# Numerical evidence for T577, Schur Q-polynomials

The archived policy manifest is
`agents/runs/20261028T093728Z-build-skills/manifest.json`.
Both skill-local.md and skill-live.md have verified SHA-256
`e37efb7e2ededf6c1d7134c673f152f648f26e0267c556aebdf3ec65765d8b4c`.
The source snapshot's canonical skill matches. See skill-evidence.json.

The computation is exact over ZZ[x1,...,xn]. No approximation, working
precision, or 100-digit padding is applicable. SageMath 10.9 and NumberDB
0.1.10 were reported by the isolated Sage runtime.

The generator's Pfaffian construction is independently compared with a
backtracking enumeration of marked shifted semistandard tableaux, accumulating
an exact integer coefficient for each monomial. This algorithm shares no
one-row series or Pfaffian arithmetic with the generator. Before comparing
entries, its empty, linear, quadratic, (2,1) and impossible one-variable
shape controls are checked against explicit known polynomials. Sage's
Hall-Littlewood implementation supplies a second independent comparison.

Pre-submission results (`checks-before.log`):

* All 40 entries agree coefficient by coefficient with marked shifted tableaux.
* All 40 agree with Sage Hall-Littlewood Q at t=-1 and with 2^ell times P.
* All 91 adjacent-variable transpositions preserve their polynomial.
* All 40 have the declared degree and positive integer coefficients divisible
  by 2^ell. Their symmetric-function power-sum expansions have only odd parts.
* All 40 pass zero-alphabet stability and opposite-pair cancellation.
* All 6 included staircase shapes agree with 2^ell times Sage's Schur function.
* All 15 overlaps with the published T261 agree after t=-1 and 2^ell scaling.
* All 3 overlapping staircase entries in published T122 agree after scaling.
* Boundary formulas and the Programs example (n=3, lambda=(3,2)) pass.
* Enumeration independently checked using subsets, including entry order.

The unmodified campaign dry_run.py and check.py report 40 exact entries,
no disguised decimals, no prose issues and named Sage imports. The maximum
written value is 1299 characters, at n=4, lambda=(5,1); the entries estimate
is about 15.8 KiB (tuple-key measure: 15.7197 KiB). All 47 proposed entries
were measured before reducing the five-variable degree bound; their maximum
was 3542 characters. No partial degree slice was kept.

The only failed preliminary check was an import failure in Sage's
Hall-Littlewood conversion: matrix construction referenced a Laurent
polynomial module that had not been initialized. An explicit named import
of sage.rings.polynomial.laurent_polynomial_ring fixed it. This was not a
numerical disagreement. The original failure is preserved in
checks-initial-import-failure.log.

Reproduction: download generate.py, checks.py and check.py together and run
`sage -python checks.py`. After authenticating to read the private draft, run
`sage -python generate.py` for the complete generator verification and
`sage -python checks.py --stored` for independent checks of stored entries.
The builder will preserve the actual readback and audit results in its handoff;
the subsequent independent critique is a separate stage.