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3780 bytes, as of the version from 2026-10-28 08:16 (current). Recorded here, not run.
# Numerical evidence for T556
The exact archived instruction versions are identified in
`skill-manifest.json`, copied from the run manifest at
`/home/ubuntu/numberdb-campaign-20261026-hundred/repo-balanced-coverage/agents/runs/20261027T182939Z-build-skills/manifest.json`. Both versions have SHA-256
`e37efb7e2ededf6c1d7134c673f152f648f26e0267c556aebdf3ec65765d8b4c`.
The default enumeration contains n=0,...,12, with monic integer polynomials
in the unscaled indeterminate x. The largest degree is 78. The longest entry
has 1076 characters; the measured block is 3.8291 KB. A pilot through n=16
found Q_13 has 1358 characters and Q_14 has 1922, motivating the cutoff.
Exact integer coefficients require no numerical error estimate or decimal
precision target.
The generator's differential recurrence uses exact polynomial division in
ZZ[x] and asserts zero remainder. The independent checker builds coefficients
of exp(x*t-4*t^3/3) by a finite factorial sum in QQ[x], then evaluates the
normalized determinant in Bertola and Bothner, equations (1.3)-(1.4), using
Bareiss elimination. It shares no recurrence code with the generator. All
13 expanded polynomials agree coefficient by coefficient.
For n=1,...,12, clearing denominators verifies w''=2w^3+xw+n for
w=Q_(n-1)'/Q_(n-1)-Q_n'/Q_n. All 11 recurrence triples entirely present in
the table agree. Every polynomial has the claimed degree and leading
coefficient, and every coefficient is integral. Q_0 and Q_1 and all seven
polynomials in Roffelsen's Table 1 (Q_2,...,Q_8) agree exactly. The table's
standalone program reproduces all 13 entries.
Controls ran before comparisons: determinant of the empty matrix is 1,
a row-swap matrix has determinant -1, a singular matrix has determinant 0,
and a polynomial matrix has known determinant x^3+1. The known solution
w=-1/x passes the differential-equation check, while an intentionally
altered Q_2 fails. A suspicious Q_5 reference coefficient is also explicitly
rejected by both the defining recurrence and the differential equation.
The initial comparison exposed a discrepancy in the retrieved DLMF text:
its x^3 coefficient in Q_5 was -313600; exact recurrence and determinant give
-3136000, agreeing with Roffelsen. Substitution of the retrieved coefficient
yields recurrence residual 2822400*x^3*Q_3, so this is not a convention change.
Direct access restrictions prevented distinguishing a live source typo from
a text-extraction error. A separate Q_8 mismatch was a copied reference
literal with three extra zeros; checking the source resolved that error in
the checker. Both failures happened before filling the table and are retained
in the run logs. See source-notes.md for URLs and access limitations.
After filling, numberdb verify(sample=None) reported 13/13 matched, zero
differing, missing or extra entries. The independent determinant, published
initial values, degree, coefficient and Painlevé II checks all ran again on
values read from the database. The final site audit reports clean=true and
no findings. Its only initial finding, an omitted installation command in
the generator docstring, was fixed without changing arithmetic or selection.
Reproduction: install numberdb in SageMath, download generate.py, checks.py
and check.py, and run `sage -python checks.py`. To check a readback export,
place the n-to-polynomial mapping in values.json beside the checker. Run
`sage -python generate.py` with the author's key to run the client comparison
against the private draft. The build used SageMath 10.9 and numberdb 0.1.10.
The full report and final live source hashes are in numerical-evidence.json.
These are build evidence, not an independent critique verdict. The controller
must arrange the separate critique and argument-selection sidecar.