# Argument selection for T587

The sole free parameter is the integer zero index n >= 1. It selects the nth
strictly positive zero, in increasing order, of kei_0(x) = Im K_0(x exp(i pi/4)),
with the principal branch of K_0. The stored quantity is its dimensionless
location x_n. The normalization is x, without a factor of sqrt(2).

Classification: discrete_index. These are intrinsic roots of one named
function, not values at a grid of freely chosen arguments. Order zero is fixed
as part of the named object. It also corresponds to the zero angular mode of
the radial equation x^2 w'' + x w' - i x^2 w = 0 (DLMF 10.61.3).
There is no free order, coupling, scale, or probability parameter.

The enumeration contains every n from 1 through 20, once, in increasing order.
Twenty is a noncanonical practical reference depth. It covers the first few
nodes and a further sequence of oscillations, permits full certified counting
from the origin, and keeps each written entry near 100 characters. It is not
chosen to fill a size quota, and no claim is made that every node has a separate
published application. DLMF 10.75(xii) documents a twenty-root table in Zhang
and Jin (1996), p. 323, as a reference-coverage precedent. That book page was
not directly consulted. The first five controls were read directly from
Abramowitz and Stegun, p. 381.

Breadth was assessed before depth: the family proposal separates the four
Kelvin components; this table covers one of them. Existing T509 holds function
values, not root locations. Higher angular orders and derivative zeros are
different extensions and are deferred. There is no continuous exploratory
reference sample or mixed group, so no reference_policy is applicable. The
parameter declaration gives n >= 1; the completeness note gives 1 <= n <= 20.

Checked sources:
- https://dlmf.nist.gov/10.61, equations 10.61.2 and 10.61.3 (definition and ODE).
- https://dlmf.nist.gov/10.68, equation 10.68.4 (phase interpretation).
- https://dlmf.nist.gov/10.75#xii (historical root-table coverage).
- https://personal.math.ubc.ca/~cbm/aands/page_381.htm (first five roots).

Policy provenance: agents/runs/20261028T182642Z-build-skills/manifest.json.
Both archived versions have SHA-256
 e37efb7e2ededf6c1d7134c673f152f648f26e0267c556aebdf3ec65765d8b4c.
This is the builder's justification, not the required independent critique.