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numthy · PointCountCertificate

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

Worked example

The complete checked program Enumerate and certify rational points on a finite-field hyperbola uses this API directly.

# SPDX-License-Identifier: MIT
# Copyright (c) 2026 Benjamin Antieau and Deven Manam
"""Enumerate and certify rational points on a finite-field hyperbola."""

from noether.commalg import AffineSpace
from noether.numthy import FiniteField


def main() -> None:
    field = FiniteField.prime(5)
    plane = AffineSpace(field, 2, names=["x", "y"])
    x, y = plane.coordinate_ring.generators()
    hyperbola = plane.closed_subscheme(plane.coordinate_ring.ideal([x * y - 1]))
    count, certificate = hyperbola.point_count(shards=3)
    certificate.verify(hyperbola)
    assert count == 4
    assert len(list(hyperbola.rational_points())) == count
    print(f"xy=1 over F_5 has {count} rational points")


if __name__ == "__main__":
    main()

Run it with .venv/bin/python examples/18_finite_field_schemes.py.

numthy.PointCountCertificate

Checkable deterministic transcript for a finite-field point count.

class PointCountCertificate

noether::numthy::PointCountCertificate · Direct Rust declaration

pub struct PointCountCertificate { /* 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/PointCountCertificate@1

Executable usage: validation/unit/commalg/test_acceptance_workload.py:139 · validation/unit/commalg/test_acceptance_workload.py:143 · validation/unit/commalg/test_acceptance_workload.py:21 · validation/unit/commalg/test_scheme_facade.py:22 · validation/unit/commalg/test_scheme_facade.py:46 · validation/unit/commalg/test_scheme_facade.py:50

numthy.PointCountCertificate.candidate_count

Return the candidate count recorded by this PointCountCertificate.

Detailed contract

Number of canonical ambient candidates covered.

(self) -> int

noether::numthy::PointCountCertificate::candidate_count · Direct Rust declaration

pub fn candidate_count(&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/PointCountCertificate@1

Executable usage: validation/unit/commalg/test_acceptance_workload.py:137 · validation/unit/commalg/test_scheme_facade.py:142 · validation/unit/commalg/test_scheme_facade.py:41 · validation/corpus/commalg/finite-projective-variety-point-count.py:23 · validation/corpus/commalg/finite-projective-variety-point-count.py:27 · validation/corpus/commalg/klein-quartic-f43-point-count.py:19

numthy.PointCountCertificate.from_json

Deserialize and revalidate a PointCountCertificate from typed JSON.

Detailed contract

Restore and structurally validate a deterministic transcript.

(cls, value: str) -> PointCountCertificate

noether::numthy::PointCountCertificate::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/PointCountCertificate@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.PointCountCertificate.model_checksum

Return the model checksum recorded by this PointCountCertificate.

Detailed contract

Stable checksum of the exact scheme presentation.

(self) -> int

noether::numthy::PointCountCertificate::model_checksum · Direct Rust declaration

pub fn model_checksum(&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/PointCountCertificate@1

Executable usage: validation/unit/commalg/test_scheme_facade.py:42

numthy.PointCountCertificate.point_count

Return the point count recorded by this PointCountCertificate.

Detailed contract

Total rational-point count certified by the shards.

(self) -> int

noether::numthy::PointCountCertificate::point_count · Direct Rust declaration

pub fn point_count(&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/PointCountCertificate@1

Executable usage: examples/18_finite_field_schemes.py:14 · validation/unit/commalg/test_acceptance_workload.py:135 · validation/unit/commalg/test_acceptance_workload.py:149 · validation/unit/commalg/test_scheme_facade.py:137 · validation/unit/commalg/test_scheme_facade.py:138 · validation/unit/commalg/test_scheme_facade.py:178

numthy.PointCountCertificate.shard_candidate_counts

Return the shard candidate counts recorded by this PointCountCertificate.

Detailed contract

Candidate counts scanned by each shard.

(self) -> list[int]

noether::numthy::PointCountCertificate::shard_candidate_counts · Direct Rust declaration

pub fn shard_candidate_counts(&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/PointCountCertificate@1

Executable usage: validation/unit/commalg/test_acceptance_workload.py:137 · validation/unit/commalg/test_scheme_facade.py:142 · validation/unit/commalg/test_scheme_facade.py:43 · validation/corpus/commalg/finite-projective-variety-point-count.py:24 · validation/corpus/commalg/klein-quartic-f43-point-count.py:20

numthy.PointCountCertificate.shard_checksums

Return the shard checksums recorded by this PointCountCertificate.

Detailed contract

Stable evaluation checksums for each shard.

(self) -> list[int]

noether::numthy::PointCountCertificate::shard_checksums · Direct Rust declaration

pub fn shard_checksums(&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/PointCountCertificate@1

Executable usage: validation/unit/commalg/test_acceptance_workload.py:142 · validation/unit/commalg/test_scheme_facade.py:49

numthy.PointCountCertificate.shard_count

Return the shard count recorded by this PointCountCertificate.

Detailed contract

Number of deterministic disjoint shards.

(self) -> int

noether::numthy::PointCountCertificate::shard_count · Direct Rust declaration

pub fn shard_count(&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/PointCountCertificate@1

Executable usage: validation/unit/commalg/test_scheme_facade.py:159 · validation/unit/commalg/test_scheme_facade.py:165

numthy.PointCountCertificate.shard_point_counts

Return the shard point counts recorded by this PointCountCertificate.

Detailed contract

Rational-point counts found by each shard.

(self) -> list[int]

noether::numthy::PointCountCertificate::shard_point_counts · Direct Rust declaration

pub fn shard_point_counts(&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/PointCountCertificate@1

Executable usage: validation/unit/commalg/test_scheme_facade.py:44 · validation/corpus/commalg/finite-projective-variety-point-count.py:25 · validation/corpus/commalg/finite-projective-variety-point-count.py:29 · validation/corpus/commalg/klein-quartic-f43-point-count.py:21 · validation/corpus/commalg/klein-quartic-f43-point-count.py:24

numthy.PointCountCertificate.to_json

Serialize this PointCountCertificate to its canonical typed JSON representation.

Detailed contract

Serialize the deterministic transcript.

(self) -> str

noether::numthy::PointCountCertificate::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/PointCountCertificate@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.PointCountCertificate.verify

Replay and verify the mathematical evidence carried by this PointCountCertificate.

Detailed contract

Re-run the same deterministic partition and compare every transcript field.

(self, scheme: Scheme[FiniteField]) -> None

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

pub fn verify(&self, scheme: &Scheme<FiniteField>) -> Result<(), SchemeError>
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/PointCountCertificate@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