numthy · QadicField¶

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

Worked example¶

The complete checked program Lift a simple square root from F_5 to fixed-precision Z_5 uses this API directly.

# SPDX-License-Identifier: MIT
# Copyright (c) 2026 Benjamin Antieau and Deven Manam
"""Lift a simple square root from F_5 to fixed-precision Z_5."""

from noether.commalg import PolynomialRing
from noether.numthy import PadicRing


def main() -> None:
    integers = PadicRing(5, absolute_precision=6)
    polynomials = PolynomialRing(integers, ["x"])
    x = polynomials.generator("x")
    equation = x**2 - 6
    root = integers.hensel_lift(equation, integers.residue_field(1))
    assert equation.evaluate([root]) == integers.zero()
    print(f"sqrt(6) in Z_5 mod 5^6 is represented by {int(root)}")


if __name__ == "__main__":
    main()

Run it with .venv/bin/python examples/10_padic_hensel_lifting.py.

`numthy.QadicField`¶

A fixed-precision unramified p-adic model, with inversion but without exact-field claims.

PythonRust

QadicField(residue_field: FiniteField, absolute_precision: int)

noether::numthy::QadicField · Direct Rust declaration

pub struct QadicField { /* 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/QadicField@1

Executable usage: validation/unit/commalg/test_context_tower_workload.py:120 · validation/unit/commalg/test_context_tower_workload.py:26 · validation/unit/commalg/test_context_tower_workload.py:38 · validation/unit/commalg/test_context_tower_workload.py:58 · validation/unit/commalg/test_polynomial_facade.py:28 · validation/unit/commalg/test_polynomial_facade.py:355

`numthy.QadicField.call`¶

Coerce the supplied input through this context or mathematical map.

PythonRust

(self, value: int) -> QadicElement

noether::numthy::QadicField · 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/QadicField@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.QadicField.absolute_precision`¶

Return the absolute precision recorded by this QadicField.

Detailed contract

Immutable absolute precision cap N.

PythonRust

(self) -> int

noether::numthy::QadicField::absolute_precision · Direct Rust declaration

pub fn absolute_precision(&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/QadicField@1

Executable usage: examples/10_padic_hensel_lifting.py:10 · examples/11_qadic_witt_vectors.py:10 · examples/31_local_fields_and_precision.py:11 · validation/unit/commalg/test_acceptance_workload.py:26 · validation/unit/commalg/test_acceptance_workload.py:27 · validation/unit/commalg/test_context_tower_workload.py:111

`numthy.QadicField.coefficient_field`¶

Return the coefficient field recorded by this QadicField.

Detailed contract

Base field Q_p supplying the power-basis coefficients.

PythonRust

(self) -> PadicField

noether::numthy::QadicField::coefficient_field · Direct Rust declaration

pub fn coefficient_field(&self) -> &PadicField

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/QadicField@1

Executable usage: validation/unit/linalg/test_semilinear_facade.py:53 · validation/unit/numthy/test_padic_local_field_facade.py:49 · validation/unit/numthy/test_padic_local_field_facade.py:78 · validation/unit/numthy/test_qadic_field_facade.py:145 · validation/unit/numthy/test_qadic_field_facade.py:146 · validation/unit/numthy/test_qadic_field_facade.py:31

`numthy.QadicField.coefficient_frobenius`¶

Compute coefficient frobenius for this QadicField.

Detailed contract

Checked lift of residue-field Frobenius on this unramified field.

PythonRust

(self, exponent: int) -> QadicFieldFrobenius

noether::numthy::QadicField::coefficient_frobenius · Direct Rust declaration

pub fn coefficient_frobenius(&self, exponent: u64) -> QadicFieldFrobenius
fn coefficient_frobenius(&self, exponent: u64) -> Self::Frobenius

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/QadicField@1

Executable usage: validation/unit/commalg/test_power_series_facade.py:279 · validation/unit/commalg/test_power_series_facade.py:286 · validation/unit/numthy/test_padic_field_facade.py:109 · validation/unit/numthy/test_padic_ring_facade.py:132 · validation/unit/numthy/test_padic_ring_facade.py:138 · validation/unit/numthy/test_qadic_field_facade.py:125

`numthy.QadicField.divide_uniformizer_exact`¶

Compute divide uniformizer exact for this QadicField.

Detailed contract

Divide by p^exponent and lower absolute precision by exponent.

