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5002 bytes, as of the version from 2026-10-27 01:18 (current). Recorded here, not run.
# T533 build handoff
Draft: https://numberdb.org/T533 (Bessel Rayleigh sums).
Family: numberdb-data#216. Builder run: 20261026T235354Z.
The draft has 96 exact rational entries and remains private, unoffered and
unpublished. The controller must run the separate critique and repair stages,
including independent argument-selection review, before any offer.
No independent critique sidecar was created by this build. The controller
will mark the proposal queue after its stages succeed.
## Selection and convention
Ordered parameters are (nu,n), with full domains rational nu>=0 and integer
n>=1. The finite selection is nu=0,1/2,1,3/2,2,5/2,3,7/2 and n=1,...,12,
ordered first by nu, then n. The order family derives from the first four
angular sectors of the unit disk and three-dimensional ball. The quantity is
sum j_(nu,k)^(-2n), using strictly positive cylindrical J_nu roots, unit radius,
and no angular multiplicities. argument-selection.md records the derivation,
checked sources and editorial prefix bounds. The proposal's conventions were
retained; no unresolved convention was guessed.
## Checks actually performed
- Repository dry_run.py: 96 exact entries, no prose or disguised-decimal
warnings; longest written fraction 52 characters; measured block 5.5 KiB.
- Separate Fraction implementation of Kishore's convolution recurrence:
96 matches. The generator instead extracts logarithmic-derivative
coefficients from the normalized Bessel Taylor series.
- First three moment formulas: 24 exact matches.
- Sine-spectrum specialization at nu=1/2: 12 exact Bernoulli matches and
12 matches to zeta(2n)/pi^(2n) evaluated at 120 decimal digits.
- Positivity: 96; strict decrease: 88; strict log convexity: 80.
- At 80 decimal digits, 32 independently computed Bessel roots per order
give partial sums compatible with all 96 values and an explicit bound for
the infinite tail. Known half-order roots k*pi and sum 1/6 are controls.
Perturbed-value controls reject incorrect answers.
- Read-back verify(sample=None): 96/96 matched, no differing, missing or
extra entries. Every independent check was repeated on stored values.
- The displayed Programs snippet was separately executed on all 96 entries.
- Site audit: clean, no findings. Two initial cross-reference findings and
a subsequent first-mention link finding were corrected in prose-only
revisions. All 96 fractions and their order were confirmed unchanged.
- Anonymous API access cannot read the table; the bot can read all entries.
The numerical root indices are not certified, and the 32-root sums are not
100-digit certificates. Exact rational coefficient arithmetic plus the Bessel
canonical product establish the stated rigour. No decimal precision claim
is made on the stored fractions. The numbers were filled once. The final
state has four document revisions: claim, fill and two audit prose fixes.
## Instruction and selection evidence
Archived manifest: agents/runs/20261026T235354Z-build-skills/manifest.json.
Manifest SHA-256: d03b756eaa1cd22ce39c8cee78fb3a44b052146786d6371925ac7bc9dec564f3.
Both archived skills, the canonical local skill and the freshly fetched live
skill have SHA-256
b4d51ded0d52cfe3dcc0859de0845c3879b0c2363273ba7e809a7e06ded40b18.
The manifest is attached as skill-manifest.json, and skill-read-evidence.json
records every version read. These exact instructions governed the selection.
Final selection SHA-256: 658f5827b3ca6c6b0041b2b96035642127e360092eb71216a145bd89c76a203d.
Read from the database with:
python3 agents/argument_review.py fingerprint T533 --generator generate.py --generator argument-selection.md --generator checks.py --generator check.py
This fingerprint is not an acceptance. generate.py alone controls selection
and values. All four implementation and justification attachments are included
for a conservative review boundary. The critic must obtain a fresh fingerprint
and use its own NUMBERDB_RUN_ID as reviewer_run.
## Artifacts and remaining review
Attached reproducibility files include generate.py, argument-selection.md,
checks.py, check.py, dry_run.py, dry-run.log, checks-generated.json,
checks-stored-clean.json, skill-manifest.json, skill-read-evidence.json,
source-notes.md, audit-final.json, selection-fingerprint-final.json,
numerical-evidence.json and handoff.md. Raw logs, API searches, initial audits
and document versions remain in this run's build-work directory.
The lesson artifact 20261026T235354Z-build-lesson.md records missing Sage
rational-coercion machinery and the exact integer-parsing remedy. No skill,
pipeline source, or published table was changed; no Git, deployment, offer,
publication or human-review actions were performed.
The absent spectral zeta tag is proposed for T531 Airy zeta values, T532 Airy
derivative zeta values, T533 Bessel Rayleigh sums, and the queued Bessel
derivative Rayleigh table. Existing special values, special functions,
analysis and physics tags were used. No tag was created.