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# Argument selection: Rayleigh sums of Bessel derivatives (T535)
The stored quantity is the sum of inverse even powers of the strictly positive
zeros of the conventional cylindrical Bessel derivative J'_nu. A zero at the
origin is excluded. Parameters are ordered `(nu, n)` with types `Q, Z` and
domains nu >= 0, n >= 1. The domain is larger than this initial selection.
## Bessel order nu: documented application
The selection is exactly nu = 0, 1, 2, 3. On the unit disk, separation of the
Laplacian into angular harmonics exp(i ell theta) gives the regular radial
factor J_ell(sqrt(lambda) r). Requiring its radial derivative to vanish at r=1
selects J'_ell(sqrt(lambda))=0. Thus nu=ell, a nonnegative angular harmonic
order, and lambda=(j'_(ell,k))^2. Each radial spectrum is counted once, with
no factor of two for the nonzero angular harmonics. The constant Neumann
mode is removed before taking inverse powers.
DLMF 10.73(i), especially the separated cylindrical solutions after 10.73.1,
supports the angular family. Applying the Neumann condition to that radial
factor gives the derivative equation directly. DLMF 10.21(i), (iii) and
10.21.1 give the root conventions, product and order-zero identity.
The bound ell<=3 is a noncanonical practical prefix of this application
family, covering the axisymmetric sector and the first three nonaxisymmetric
sectors. It is not a rational-denominator sample or a claim that larger modes
lack interest. These are four initial angular sectors, not four independently
famous arguments. No other rational orders are included. In particular,
half-integer J' orders would not be Neumann-ball spectra, since differentiating
the spherical radial prefactor changes the boundary equation. Their inclusion
would need its own justification, rather than importing the neighboring
Dirichlet table's half-integer slice.
## Moment order n: discrete index
The selection is every integer 1<=n<=12 for each selected angular sector.
These are traces of integer powers of the inverse radial operator, or
coefficients of the logarithm of its spectral determinant. The roots grow
linearly, so all these sums converge. The lower bound is the first convergent
integer moment; the upper bound is a noncanonical reference prefix. It covers
low-order sum rules and higher moments that emphasize the lowest eigenvalue.
Gupta and Muldoon, section 3 and equations (16)-(25), treats this family of
integer moments. DLMF 10.21.16 gives its canonical product.
## Breadth and finite extent
Four angular sectors and twelve moments give 48 entries. Both finite bounds
are editorial choices within meaningful families; the entry count is a result,
not a target. Keeping every moment through twelve within each sector permits
comparisons of the full prefix. The order-zero sector requires separate
normalization; all three positive sectors use the same derivative product.
We assess both, rather than allowing the distinguished order-zero identity to
justify unrelated rational orders. No continuous exploratory sample, decimal
mesh, or extra parameter product is present. Measurement of the longest exact
fraction is recorded in numerical-evidence.md before filling the draft.
The generator enumeration and table completeness note must match these bounds.
## Checked sources
- https://dlmf.nist.gov/10.73#i : cylindrical separation of variables.
- https://dlmf.nist.gov/10.21#i : strictly positive roots and interlacing;
equation 10.21.1 includes zero in DLMF's order-zero derivative indexing.
- https://dlmf.nist.gov/10.21.E16 : derivative product for nu>0.
- https://dlmf.nist.gov/10.2.E2 : conventional cylindrical Bessel series.
- https://arxiv.org/abs/math/9910128 : Gupta and Muldoon, section 3,
equations (16)-(22), (24)-(25). The general recurrence (21), rather than
the inconsistent shortened summation limits printed in (23), is used
in the independent checks.
## Policy provenance and independent review
Read against the coordinated archived local/live skills in
`agents/runs/20261027T024416Z-build-skills/manifest.json`.
Both SHA-256 values are
`e37efb7e2ededf6c1d7134c673f152f648f26e0267c556aebdf3ec65765d8b4c`.
This is the builder's justification, not an independent acceptance. A separate
critique actor must review both parameters and numerical evidence, and obtain
fresh live attachment and selection fingerprints with argument_review.py.