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3618 bytes, as of the version from 2026-10-28 03:22 (current). Recorded here, not run.
# Argument selection for T569
The table holds one named quantity, the Hankel asymptotic coefficient polynomial
$a_k(\nu)$ of DLMF 10.17.1. Its only free table parameter is the integer $k\geq0$,
the order of a correction in the fixed-order, large-argument expansion of the
modified Bessel function $K_\nu(z)$ (DLMF 10.40.2). Its role is `discrete_index`.
Both $\nu$ and $z$ are mathematical variables used to define the coefficients;
$\nu$ remains a polynomial indeterminate and $z$ is not stored as a parameter.
No choices of numerical $\nu$ or $z$ enter the enumeration.
The default enumeration is exactly $k=0,1,\ldots,20$ in increasing order. Each
entry retains the entire expanded polynomial in $\mathbb Q[\nu]$, with reduced
rational coefficients. The external signs and phase factors of other Bessel
expansions are not absorbed. This is a complete initial prefix of successive
asymptotic corrections, not a value grid or a collection limited to famous
specializations. The infinite index domain is specified in Parameters; the
finite extent belongs in complete-note.
The cutoff 20 is a practical editorial choice, not an intrinsic terminal index.
It gives degree at most 40. Measured with agents/table-build/check.py, the
21-entry block is 9.638 KiB and its longest expanded entry has 1,213 characters.
That entry is near the 1,100–1,300-character readability guidance, so the range
is retained rather than extended just because the byte limit has room. All
orders up to the cutoff are included. At family level, the Debye function and
derivative corrections and the Hankel derivative corrections are distinct
quantities for separate tables; breadth across those quantities is preferable
to extending only this prefix. This table samples none of their arguments.
There are no mixed parameter roles, exploratory reference samples, arbitrary
decimal meshes or multidimensional products. The half-integers used in checks
are interpolation nodes, not table entries: the Bessel-polynomial relation
selects them, and their count proves equality of all stored polynomials.
Checked supporting sources:
- https://dlmf.nist.gov/10.17#E1: the indexed coefficient family and normalization.
- https://dlmf.nist.gov/10.40#E2: the expansion using these exact polynomials.
- https://dlmf.nist.gov/10.49#E1 and #E12: half-integer coefficients and their
relation to finite spherical-Bessel expressions.
- The live Bessel_polynomials table: its independently indexed recurrence and
its exact stored polynomials through n=20 were checked against our controls.
Policy evidence: agents/runs/20261028T022135Z-build-skills/manifest.json.
Both archived local and live skills were read, hashed, and found byte-identical:
SHA-256 e37efb7e2ededf6c1d7134c673f152f648f26e0267c556aebdf3ec65765d8b4c.
This is the builder's rationale, not an independent acceptance. A separate
critique must check the live enumeration and source hashes and supply its own
argument-review sidecar. No offer, review or publication is performed here.
## Storage spelling
The API rejected the two-letter name `nu` before storing any entries. Its
polynomial grammar permits only one Latin letter, optionally followed by
digits. The attached generator therefore uses `v` as the serialization name
for the same order $\nu$, and the table explains $v=\nu$ explicitly. This is
a bijective variable rename, not a substitution by $4\nu^2$, a numerical
sample, a parameter change or a renormalization. All mathematical formulas
and the column header retain $\nu$. This representation adjustment to the
proposal must be considered by the independent critique.