# Sources checked in the T533 build

Instruction manifest: agents/runs/20261026T235354Z-build-skills/manifest.json.
Both archived versions and the newly fetched local/live copies have SHA-256
b4d51ded0d52cfe3dcc0859de0845c3879b0c2363273ba7e809a7e06ded40b18.
The local and live instruction texts are identical.

The following sources were read through the web tool. A direct urllib
attempt to archive DLMF HTML returned HTTP 403. The pages were accessible
through the web tool, so this did not block research; no missing source is
being represented as read.

- DLMF 10.21.55 defines sigma_n(nu) with exponent -2n and n>=1.
- DLMF 10.21.15 gives the product over positive roots for nu>=0.
- DLMF 10.2.2 gives the Bessel series used in the generator.
- DLMF 10.73(i) and (ii) describe cylindrical and spherical separation;
  10.47.3 gives the spherical-to-cylindrical order shift.
- DLMF 10.16.1 gives J_(1/2) in terms of sin and J_(-1/2) in terms of cos.
- DLMF 25.6.2 gives the even-zeta/Bernoulli identity.
- DLMF 10.21(iv) discusses monotonicity of positive roots with order. Together
  with the cosine zeros this supplies the lower bound used for the
  supplementary tail estimate. The root calculation remains numerical,
  not an indexed-root certificate.
- Gupta and Muldoon, arXiv:math/9910128, PDF pages 1-2, equations (1)-(4),
  independently states the definition, Bessel series, convolution recurrence
  and sigma_1. Full web extraction is retained in
  gupta-muldoon-web-evidence.txt. Identifiers are fields in the table reference.

The exact product and series give the mathematical proof of rationality.
The distinct Fraction recurrence verifies all values independently of the
coefficient extraction. Bernoulli values and even-zeta evaluations are
checks outside the Bessel family. Numerical sums over only 32 roots are
not asserted to furnish 100-digit values, even though the table is exact.

The corpus checks and corrected API lookups are recorded in screening.json
and screening-corrected.json. Initial raw API experiments used the wrong
wire format; they are not treated as negative search results. The successful
client search uses api/lookup with an encoded exact rational. Returned tables
were inspected and no Bessel Rayleigh family was found.

All 73 distinct live tags were checked across the paginated tag list. The
existing special values, special functions, analysis and physics tags fit.
The absent spectral zeta tag would serve T531 Airy zeta values, T532 Airy
derivative zeta values, T533 Bessel Rayleigh sums, and the queued Rayleigh
sums of Bessel derivatives. This is a tag proposal, not a tag created here.