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3795 bytes, as of the version from 2026-10-27 19:10 (current). Recorded here, not run.
# Argument selection for T557
This build follows the coordinated canonical NumberDB skill recorded in
`agents/runs/20261027T185658Z-build-skills/manifest.json`. Both the archived
local and live files were read and verified as SHA-256
`e37efb7e2ededf6c1d7134c673f152f648f26e0267c556aebdf3ec65765d8b4c`.
The attachment `skill-evidence.json` records these sources and hashes.
This is the builder's justification, not an independent argument review.
The table holds one named polynomial family in the monic normalization agreed
in numberdb-data family #223. The unchanged indeterminate is x; x is not a
sampled argument. Both free parameters are genuine discrete indices.
* **m, first Hermite degree (`discrete_index`).** In
W(H_m,H_{m+1},...,H_{m+n-1}), m locates the consecutive block of Hermite
degrees. It controls one dimension of the generalized Hermite family and
the integer Painlevé IV parameters alpha=2m+n+1, beta=-2n^2 of the displayed
rational solution. The positive domain selects the nonconstant family.
* **n, number of Wronskian factors (`discrete_index`).** This is the length of
the consecutive block and the determinant order. It is an independent
discrete family index, not a sampling index for a continuous coupling.
Its positive domain again excludes the identically unit boundary.
The joint finite selection contains every positive pair with m*n <= 50,
ordered by (m*n,m,n). Its 207 members cover all shapes through degree 50,
including m=1 and n=1, both orientations, and balanced cases through (7,7).
We do not retain only the classical n=1 slice or impose a square cutoff that
would discard small-degree elongated shapes while retaining much larger
polynomials. No continuous input, decimal mesh, reference probability or
quantile is involved. All selected groups have the same discrete-index role.
The degree bound is a practical, noncanonical reference choice. The degree-40
trial had 158 entries and a longest polynomial of 597 characters. Degree 50
has 207 entries with a longest polynomial of 920 characters at (1,50), and
about 77.4 KiB by the supplied measurement helper. This extends the coverage
to the next balanced shape while keeping all expanded polynomials readable.
No entry count or storage threshold was used as a target. Coefficients are
exact rationals, so the 100-digit approximation goal does not apply.
The sources also extend this family to m=0 or n=0, giving 1 throughout both
axes. These repeated unit boundary cases are stated in the table's comments
and used by the checks; they do not add positive-degree identification targets.
The parameters describe the infinite positive-index subfamily, while the
completeness note, generator, and checks all state the finite joint bound.
Supporting sources actually checked:
* P. A. Clarkson, *The Fourth Painlevé Transcendent* (2008), Section 5.1,
Theorem 5.2 and equations (5.4), (5.5), (5.7), (5.8), and (5.9):
<https://kar.kent.ac.uk/id/document/3149>. PDF pages 9 and 10 (printed 8 and 9)
were read visually in addition to extracted text.
* R. Buckingham, *Large-Degree Asymptotics of Rational Painlevé-IV Functions
Associated to Generalized Hermite Polynomials*, IMRN 2020, 5534–5577,
equations (1)–(8), DOI <https://doi.org/10.1093/imrn/rny172>.
The sources justify the family, indices, scaling, and relations. The finite
bound 50 is our reference choice, not a cutoff claimed by those papers.
The independent critique must assess both indices and this joint selection,
check the live default enumeration, and fingerprint `generate.py`, plus any
other source files the critic considers selection-relevant. The generator has
no local arithmetic or input-data dependencies; `checks.py` and `check.py`
are verification attachments, not generation dependencies.