDX-License-Identifier: MIT
# Copyright (c) 2026 Benjamin Antieau and Deven Manam
"""Compute a fast Smith form without transformation matrices."""

from noether import Z
from noether.linalg import Matrix


def main() -> None:
    matrix = Matrix.from_rows(Z, [[2, 4, 4], [6, 6, 12], [10, 4, 16]])
    smith = matrix.smith_normal_form()
    smith.verify()
    assert all(
        right % left == 0
        for left, right in zip(smith.invariant_factors, smith.invariant_factors[1:])
    )
    print(f"invariant factors: {smith.invariant_factors}")


if __name__ == "__main__":
    main()

Run it with .venv/bin/python examples/04_smith_normal_form.py.

numthy.PartialIntegerFactorization

A deterministic bounded factorization with an honestly classified cofactor.

class PartialIntegerFactorization

noether::numthy::PartialIntegerFactorization · Direct Rust declaration

pub struct PartialIntegerFactorization { /* private fields */ }
Contract Value
Guarantee checked deterministic result
Determinism deterministic
Availability default CPU build
Domain implementation crates noether, flint-core
GIL attached / bounded
Semantic ID noether/numthy/PartialIntegerFactorization@1

Executable usage: validation/unit/numthy/test_integer_arithmetic_facade.py:12 · validation/unit/numthy/test_integer_arithmetic_facade.py:67 · validation/unit/numthy/test_integer_arithmetic_facade.py:73

numthy.PartialIntegerFactorization.classification

Return the classification recorded by this PartialIntegerFactorization.

Detailed contract

Honest proof status of the remaining cofactor.

(self) -> Literal['unit', 'proved_prime', 'proved_composite', 'unclassified']

noether::numthy::PartialIntegerFactorization::classification · Direct Rust declaration

pub fn classification(&self) -> CofactorClassification
Contract Value
Guarantee checked deterministic result
Determinism deterministic
Availability default CPU build
Domain implementation crates noether, flint-core
GIL attached / bounded
Semantic ID noether/numthy/PartialIntegerFactorization@1

Executable usage: validation/unit/numthy/test_integer_arithmetic_facade.py:70 · validation/unit/numthy/test_integer_arithmetic_facade.py:76 · validation/unit/numthy/test_integer_arithmetic_facade.py:80 · validation/corpus/homalg/codepth-three-resolution.py:3

numthy.PartialIntegerFactorization.cofactor

Return the cofactor recorded by this PartialIntegerFactorization.

Detailed contract

Positive remaining cofactor.

(self) -> int

noether::numthy::PartialIntegerFactorization::cofactor · Direct Rust declaration

pub fn cofactor(&self) -> &Integer
Contract Value
Guarantee checked deterministic result
Determinism deterministic
Availability default CPU build
Domain implementation crates noether, flint-core
GIL attached / bounded
Semantic ID noether/numthy/PartialIntegerFactorization@1

Executable usage: validation/unit/commalg/test_polynomial_facade.py:481 · validation/unit/commalg/test_polynomial_facade.py:483 · validation/unit/numthy/test_integer_arithmetic_facade.py:69 · validation/unit/numthy/test_integer_arithmetic_facade.py:77 · validation/unit/test_validity.py:45 · validation/corpus/linalg/restricted-grassmannian-secant-rank.py:11

numthy.PartialIntegerFactorization.factors

Return the factors recorded by this PartialIntegerFactorization.

Detailed contract

Strictly increasing proved prime factors found within the work bound.

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

noether::numthy::PartialIntegerFactorization::factors · Direct Rust declaration

pub fn factors(&self) -> &[(Integer, u64)]
Contract Value
Guarantee checked deterministic result
Determinism deterministic
Availability default CPU build
Domain implementation crates noether, flint-core
GIL attached / bounded
Semantic ID noether/numthy/PartialIntegerFactorization@1

Executable usage: examples/04_smith_normal_form.py:17 · examples/07_integer_factorization.py:12 · validation/unit/commalg/test_context_tower_workload.py:87 · validation/unit/commalg/test_polynomial_facade.py:126 · validation/unit/commalg/test_polynomial_facade.py:147 · validation/unit/commalg/test_polynomial_facade.py:158

numthy.PartialIntegerFactorization.from_json

Deserialize and revalidate a PartialIntegerFactorization from typed JSON.

Detailed contract

Deserialize and recompute the bounded result before accepting it.

(cls, value: str) -> PartialIntegerFactorization

noether::numthy::PartialIntegerFactorization::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
GIL attached / bounded
Semantic ID noether/numthy/PartialIntegerFactorization@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

numthy.PartialIntegerFactorization.primality_proof_bits

Return the primality proof bits recorded by this PartialIntegerFactorization.

Detailed contract

Largest cofactor bit length admitted to deterministic primality proof.

