Datasets:

PureOne
/

AELYRION-NOULITH-III

Function in `noulith.third`	Meaning / boundary
`all_components(m)`	Star, triangle, and twin descriptors; theorem domain m>=6
`component_equations(m, component)`	Exact integer matrix in ascending logarithmic edge order
`intersection_dimension(m, components)`	Dimension computed from stacked exact equations
`component_membership(m, q)`	Components containing a coefficient point under the implemented relations
`certify_scalar(m, q)`	Exact rational certificate, m>=6 and abs(q+1)<1/20
`predicted_rank(m, q)`	Third-deficit rank from classification conditions
`direct_block_rank(m, q, r)`	Independent all-subset calculation; exponential in m
`optimal_calibration_subfamily(m)`	Attaining selection of twin outlier pairs, m>=6
`calibration_classes(m, selected_twins)`	Equivalence classes of edge scales constrained by the selected twin subspaces
`example_point(m, kind, permutation=None)`	Nearby rational example point
`q_from_exponents(exponents, base=Rational(1001,1000))`	Rational coefficients from directed antisymmetric integer exponents

Named examples: origin, star, triangle, twin, uniform_star, star_twin, edge, double_star, generic.

Verify a raw witness

from noulith.algebra import TwistedAlgebra
from noulith.third import example_point, certify_scalar

m = 6
q = example_point(m, "generic")
certificate = certify_scalar(m, q)
assert certificate.visible
blocks = certificate.full_partition(m)
algebra = TwistedAlgebra(m, q)
raw = algebra.isolated_response(blocks)

assert len(blocks) == m - 3
assert raw == algebra.isolated_unit(blocks) * certificate.normalized_response
assert raw != 0

This checks the actual sparse algebra, including its crossing factor. It should not be replaced by a rounded near-zero test.

Scope

The local analytic theorem and strict real-rational implementation neighborhood are distinct. Passing an arbitrary dictionary to a low-level routine does not extend a theorem. A selected calibration pair represents an entire twin-subspace constraint, not a detector sample. Do not apply the third-deficit theorem at m=5 or the generic core formula at r=0.