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numthy · QadicIntegralEmbedding

17 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.QadicIntegralEmbedding

The canonical fixed-cap embedding of a Witt ring in its fraction field.

class QadicIntegralEmbedding

noether::numthy::QadicIntegralEmbedding · Direct Rust declaration

pub struct QadicIntegralEmbedding { /* 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/QadicIntegralEmbedding@1

Executable usage: validation/unit/numthy/test_qadic_field_facade.py:17 · validation/unit/numthy/test_qadic_field_facade.py:84 · validation/unit/numthy/test_qadic_field_facade.py:90

numthy.QadicIntegralEmbedding.__call__

Coerce the supplied input through this context or mathematical map.

(self, value: QadicElement) -> QadicElement

noether::numthy::QadicIntegralEmbedding · 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/QadicIntegralEmbedding@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.QadicIntegralEmbedding.apply

Apply this QadicIntegralEmbedding to the supplied value.

Detailed contract

Embed an integral value after checking its exact source context.

(self, value: QadicElement) -> QadicElement

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

pub fn apply( &self, value: &QadicElement, ) -> Result<QadicElement, QadicEmbeddingError>
fn apply(&self, value: &QadicElement) -> Result<QadicElement, 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/QadicIntegralEmbedding@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.QadicIntegralEmbedding.apply_matrix

Compute apply matrix for this QadicIntegralEmbedding.

(self, value: Matrix[QadicRing]) -> Matrix[QadicField]

noether::numthy::QadicIntegralEmbedding · 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/QadicIntegralEmbedding@1

Executable usage: validation/unit/numthy/test_padic_field_facade.py:86 · validation/unit/numthy/test_padic_field_facade.py:95 · validation/unit/numthy/test_padic_local_field_facade.py:148 · validation/unit/numthy/test_padic_local_field_facade.py:161 · validation/unit/numthy/test_padic_ring_facade.py:102 · validation/unit/numthy/test_qadic_field_facade.py:113

numthy.QadicIntegralEmbedding.apply_polynomial

Compute apply polynomial for this QadicIntegralEmbedding.

(self, value: Polynomial[QadicRing]) -> Polynomial[QadicField]

noether::numthy::QadicIntegralEmbedding · 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/QadicIntegralEmbedding@1

Executable usage: validation/unit/numthy/test_padic_field_facade.py:84 · validation/unit/numthy/test_padic_field_facade.py:92 · validation/unit/numthy/test_padic_local_field_facade.py:129 · validation/unit/numthy/test_padic_local_field_facade.py:158 · validation/unit/numthy/test_padic_local_field_facade.py:164 · validation/unit/numthy/test_padic_ring_facade.py:99

numthy.QadicIntegralEmbedding.apply_power_series

Compute apply power series for this QadicIntegralEmbedding.

(self, value: PowerSeries[QadicRing]) -> PowerSeries[QadicField]

noether::numthy::QadicIntegralEmbedding · 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/QadicIntegralEmbedding@1

Executable usage: validation/unit/commalg/test_power_series_facade.py:315 · validation/unit/commalg/test_power_series_facade.py:320 · validation/unit/numthy/test_padic_local_field_facade.py:135 · validation/unit/numthy/test_padic_local_field_facade.py:159 · validation/unit/numthy/test_padic_local_field_facade.py:167 · validation/unit/numthy/test_padic_ring_facade.py:100

numthy.QadicIntegralEmbedding.apply_vector

Compute apply vector for this QadicIntegralEmbedding.

(self, value: Vector[QadicRing]) -> Vector[QadicField]

noether::numthy::QadicIntegralEmbedding · 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/QadicIntegralEmbedding@1

Executable usage: validation/unit/numthy/test_padic_field_facade.py:85 · validation/unit/numthy/test_padic_field_facade.py:94 · validation/unit/numthy/test_padic_local_field_facade.py:147 · validation/unit/numthy/test_padic_local_field_facade.py:160 · validation/unit/numthy/test_padic_ring_facade.py:101 · validation/unit/numthy/test_qadic_field_facade.py:112

numthy.QadicIntegralEmbedding.from_json

Deserialize and revalidate a QadicIntegralEmbedding from typed JSON.

Detailed contract

Deserialize and revalidate the embedding endpoints.

(cls, value: str) -> QadicIntegralEmbedding

noether::numthy::QadicIntegralEmbedding::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/QadicIntegralEmbedding@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.QadicIntegralEmbedding.retract

Compute retract for this QadicIntegralEmbedding.

Detailed contract

Retract a field value after certifying nonnegative valuation.