(self) -> int

noether::numthy::PartialIntegerFactorization::primality_proof_bits · Direct Rust declaration

pub fn primality_proof_bits(&self) -> u64
Contract Value
Guarantee checked deterministic result
Determinism deterministic
Availability default CPU build
Domain implementation crates noether, flint-core
GIL attached / bounded
Semantic ID noether/numthy/PartialIntegerFactorization@1

Executable usage: validation/unit/infra/test_concurrency.py:122 · validation/unit/numthy/test_integer_arithmetic_facade.py:65 · validation/unit/numthy/test_integer_arithmetic_facade.py:75 · validation/unit/numthy/test_integer_arithmetic_facade.py:79

numthy.PartialIntegerFactorization.source

Return the source recorded by this PartialIntegerFactorization.

Detailed contract

Original nonzero integer.

(self) -> int

noether::numthy::PartialIntegerFactorization::source · Direct Rust declaration

pub fn source(&self) -> &Integer
Contract Value
Guarantee checked deterministic result
Determinism deterministic
Availability default CPU build
Domain implementation crates noether, flint-core
GIL attached / bounded
Semantic ID noether/numthy/PartialIntegerFactorization@1

Executable usage: examples/09_finite_field_isomorphisms.py:11 · examples/09_finite_field_isomorphisms.py:13 · examples/09_finite_field_isomorphisms.py:9 · examples/30_modules_and_resolutions.py:34 · examples/30_modules_and_resolutions.py:37 · validation/unit/category/test_acceptance_workload.py:25

numthy.PartialIntegerFactorization.to_json

Serialize this PartialIntegerFactorization to its canonical typed JSON representation.

Detailed contract

Serialize the complete bounded result as canonical typed JSON.

(self) -> str

noether::numthy::PartialIntegerFactorization::to_json · Direct Rust declaration

pub fn to_json(&self) -> Result<String, NoetherError>
Contract Value
Guarantee checked deterministic result
Determinism deterministic
Availability default CPU build
Domain implementation crates noether, flint-core
GIL attached / bounded
Semantic ID noether/numthy/PartialIntegerFactorization@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

numthy.PartialIntegerFactorization.trial_bound

Return the trial bound recorded by this PartialIntegerFactorization.

Detailed contract

Inclusive upper bound used for deterministic trial division.

(self) -> int

noether::numthy::PartialIntegerFactorization::trial_bound · Direct Rust declaration

pub fn trial_bound(&self) -> u64
Contract Value
Guarantee checked deterministic result
Determinism deterministic
Availability default CPU build
Domain implementation crates noether, flint-core
GIL attached / bounded
Semantic ID noether/numthy/PartialIntegerFactorization@1

Executable usage: validation/unit/infra/test_concurrency.py:121 · validation/unit/numthy/test_integer_arithmetic_facade.py:65 · validation/unit/numthy/test_integer_arithmetic_facade.py:75 · validation/unit/numthy/test_integer_arithmetic_facade.py:79

numthy.PartialIntegerFactorization.unit

Return the unit recorded by this PartialIntegerFactorization.

Detailed contract

Multiplicative unit, either -1 or 1.

(self) -> int

noether::numthy::PartialIntegerFactorization::unit · Direct Rust declaration

pub fn unit(&self) -> i8
Contract Value
Guarantee checked deterministic result
Determinism deterministic
Availability default CPU build
Domain implementation crates noether, flint-core
GIL attached / bounded
Semantic ID noether/numthy/PartialIntegerFactorization@1

Executable usage: validation/unit/commalg/test_local_algebra.py:42 · validation/unit/commalg/test_local_algebra.py:44 · validation/unit/commalg/test_local_algebra.py:45 · validation/unit/commalg/test_polynomial_ideal_facade.py:62 · validation/unit/commalg/test_polynomial_ideal_facade.py:68 · validation/unit/commalg/test_polynomial_ideal_facade.py:69

numthy.PartialIntegerFactorization.value

Return the value recorded by this PartialIntegerFactorization.

Detailed contract

Reconstruct the exact source integer.

(self) -> int

noether::numthy::PartialIntegerFactorization::value · Direct Rust declaration

pub fn value(&self) -> Integer
Contract Value
Guarantee checked deterministic result
Determinism deterministic
Availability default CPU build
Domain implementation crates noether, flint-core
GIL attached / bounded
Semantic ID noether/numthy/PartialIntegerFactorization@1

Executable usage: examples/09_finite_field_isomorphisms.py:13 · examples/09_finite_field_isomorphisms.py:14 · examples/13_chinese_remainder_theorem.py:12 · examples/13_chinese_remainder_theorem.py:13 · examples/13_chinese_remainder_theorem.py:14 · examples/17_weighted_power_series.py:18

numthy.PartialIntegerFactorization.verify

Replay and verify the mathematical evidence carried by this PartialIntegerFactorization.

Detailed contract

Recompute the bounded operation and validate every proof-status claim.

(self) -> bool

noether::numthy::PartialIntegerFactorization::verify · Direct Rust declaration

pub fn verify(&self) -> bool
Contract Value
Guarantee checked deterministic result
Determinism deterministic
Availability default CPU build
Domain implementation crates noether, flint-core
GIL attached / bounded
Semantic ID noether/numthy/PartialIntegerFactorization@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