# Numerical evidence for T581

Build run: 20261028T121807Z-build. Governing manifest:
`agents/runs/20261028T121807Z-build-skills/manifest.json`.
Archived local and live skill SHA-256, both identical:
`e37efb7e2ededf6c1d7134c673f152f648f26e0267c556aebdf3ec65765d8b4c`.
The attached skills-evidence.json records both source filenames and the manifest.

All 37 values are exact integer polynomials. Decimal precision and error balls
are not applicable; coefficients have no numerical rounding error. Runtime:
SageMath 10.9 and numberdb 0.1.10, via the campaign's agents/sage.sh wrapper.

The attached generate.py sums weighted chains by the first nonempty flat.
It uses symmetry for uniforms and braids, and actual flat masks for all upper
intervals of the named objects. The attached checks.py shares no polynomial
computation with it. It uses the characteristic convolution; for named objects,
every ground-set subset is ranked, flats are found by rank increase, and each
interval characteristic polynomial is obtained from an alternating subset sum.
For braids it enumerates actual partitions, not the generator's Stirling
recursion. The mathematical identities are Proposition 3.5 and Theorem 3.12 of
FMSV, arXiv:2212.03190v3, https://arxiv.org/html/2212.03190v3 .

Before comparisons, controls verify 10 specified answers, including rank zero,
rank one, rank two, U(3,3), U(3,4), K(4), Fano, and three derangement
polynomials. Seven further controls compare Boolean matroids with explicitly
counted permutation descents. Partition enumeration returns Bell(8)=4140.
The controls are not allowed to silently return zero.

Results before filling:

- All 37 complete polynomials agree with the characteristic convolution.
- All 24 uniform polynomials also agree with FMSV Theorem 3.21, with the
  derangement polynomials obtained by explicitly counting permutations through
  size seven, rather than by a polynomial recurrence.
- Every value has degree rank-1, constant and leading coefficients one,
  nonnegative coefficients, palindromicity and unimodality. Its linear
  coefficient is |L(M)|-|E|-1, as also follows from the degree-one presentation.
- The standard check.py exactness, agrees_with, and measure helpers pass.
- The standard dry_run.py reports 37 entries, longest 79 characters at U(8,12),
  and about 3.0 KiB including parameter labels. Measuring the keyed polynomials
  alone reports 2.711 KiB. No exact-in-disguise warnings occur.
- The full independent check took 0.757 seconds, excluding wrapper startup.
- Named flat counts: Fano 16, NonFano 18, Pappus 29, NonPappus 31, Vamos 79,
  R10 162, TicTacToe 225, K5dual 314.

These are builder checks, not the campaign's independent critique. The critic
must independently inspect coefficients and argument-selection.md, and create
its own fingerprint-bound sidecar. The generator imports no research files;
all of its selection logic is in generate.py. checks.py imports the attached
standard check.py. The post-fill verification and audit outcomes are recorded
in the build handoff and retained run logs.