PythonRust

(self, value: QadicElement, exponent: int) -> QadicElement

noether::numthy::QadicField::divide_uniformizer_exact · Direct Rust declaration

pub fn divide_uniformizer_exact( &self, value: &QadicElement, exponent: u64, ) -> Result<QadicElement, QadicUniformizerDivisionError>

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/QadicField@1

Executable usage: validation/unit/numthy/test_padic_field_facade.py:100 · validation/unit/numthy/test_padic_ring_facade.py:147 · validation/unit/numthy/test_padic_ring_facade.py:153 · validation/unit/numthy/test_qadic_field_facade.py:92 · validation/unit/numthy/test_qadic_ring_facade.py:84 · validation/unit/numthy/test_qadic_ring_facade.py:89

`numthy.QadicField.element_from_coefficients`¶

Compute element from coefficients for this QadicField.

Detailed contract

Construct an element from ascending Q_p coordinates in the declared q-adic power basis. Missing high coordinates are zero, and at most residue_degree coordinates may be supplied. Every coordinate must belong to this context’s exact coefficient field.

PythonRust

(self, coefficients: list[PadicElement]) -> QadicElement

noether::numthy::QadicField::element_from_coefficients · Direct Rust declaration

pub fn element_from_coefficients( &self, coefficients: &[PadicElement], ) -> Result<QadicElement, InvalidQadicCoordinatesError>

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/QadicField@1

Executable usage: examples/09_finite_field_isomorphisms.py:13 · validation/unit/commalg/test_base_change.py:92 · validation/unit/commalg/test_context_tower_workload.py:157 · validation/unit/linalg/test_semilinear_facade.py:37 · validation/unit/numthy/test_finite_field_facade.py:111 · validation/unit/numthy/test_finite_field_facade.py:144

`numthy.QadicField.from_json`¶

Deserialize and revalidate an unramified fixed-precision field context.

PythonRust

(cls, value: str) -> QadicField

noether::numthy::QadicField::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/QadicField@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.QadicField.generator`¶

Compute generator for this QadicField.

Detailed contract

Distinguished Teichmüller lift of the residue-field generator.

PythonRust

(self) -> QadicElement

noether::numthy::QadicField::generator · Direct Rust declaration

pub fn generator(&self) -> QadicElement

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/QadicField@1

Executable usage: examples/08_finite_field_extensions.py:10 · examples/08_finite_field_extensions.py:11 · examples/08_finite_field_extensions.py:12 · examples/08_finite_field_extensions.py:13 · examples/08_finite_field_extensions.py:14 · examples/10_padic_hensel_lifting.py:12

`numthy.QadicField.integral_embedding`¶

Compute integral embedding for this QadicField.

Detailed contract

Canonical embedding of this field’s fixed-cap Witt ring.

PythonRust

(self) -> QadicIntegralEmbedding

noether::numthy::QadicField::integral_embedding · Direct Rust declaration

pub fn integral_embedding(&self) -> QadicIntegralEmbedding

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/QadicField@1

Executable usage: validation/unit/commalg/test_power_series_facade.py:308 · validation/unit/numthy/test_padic_field_facade.py:59 · validation/unit/numthy/test_padic_field_facade.py:77 · validation/unit/numthy/test_qadic_field_facade.py:102 · validation/unit/numthy/test_qadic_field_facade.py:78

`numthy.QadicField.integral_ring`¶

Return the integral ring recorded by this QadicField.

Detailed contract

Fixed-cap Witt ring canonically embedded in this field context.

PythonRust

(self) -> QadicRing

noether::numthy::QadicField::integral_ring · Direct Rust declaration

pub fn integral_ring(&self) -> &QadicRing

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/QadicField@1

Executable usage: validation/unit/numthy/test_padic_field_facade.py:66 · validation/unit/numthy/test_qadic_field_facade.py:85

`numthy.QadicField.inverse`¶

Compute inverse for this QadicField.

Detailed contract

Invert a value that is nonzero at the available precision. This is finite-precision arithmetic, not a proof that the represented value is nonzero in the complete unramified field. Consequently QadicField deliberately does not implement the exact crate::Field trait.

