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3280 bytes, as of the version from 2026-10-28 08:16 (current). Recorded here, not run.
# Argument selection for T556
This build follows the coordinated archived skill manifest
`../20261027T182939Z-build-skills/manifest.json`. Both archived versions were
read in full and SHA-256 checked; both have digest
`e37efb7e2ededf6c1d7134c673f152f648f26e0267c556aebdf3ec65765d8b4c`.
This is the builder's justification, not an independent critique acceptance.
The sole free parameter is `n`, of type `Z`, with domain n >= 0. Its category
is `discrete_index`: it indexes the monic Yablonskii–Vorob’ev polynomials,
which give rational solutions of Painlevé II at integer equation parameter
alpha=n through w=(log(Q_(n-1)/Q_n))'. The indeterminate x is part of each
exact polynomial, not a sampled numerical argument. There are no additional
couplings, sampled arguments, probability parameters or quantity selectors.
The selection is every n from 0 through 12, in increasing order. It preserves
both initial polynomials and every intervening index, including less familiar
members. It supplies an initial reference segment for identifying the expanded
polynomials that occur in rational Painlevé solutions. It does not restrict
coverage to the five small polynomials printed in the handbook, nor add other
Painlevé families as extra quantities in the same table. The generalized
hierarchies and their deformation parameters define other objects and are not
hidden free parameters of this classical family.
The upper bound is a noncanonical readability choice, not a mathematical
boundary. A pilot through n=16 measured exact expanded strings: n=11 has
779 characters, n=12 has 1076, n=13 has 1358, n=14 has 1922, n=15 has 2518,
and n=16 has 3092. The retained family reaches degree 78 at n=12. Its full
block is about 3.8 KB, so storage limits are irrelevant to the chosen extent.
No initial segment is extended merely to reach an entry-count target. There
is one discrete family here, so the breadth review keeps all its initial
members; increasing degree further would deepen the same family while
reducing readability. Negative Painlevé integer parameters introduce no new
polynomials, since w(x;-n)=-w(x;n).
The family conventions are Q_0=1, Q_1=x, and
Q_(n+1) Q_(n-1)=x Q_n^2-4(Q_n Q_n''-(Q_n')^2), with no scaling of x. Parameters
state the infinite domain; the completeness note states 0 <= n <= 12 and
this reason. MAX_N and default enumerate in generate.py, the Programs
example, and the stored entry identities must all agree on that segment.
Checked sources:
- NIST DLMF section 32.8(ii), equations 32.8.5–32.8.10, gives the recurrence,
initial values and the rational-solution/Wronskian relationship:
https://dlmf.nist.gov/32.8.ii .
- Marco Bertola and Thomas Bothner, *Zeros of large degree Vorob'ev-Yablonski
polynomials via a Hankel determinant identity*, introduction, equations
(1.1)–(1.4), gives the integer Painlevé parameter, degrees, integer
coefficients, sign symmetry and normalized determinant:
https://arxiv.org/abs/1401.1408 .
The cutoff is our reference choice; neither source is claimed to prescribe it.
The separate critic must inspect the selection and all current generator
attachments, perform numerical checks, and create its own fingerprint-bound
argument-review sidecar. This build supplies no such acceptance.