# T558 argument selection

The sole table parameter is `n`, a nonnegative integer indexing the Umemura
polynomials for rational Painlevé III solutions. Its role is `discrete_index`.
The auxiliary initial value S_{-1}=1 belongs to the recurrence, not to a second
stored copy of the unit polynomial. The initial conditions and derivative
normalization are those in Clarkson, Definition 3.2, and Clarkson–Law–Lin,
Theorem 2.3. A convention with S_0=S_1=1 would require an index shift and is
not used here.

Both x and m are exact indeterminates. The storage letter m denotes the
literature's parameter mu; x is Clarkson's x and Clarkson–Law–Lin's z without
rescaling. Only the polynomial spelling changed from the empty draft's
notation. The site's Z[] parser accepts single-letter variable names and does
not accept the word mu. No table parameter was added, removed, or renamed.
Keeping the complete bivariate polynomial covers every specialization at once;
there is no hidden grid, mesh, quantile selection, or numerical argument.

The default enumeration and stored selection are exactly n=0,1,2,3,4,5,
in that order. This contiguous initial reference segment identifies low-order
polynomial factors of rational Painlevé III solutions while retaining the full
symbolic dependence on the coupling. A separate publication for every index
is not required: the family and its indexing are the mathematical rationale.
Breadth is provided by the whole bivariate object and all intervening indices,
rather than by deeper computations on only the m=0 or m=1 slice.

The bound n<=5 is a noncanonical readability choice, not a quota or an intrinsic
end to the family. Growth was measured through n=9. Expanded strings at
n=0,...,9 occupy 1, 5, 33, 126, 361, 898, 2014, 4135, 7950, and 14449 ASCII
bytes. S_5 has degree 15 and 62 terms; S_6 more than doubles the largest
stored entry. The six selected values total 1424 bytes before record
metadata. They cover the five nonconstant polynomials explicitly printed in
Clarkson–Law–Lin, Table 1, together with S_0. Every selected polynomial has an
independent exact Wronskian check. The numerical trials through S_7 are
preserved as growth and verification evidence; they do not enlarge the selection.
The generator remains extendible by its maximum argument. The program's
stored-range example uses range(6); its separate umemura(6) call demonstrates
the first additional polynomial. Completeness remains no.

Checked primary sources:

- Peter A. Clarkson, Classical solutions of the degenerate fifth Painlevé
  equation, J. Phys. A 56 (2023), 134002, DOI 10.1088/1751-8121/acbef1.
  Definition 3.2 and Remark 3.3, equations (21)–(23), establish the exact
  recurrence, index normalization, Wronskian orientation, scale and rational
  solution. Remark 5.4 gives the parameter-shift bilinear identity.
- Peter A. Clarkson, Chun-Kong Law and Chia-Hua Lin, A Constructive Proof for
  the Umemura Polynomials of the Third Painlevé Equation, SIGMA 19 (2023),
  080, DOI 10.3842/SIGMA.2023.080, arXiv:1609.00495.
  Theorems 2.3 and 2.5, Remark 2.4(3), Remarks 2.6(1)–(4), and Table 1
  establish the family, degree, coefficient generating function, determinant
  normalization, Bessel specialization, and low-order controls. The unsigned
  symmetry printed in Remark 2.4(2) is not adopted; S_1 already disproves it.

Policy: agents/runs/20261028T224818Z-repair-skills/manifest.json.
Archived local and live skills each have SHA-256
28ecbeeccb0c5e5d6eb541b5a5b0f88aec47b154374d909085116d34f5629199,
matching the canonical skill in this source snapshot.

This is the builder's justification. The selection and generator have changed;
a fresh independent critique and its argument-review sidecar are required.
This build neither supplies nor accepts an independent review verdict.