Skip to content

root · IntegerRationalEmbedding

8 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.

IntegerRationalEmbedding

The canonical embedding of the integers into the rationals.

class IntegerRationalEmbedding

noether::IntegerRationalEmbedding · Direct Rust declaration

pub struct IntegerRationalEmbedding;
Contract Value
Guarantee checked deterministic result
Determinism deterministic
Availability default CPU build
Domain implementation crates noether, flint-core
GIL attached / bounded
Semantic ID noether/root/IntegerRationalEmbedding@1

Executable usage: validation/unit/test_coefficient_maps.py:12 · validation/unit/test_coefficient_maps.py:28

IntegerRationalEmbedding.__call__

Coerce the supplied input through this context or mathematical map.

(self, value: int) -> _Fraction

noether::root::IntegerRationalEmbedding · 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/root/IntegerRationalEmbedding@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

IntegerRationalEmbedding.apply

Apply this IntegerRationalEmbedding to the supplied value.

(self, value: int) -> _Fraction

noether::root::IntegerRationalEmbedding::apply · Direct Rust declaration

fn apply(&self, value: &Integer) -> Result<Rational, 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/root/IntegerRationalEmbedding@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

IntegerRationalEmbedding.from_json

Deserialize and revalidate a IntegerRationalEmbedding from typed JSON.

Detailed contract

Deserialize and revalidate the canonical singleton map.

(cls, value: str) -> IntegerRationalEmbedding

noether::root::IntegerRationalEmbedding::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/root/IntegerRationalEmbedding@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

IntegerRationalEmbedding.source

Return the source recorded by this IntegerRationalEmbedding.

(self) -> IntegerRing

noether::root::IntegerRationalEmbedding::source · Direct Rust declaration

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/root/IntegerRationalEmbedding@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

IntegerRationalEmbedding.target

Return the target recorded by this IntegerRationalEmbedding.

(self) -> RationalField

noether::root::IntegerRationalEmbedding::target · Direct Rust declaration

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/root/IntegerRationalEmbedding@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

IntegerRationalEmbedding.to_json

Serialize this IntegerRationalEmbedding to its canonical typed JSON representation.

Detailed contract

Serialize this canonical singleton map.

(self) -> str

noether::root::IntegerRationalEmbedding::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/root/IntegerRationalEmbedding@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

IntegerRationalEmbedding.verify

Replay and verify the mathematical evidence carried by this IntegerRationalEmbedding.

Detailed contract

Verify the fixed endpoints of this canonical embedding.

(self) -> None

noether::root::IntegerRationalEmbedding::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
GIL attached / bounded
Semantic ID noether/root/IntegerRationalEmbedding@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