PythonRust

(self, value: QadicElement) -> QadicElement

noether::numthy::QadicField::inverse · Direct Rust declaration

pub fn inverse( &self, value: &QadicElement, ) -> Result<QadicElement, InversionError>

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/QadicField@1

Executable usage: examples/distributed/f2_linear_algebra.py:37 · examples/distributed/f2_linear_algebra.py:39 · validation/unit/commalg/test_context_tower_workload.py:121 · validation/unit/commalg/test_context_tower_workload.py:122 · validation/unit/commalg/test_context_tower_workload.py:75 · validation/unit/commalg/test_context_tower_workload.py:77

`numthy.QadicField.one`¶

Compute one for this QadicField.

PythonRust

(self) -> QadicElement

noether::numthy::QadicField::one · Direct Rust declaration

fn one(&self) -> Self::Element

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/QadicField@1

Executable usage: examples/02_ball_arithmetic.py:13 · examples/11_qadic_witt_vectors.py:11 · examples/24_homology_and_persistence.py:3 · examples/31_local_fields_and_precision.py:27 · validation/unit/combinatorics/test_poset.py:3 · validation/unit/commalg/test_acceptance_workload.py:37

`numthy.QadicField.precision_restriction`¶

Compute precision restriction for this QadicField.

Detailed contract

Construct a first-class restriction to a coarser fixed-cap field.

PythonRust

(self, absolute_precision: int) -> QadicFieldRestriction

noether::numthy::QadicField::precision_restriction · Direct Rust declaration

pub fn precision_restriction( &self, target_absolute_precision: u64, ) -> Result<QadicFieldRestriction, QadicRestrictionError>

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/QadicField@1

Executable usage: examples/31_local_fields_and_precision.py:30 · validation/unit/commalg/test_power_series_facade.py:319 · validation/unit/homalg/test_padic_homology_facade.py:204 · validation/unit/numthy/test_padic_field_facade.py:58 · validation/unit/numthy/test_padic_field_facade.py:78 · validation/unit/numthy/test_padic_local_field_facade.py:101

`numthy.QadicField.prime`¶

Return the prime recorded by this QadicField.

Detailed contract

Residue characteristic p.

PythonRust

(self) -> int

noether::numthy::QadicField::prime · Direct Rust declaration

pub fn prime(&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/QadicField@1

Executable usage: examples/07_integer_factorization.py:12 · examples/12_number_field_ideals.py:16 · examples/12_number_field_ideals.py:19 · examples/18_finite_field_schemes.py:10 · examples/24_homology_and_persistence.py:22 · examples/33_simplicial_cohomology_operations.py:24

`numthy.QadicField.residue`¶

Compute residue for this QadicField.

Detailed contract

Reduce an integral value to F_q; nonintegral values have no residue.

PythonRust

(self, value: QadicElement) -> FiniteFieldElement

noether::numthy::QadicField::residue · Direct Rust declaration

pub fn residue( &self, value: &QadicElement, ) -> Result<Option<FiniteFieldElement>, ContextMismatchError>

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/QadicField@1

Executable usage: examples/11_qadic_witt_vectors.py:11 · examples/11_qadic_witt_vectors.py:12 · examples/11_qadic_witt_vectors.py:13 · examples/11_qadic_witt_vectors.py:15 · examples/13_chinese_remainder_theorem.py:3 · examples/31_local_fields_and_precision.py:10

`numthy.QadicField.residue_degree`¶

Return the residue degree recorded by this QadicField.

Detailed contract

Residue degree [F_q : F_p].

PythonRust

(self) -> int

noether::numthy::QadicField::residue_degree · Direct Rust declaration

pub fn residue_degree(&self) -> usize

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/QadicField@1

Executable usage: validation/unit/numthy/test_acceptance_workload.py:59 · validation/unit/numthy/test_acceptance_workload.py:68 · validation/unit/numthy/test_acceptance_workload.py:69 · validation/unit/numthy/test_acceptance_workload.py:72 · validation/unit/numthy/test_number_field_facade.py:213 · validation/unit/numthy/test_number_field_facade.py:234

`numthy.QadicField.residue_field`¶

Return the residue field recorded by this QadicField.

Detailed contract

Residue field F_q, including its normalized defining presentation.

PythonRust

(self) -> FiniteField

noether::numthy::QadicField::residue_field · Direct Rust declaration

pub fn residue_field(&self) -> &FiniteField

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/QadicField@1

Executable usage: examples/10_padic_hensel_lifting.py:14 · examples/11_qadic_witt_vectors.py:10 · examples/11_qadic_witt_vectors.py:11 · examples/11_qadic_witt_vectors.py:14 · examples/11_qadic_witt_vectors.py:9 · validation/unit/commalg/test_acceptance_workload.py:25

`numthy.QadicField.restrict`¶

Compute restrict for this QadicField.

