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# Argument selection for T590
The sole free parameter is n, an integer with n>=1. It indexes the strictly
positive zeros of bei_0 in increasing order. This is a discrete_index family;
the zeros are intrinsic nodes, not continuously chosen evaluation inputs.
There are no independently varying couplings, orders, or normalizations.
Order zero and the dimensionless argument x are fixed by the named object.
DLMF 10.61.1 defines bei_0=Im J_0(x exp(3*pi*i/4))=Im I_0(x exp(pi*i/4)).
The origin is excluded, so the exact zero there never consumes index 1.
DLMF 10.65.2 and Abramowitz-Stegun p. 381 independently settle that convention.
Include every n=1,...,20. This noncanonical finite reference depth covers
several oscillations past the first few nodes, while allowing exhaustive
root counting and independent endpoint certification at 100 significant
digits. DLMF 10.75(xii) lists a first-20-zero reference in Zhang and Jin,
p. 323. We inspected DLMF's inventory, not the Zhang-Jin book; the inventory
is a precedent for coverage, not evidence of an application of every root.
The A&S scan was directly viewed and supplies the first five numerical controls.
Breadth precedes depth: the family proposal separates ber, bei, ker and kei
into four named zero tables. This table completes the regular pair's component
coverage; the corpus already holds the three other proposed zero families
(T587, T588, T589). Extending one component to hundreds of roots would add
less breadth than this fourth family. Higher orders and derivative zeros
are different mathematical scopes, deferred rather than mixed into this table.
Twenty is a practical cutoff, not an assertion of completeness of an infinite
family or a storage target. The actual written lengths and block size are
recorded in numerical evidence. No exploratory reference_sample inputs,
probability levels, parameter products, or arbitrary decimal mesh are used.
The Bessel value and real-axis zero tables do not duplicate this family:
taking an imaginary component on a rotated ray and then solving for its
zero is not a small rescaling of a real-axis Bessel zero. The screening
artifact records current text, issue and value searches, with no duplicate.
Existing tags zero and special functions are used. Kelvin functions is absent;
a proposed tag could connect T490, T491, T493, T509 and T587 through T590.
Sources checked: https://dlmf.nist.gov/10.61 (10.61.1 and 10.61.3),
https://dlmf.nist.gov/10.65 (10.65.2),
https://dlmf.nist.gov/10.75#xii, and
https://personal.math.ubc.ca/~cbm/aands/page_381.htm (scan).
Policy manifest: ../20261028T202403Z-build-skills/manifest.json.
Both archived SHA-256 values:
e37efb7e2ededf6c1d7134c673f152f648f26e0267c556aebdf3ec65765d8b4c.
Generator enumeration, completeness note and this selection must remain consistent.
A separate critique actor must independently assess the selection and bind its
sidecar to the current document and all selection-relevant attached sources.