# Argument selection for Debye polynomials U_k(p), T567

The only free table parameter is `k`, an integer with k >= 0. Its category is
`discrete_index`: it counts successive inverse-order corrections in the
uniform asymptotic expansions of modified Bessel functions. This is a named
symbolic coefficient family, not a sampled function-value grid.

The default enumeration is every integer k from 0 through 12, in increasing
order. The polynomial indeterminate p is not a table parameter or a selected
numerical argument. Every rational coefficient of each polynomial is retained.
The order zero polynomial is exactly 1, and no denominators are cleared or
external (-1)^k factors absorbed.

The source for the family, index and normalization is DLMF 10.41(ii),
particularly 10.41.3, 10.41.4, 10.41.8 and 10.41.9:
https://dlmf.nist.gov/10.41#ii . For the real-variable expansion,
p = (1+z^2)^(-1/2), z > 0, with positive square roots and real logarithms.
The stored polynomial is an element of QQ[p], valid as a symbolic polynomial
without a branch choice. Choosing p rather than z preserves the standard
coefficient object a reader encounters.

The cutoff 12 is an editorial extent, not a boundary in the mathematical
family. A complete initial prefix supports identification of successive
asymptotic corrections and includes every coefficient of the polynomials of degrees 0,3,...,36
at their actual indices. The proposal's other objects, V_k and the fixed-order
Hankel coefficients, belong in separate tables. Coverage across those distinct
regimes should precede a much longer prefix here. There are no additional
parameter families, exploratory samples, arbitrary decimal inputs or Cartesian
products in this table. The selection does not prune to the first three
polynomials printed in a handbook.

The exact length and serialized-size measurements used to confirm this bound
are recorded in numerical-evidence.json and the build handoff. The generator's
default range, Programs example and completeness note all use k=0,...,12.
All higher nonnegative indices remain part of the parameter's declared domain.

The governing skill copies are archived at
agents/runs/20261028T010346Z-build-skills/manifest.json. Both local and live
copies have SHA-256
e37efb7e2ededf6c1d7134c673f152f648f26e0267c556aebdf3ec65765d8b4c.
This builder's justification is not an independent critique acceptance.