(self, value: QadicElement) -> QadicElement

noether::numthy::QadicIntegralEmbedding::retract · Direct Rust declaration

pub fn retract( &self, value: &QadicElement, ) -> Result<QadicElement, QadicEmbeddingError>
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/QadicIntegralEmbedding@1

Executable usage: validation/unit/numthy/test_padic_field_facade.py:68 · validation/unit/numthy/test_padic_field_facade.py:71 · validation/unit/numthy/test_padic_field_facade.py:72 · validation/unit/numthy/test_qadic_field_facade.py:88 · validation/unit/numthy/test_qadic_field_facade.py:96 · validation/unit/numthy/test_qadic_field_facade.py:97

numthy.QadicIntegralEmbedding.retract_matrix

Compute retract matrix for this QadicIntegralEmbedding.

Detailed contract

Retract every matrix coefficient after certifying integrality.

(self, value: Matrix[QadicField]) -> Matrix[QadicRing]

noether::numthy::QadicIntegralEmbedding::retract_matrix · Direct Rust declaration

pub fn retract_matrix( &self, value: &Matrix<QadicField>, ) -> Result<Matrix<QadicRing>, CoefficientMappingError>
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/QadicIntegralEmbedding@1

Executable usage: validation/unit/numthy/test_padic_field_facade.py:90 · validation/unit/numthy/test_qadic_field_facade.py:117

numthy.QadicIntegralEmbedding.retract_polynomial

Compute retract polynomial for this QadicIntegralEmbedding.

Detailed contract

Retract every coefficient of a polynomial after certifying integrality.

(self, value: Polynomial[QadicField]) -> Polynomial[QadicRing]

noether::numthy::QadicIntegralEmbedding::retract_polynomial · Direct Rust declaration

pub fn retract_polynomial( &self, value: &Polynomial<QadicField>, ) -> Result<Polynomial<QadicRing>, CoefficientMappingError>
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/QadicIntegralEmbedding@1

Executable usage: validation/unit/numthy/test_padic_field_facade.py:88 · validation/unit/numthy/test_qadic_field_facade.py:115

numthy.QadicIntegralEmbedding.retract_power_series

Compute retract power series for this QadicIntegralEmbedding.

Detailed contract

Retract every retained series coefficient after certifying integrality.

(self, value: PowerSeries[QadicField]) -> PowerSeries[QadicRing]

noether::numthy::QadicIntegralEmbedding::retract_power_series · Direct Rust declaration

pub fn retract_power_series( &self, value: &PowerSeries<QadicField>, ) -> Result<PowerSeries<QadicRing>, CoefficientMappingError>
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/QadicIntegralEmbedding@1

Executable usage: validation/unit/commalg/test_power_series_facade.py:317

numthy.QadicIntegralEmbedding.retract_vector

Compute retract vector for this QadicIntegralEmbedding.

Detailed contract

Retract every vector coefficient after certifying integrality.

(self, value: Vector[QadicField]) -> Vector[QadicRing]

noether::numthy::QadicIntegralEmbedding::retract_vector · Direct Rust declaration

pub fn retract_vector( &self, value: &Vector<QadicField>, ) -> Result<Vector<QadicRing>, CoefficientMappingError>
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/QadicIntegralEmbedding@1

Executable usage: validation/unit/numthy/test_padic_field_facade.py:89 · validation/unit/numthy/test_qadic_field_facade.py:116

numthy.QadicIntegralEmbedding.source

Return the source recorded by this QadicIntegralEmbedding.

Detailed contract

Integral source context.

(self) -> QadicRing

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

pub fn source(&self) -> &QadicRing
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/QadicIntegralEmbedding@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.QadicIntegralEmbedding.target

Return the target recorded by this QadicIntegralEmbedding.

Detailed contract

Fraction-field target context.

(self) -> QadicField

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

pub fn target(&self) -> &QadicField
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/QadicIntegralEmbedding@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.QadicIntegralEmbedding.to_json

Serialize this QadicIntegralEmbedding to its canonical typed JSON representation.

Detailed contract

Canonical JSON containing both endpoint contexts.

(self) -> str

noether::numthy::QadicIntegralEmbedding::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/QadicIntegralEmbedding@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.QadicIntegralEmbedding.verify

Replay and verify the mathematical evidence carried by this QadicIntegralEmbedding.

Detailed contract

Recheck that source and target have identical presentation and precision identity.

(self) -> None

noether::numthy::QadicIntegralEmbedding::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/numthy/QadicIntegralEmbedding@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