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# Argument selection for T588
The sole free parameter is the integer zero index n >= 1, classified as
`discrete_index`. It identifies the nth strictly positive real zero of
ker_0(x) = Re K_0(x exp(pi i/4)), with the principal branch of K and the
unscaled dimensionless variable x. The default enumeration is every n from
1 through 20 in increasing order. No zeros are omitted in that prefix.
The order is fixed at zero; it is not a hidden selection of real orders.
These zeros are intrinsic to the function, rather than samples of function
values on a continuous mesh. Order zero is the radial solution with no
angular variation. Its differential equation is DLMF 10.61.3 with nu = 0.
The identification purpose is to recognize nodal locations of this decaying
radial solution, including early and later oscillations. This does not claim
that each of the twenty nodes has a separately documented application.
The cutoff twenty is a noncanonical reference depth. It includes more than
the first few nodes while permitting complete root counting and independent
high precision checks at modest cost. DLMF 10.75(xii) explicitly lists a
reference covering twenty roots (Zhang and Jin, p. 323, eight decimals).
That inventory was read; the underlying book page was not read and is not
used as numerical evidence. The first five controls were instead read
visually from Abramowitz and Stegun, p. 381.
Breadth is provided by distinct ber, bei, ker and kei families in issue 231;
this table keeps only the one named object ker_0. Higher orders and derivative
zeros change scope and are deferred, rather than deepening this component to
hundreds of roots. There are no continuous input groups, parameter couplings,
probabilities or Cartesian products. `reference_sample` is consequently not
the category of this root index: the finite depth is a reference choice, but
the arguments enumerate a genuinely discrete family.
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, equations 10.68.2 and 10.68.4, phase representation.
- https://dlmf.nist.gov/10.75#xii, twenty-root reference coverage.
- https://personal.math.ubc.ca/~cbm/aands/page_381.htm, scanned zero table.
The archived instruction manifest is
agents/runs/20261028T190129Z-build-skills/manifest.json.
Both archived versions have SHA-256
e37efb7e2ededf6c1d7134c673f152f648f26e0267c556aebdf3ec65765d8b4c.
This is the builder's justification; independent critique must assess it.