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numthy · FiniteFieldEmbedding

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

Worked example

The complete checked program Apply and multiply canonical sparse exact operators uses this API directly.

# SPDX-License-Identifier: MIT
# Copyright (c) 2026 Benjamin Antieau and Deven Manam
"""Apply and multiply canonical sparse exact operators."""

from noether import Z
from noether.linalg import SparseMatrix, Vector


def main() -> None:
    operator = SparseMatrix.from_triplets(
        Z,
        (3, 3),
        [(0, 0, 2), (0, 2, -1), (1, 1, 3), (2, 0, 4)],
    )
    vector = Vector.from_values(Z, [5, 7, 11])
    image = operator.apply(vector)
    assert image.to_values() == [-1, 21, 20]
    assert (operator @ operator).nnz == 4
    print(f"A v = {image.to_values()}; nnz(A^2) = {(operator @ operator).nnz}")


if __name__ == "__main__":
    main()

Run it with .venv/bin/python examples/05_sparse_exact_operators.py.

numthy.FiniteFieldEmbedding

An explicitly presented embedding between compatible finite fields.

FiniteFieldEmbedding(source: FiniteField, target: FiniteField, generator_image: FiniteFieldElement)

noether::numthy::FiniteFieldEmbedding · Direct Rust declaration

pub struct FiniteFieldEmbedding { /* 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/FiniteFieldEmbedding@1

Executable usage: validation/unit/numthy/test_finite_field_facade.py:14 · validation/unit/numthy/test_finite_field_facade.py:175 · validation/unit/numthy/test_finite_field_facade.py:181 · validation/unit/numthy/test_finite_field_facade.py:184

numthy.FiniteFieldEmbedding.__call__

Coerce the supplied input through this context or mathematical map.

(self, value: FiniteFieldElement) -> FiniteFieldElement

noether::numthy::FiniteFieldEmbedding · Python projection

Python projection convenience over this Rust mathematical type
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/FiniteFieldEmbedding@1

Executable usage: validation/unit/infra/test_discovery.py:57 · validation/unit/infra/test_public_api_index.py:155 · validation/unit/infra/test_public_api_index.py:169

numthy.FiniteFieldEmbedding.apply

Apply this FiniteFieldEmbedding to the supplied value.

Detailed contract

Apply the embedding after checking exact source identity.

(self, value: FiniteFieldElement) -> FiniteFieldElement

noether::numthy::FiniteFieldEmbedding::apply · Direct Rust declaration

pub fn apply( &self, value: &FiniteFieldElement, ) -> Result<FiniteFieldElement, FiniteFieldEmbeddingError>
fn apply( &self, value: &FiniteFieldElement, ) -> Result<FiniteFieldElement, Self::Error>
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/FiniteFieldEmbedding@1

Executable usage: examples/05_sparse_exact_operators.py:16 · validation/unit/commalg/test_local_algebra.py:94 · validation/unit/commalg/test_local_algebra.py:97 · validation/unit/commalg/test_polynomial_facade.py:199 · validation/unit/commalg/test_quotient_facade.py:186 · validation/unit/groups/test_weight_characters.py:39

numthy.FiniteFieldEmbedding.from_json

Deserialize and revalidate a FiniteFieldEmbedding from typed JSON.

Detailed contract

Deserialize and exactly reverify the embedding.

(cls, value: str) -> FiniteFieldEmbedding

noether::numthy::FiniteFieldEmbedding::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/FiniteFieldEmbedding@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.FiniteFieldEmbedding.generator_image

Return the generator image recorded by this FiniteFieldEmbedding.

Detailed contract

Image of the source’s distinguished power-basis generator.

(self) -> FiniteFieldElement

noether::numthy::FiniteFieldEmbedding::generator_image · Direct Rust declaration

pub fn generator_image(&self) -> &FiniteFieldElement
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/FiniteFieldEmbedding@1

Executable usage: examples/09_finite_field_isomorphisms.py:15 · validation/unit/numthy/test_finite_field_facade.py:138 · validation/unit/numthy/test_finite_field_facade.py:178 · validation/corpus/numthy/finite-field-isomorphism-f25.py:24 · validation/corpus/numthy/finite-field-isomorphism-f25.py:54

numthy.FiniteFieldEmbedding.source

Return the source recorded by this FiniteFieldEmbedding.

Detailed contract

Source field presentation.

(self) -> FiniteField

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

pub fn source(&self) -> &FiniteField
fn source(&self) -> &Self::Source
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/FiniteFieldEmbedding@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.FiniteFieldEmbedding.target

Return the target recorded by this FiniteFieldEmbedding.

Detailed contract

Target field presentation.

(self) -> FiniteField

noether::numthy::FiniteFieldEmbedding::target · Direct Rust declaration

pub fn target(&self) -> &FiniteField
fn target(&self) -> &Self::Target
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/FiniteFieldEmbedding@1

Executable usage: examples/09_finite_field_isomorphisms.py:10 · examples/09_finite_field_isomorphisms.py:11 · examples/30_modules_and_resolutions.py:33 · examples/30_modules_and_resolutions.py:38 · examples/31_local_fields_and_precision.py:27 · examples/36_affine_normalization.py:16

numthy.FiniteFieldEmbedding.to_json

Serialize this FiniteFieldEmbedding to its canonical typed JSON representation.

Detailed contract

Serialize endpoints and the exact generator image.

(self) -> str

noether::numthy::FiniteFieldEmbedding::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/FiniteFieldEmbedding@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.FiniteFieldEmbedding.verify

Replay and verify the mathematical evidence carried by this FiniteFieldEmbedding.

Detailed contract

Verify endpoints, degree compatibility, and the generator relation.

(self) -> None

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

pub fn verify(&self) -> Result<(), FiniteFieldEmbeddingError>
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/FiniteFieldEmbedding@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