Detailed contract

Explicitly discard high p-adic digits into a distinct field context.

PythonRust

(self, value: QadicElement, absolute_precision: int) -> QadicElement

noether::numthy::QadicField::restrict · Direct Rust declaration

pub fn restrict( &self, value: &QadicElement, absolute_precision: u64, ) -> Result<QadicElement, QadicRestrictionError>

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/QadicField@1

Executable usage: validation/unit/commalg/test_local_algebra.py:93 · validation/unit/groups/test_weight_characters.py:48 · validation/unit/numthy/test_padic_local_field_facade.py:100 · validation/unit/numthy/test_padic_ring_facade.py:62 · validation/unit/numthy/test_qadic_ring_facade.py:48

`numthy.QadicField.teichmuller`¶

Compute teichmuller for this QadicField.

Detailed contract

Lift a residue-field value to its Teichmüller representative.

PythonRust

(self, value: FiniteFieldElement) -> QadicElement

noether::numthy::QadicField::teichmuller · Direct Rust declaration

pub fn teichmuller( &self, value: &FiniteFieldElement, ) -> Result<QadicElement, ContextMismatchError>

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/QadicField@1

Executable usage: examples/11_qadic_witt_vectors.py:12 · validation/unit/commalg/test_acceptance_workload.py:33 · validation/unit/commalg/test_acceptance_workload.py:38 · validation/unit/commalg/test_context_tower_workload.py:101 · validation/unit/commalg/test_context_tower_workload.py:113 · validation/unit/infra/test_public_api_index.py:164

`numthy.QadicField.to_json`¶

Serialize the verified residue-field presentation and precision.

Detailed contract

Serialize the residue-field presentation and absolute precision as typed JSON.

PythonRust

(self) -> str

noether::numthy::QadicField::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/QadicField@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.QadicField.zero`¶

Compute zero for this QadicField.

PythonRust

(self) -> QadicElement

noether::numthy::QadicField::zero · Direct Rust declaration

fn zero(&self) -> Self::Element

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/QadicField@1

Executable usage: examples/08_finite_field_extensions.py:11 · examples/10_padic_hensel_lifting.py:15 · examples/40_exterior_algebra.py:26 · examples/distributed/f2_linear_algebra.py:28 · examples/distributed/f2_linear_algebra.py:29 · examples/distributed/f2_linear_algebra.py:30

numthy · QadicField¶

Worked example¶

numthy.QadicField¶

numthy.QadicField.__call__¶

numthy.QadicField.absolute_precision¶

numthy.QadicField.coefficient_field¶

numthy.QadicField.coefficient_frobenius¶

numthy.QadicField.divide_uniformizer_exact¶

numthy.QadicField.element_from_coefficients¶

numthy.QadicField.from_json¶

numthy.QadicField.generator¶

numthy.QadicField.integral_embedding¶

numthy.QadicField.integral_ring¶

numthy.QadicField.inverse¶

numthy.QadicField.one¶

numthy.QadicField.precision_restriction¶

numthy.QadicField.prime¶

numthy.QadicField.residue¶

numthy.QadicField.residue_degree¶

numthy.QadicField.residue_field¶

numthy.QadicField.restrict¶

numthy.QadicField.teichmuller¶

numthy.QadicField.to_json¶

numthy.QadicField.zero¶

`numthy.QadicField`¶

`numthy.QadicField.call`¶

`numthy.QadicField.absolute_precision`¶

`numthy.QadicField.coefficient_field`¶

`numthy.QadicField.coefficient_frobenius`¶

`numthy.QadicField.divide_uniformizer_exact`¶

`numthy.QadicField.element_from_coefficients`¶

`numthy.QadicField.from_json`¶

`numthy.QadicField.generator`¶

`numthy.QadicField.integral_embedding`¶

`numthy.QadicField.integral_ring`¶

`numthy.QadicField.inverse`¶

`numthy.QadicField.one`¶

`numthy.QadicField.precision_restriction`¶

`numthy.QadicField.prime`¶

`numthy.QadicField.residue`¶

`numthy.QadicField.residue_degree`¶

`numthy.QadicField.residue_field`¶

`numthy.QadicField.restrict`¶

`numthy.QadicField.teichmuller`¶

`numthy.QadicField.to_json`¶

`numthy.QadicField.zero`¶