# Bessel Rayleigh sums, T533: argument selection

The quantity is sigma_n(nu) = sum_{k>=1} j_(nu,k)^(-2n), using the
strictly positive zeros of conventional cylindrical J_nu. It is one radial
spectral zeta function, without angular degeneracy weights. The eigenvalue
is the square of the wave number, not the wave number itself.

## Bessel order nu (type Q): documented application

The selected orders are derived from separation of the Dirichlet Laplacian
on the unit disk (d=2) and unit three-dimensional ball (d=3). For angular
order ell, the regular radial solution has the form
r^(-(d-2)/2) J_(ell+(d-2)/2)(sqrt(lambda)*r). At r=1 the Dirichlet condition
therefore makes lambda=j_(nu,k)^2, with nu=ell+(d-2)/2. There is no adjustable
coupling or other continuous parameter.

DLMF 10.73(i) derives integer cylindrical angular modes e^(+/- i ell phi).
DLMF 10.73(ii), equation 10.73.4 and its following paragraph, describes
spherical harmonics and spherical Bessel radial solutions. DLMF 10.47.3
identifies spherical j_ell with sqrt(pi/(2z))*J_(ell+1/2). These sources,
combined with the stated unit Dirichlet boundary, select exactly the integer
and positive half-integer order families. The derivation of the disk/ball
boundary equation here is an inference from those separated solutions.

This table takes ell=0,1,2,3 in each dimension: the first four angular
sectors, giving nu=0,1/2,1,3/2,2,5/2,3,7/2 in increasing order. The cutoff
ell=3 is an editorial reference prefix, not a distinguished endpoint or a
claim that a source selects precisely four sectors. The scalar rational
values do not justify these orders; their spectral application does.

Unit radius fixes scale: at radius R the eigenvalues become j_(nu,k)^2/R^2
and the inverse-power trace is multiplied by R^(2n). No radius parameter is
stored. Angular multiplicities (two for nonzero disk angular orders and
2*ell+1 on the ball) are not included. Positive roots alone are inverted.
The mathematical parameter domain is rational nu>=0; other rational orders
are not sampled just because the computation allows them.

Sources:
- https://dlmf.nist.gov/10.73#i
- https://dlmf.nist.gov/10.73#ii
- https://dlmf.nist.gov/10.47.E3

## Moment order n (type Z): intrinsic discrete index

The integer n indexes powers of the inverse radial operator, equivalently
coefficients of its logarithmic spectral determinant. DLMF 10.21.55 defines
this Rayleigh function with exponent -2n. The product in 10.21.15 yields
-log(F_nu(t)) = sum_{n>=1} sigma_n(nu)*t^n/n near t=0. The roots grow linearly
with k, so all n>=1 converge. This is an operator moment order, not a
continuous argument disguised as an index.

Every n=1,...,12 is included for each selected nu. Twelve is an editorial
prefix for the moment sequence, as proposed by family #216; it is not a
mathematical upper bound. No entries are suppressed for being small rational
numbers. The finite selection has 96 entries and is ordered by nu, then n.

Sources:
- https://dlmf.nist.gov/10.21.E55
- https://dlmf.nist.gov/10.21.E15
- Gupta and Muldoon, arXiv:math/9910128, equations (1)-(4).

## Instructions and review boundary

The archived instruction manifest is
agents/runs/20261026T235354Z-build-skills/manifest.json.
Both versions in that manifest, the canonical local skill, and the live skill
fetched in this run have SHA-256
b4d51ded0d52cfe3dcc0859de0845c3879b0c2363273ba7e809a7e06ded40b18.
The full records are in skill-read-evidence.json. This is the builder's
justification, not an independent argument-selection acceptance. The separate
critic must inspect both parameters, the enumeration, completeness, and
numerical evidence. generate.py is the only source controlling the selection
and the produced values; it imports no local helper. checks.py and check.py
are validation code, not generator dependencies.