# Build handoff: T531, Airy zeta values

Built https://numberdb.org/T531 (slug `Airy_zeta_values`) from proposal
“Airy zeta values” in numberdb-data family #216. The draft is populated and
private. No offer, review or publication action was taken. The controller
must run the separate independent critique and any repair before offering it.
No argument-review acceptance sidecar was written by this build.

## What is stored

Nineteen positive-spectrum Airy zeta values, one for every integer order
2 through 20, at 100 significant digits and `rigour: proven`.
The parameter domain remains all integers n>=2. The normalization is
H=-d^2/dx^2+x on [0,infinity), with a Dirichlet boundary condition at zero
and decay at infinity; every eigenvalue is counted once.
The values sum inverse powers of -a_k, not powers of negative a_k.

The family conventions were retained. The upper endpoint 20 is explicitly
an editorial prefix, not an intrinsically special order. No continuous
parameter or sampling grid was introduced. `argument-selection.md` gives
the mathematical and bibliographic justification for the only parameter n.
The unrelated inverse-Ai-prime spectrum was not combined with this table.

## Corpus and source checks

The required `already_here` and `already_asked` checks were run and preserved
in `screening.json`. Airy, spectral, Rayleigh and Gamma searches were read.
Three valid `/api/lookup?text=...` queries at orders 2, 3 and 4 gave no matches;
see `lookups.json`. Related Airy and Gamma documents were downloaded to read
their actual addresses and conventions. Two paginated tag pages yielded 73
existing tags. All used tags existed.

The proposed tag **spectral zeta** is absent. It would serve all four members
of family #216: Airy zeta values, Airy derivative zeta values, Bessel Rayleigh
sums and Rayleigh sums of Bessel derivatives. It is proposed for the family
rather than created for this lone draft.

Primary sources read: Crandall (1996), Section 4, including equations
4.12-4.22; Voros arXiv:math-ph/9811001v2, equations 5, 9, 29, 31, 42.2 and
43.3; DLMF 9.2 and 9.9. The source PDFs are retained. Crandall's low-order
quantum-bouncer formulas use eigenvalues (-a_k)/2, so their values were
divided by 2^n. The final document explains this scale conversion and the
negative-root sign convention. There is no GitHub request provenance in the
mathematical prose. The controller owns the queue's new-build bookkeeping.

## Numerical validation and database checks

- `dry_run.py` passes all 19 values; no float coefficients or suspicious
  exact-in-disguise entries were found.
- 429 working bits (96 guard bits), with minimum measured relative accuracy
  402 bits and maximum relative radius 4.2619331e-122.
- All entries agree with a 629-bit rerun and an independently implemented
  700-bit Airy Taylor quotient. Every stored decimal interval strictly
  contains the corresponding independent high-precision ball.
- Published low-order identities for n=2,...,6 all agree, using an alternate
  Gamma expression for the initial constant.
- A separate sum over 64 Airy zeros with explicit tail bounds passes for
  every order. Its known-answer nonzero control passed. Its root indexing
  is numerical, so it is a sanity check, not the 100-digit certification.
- Positivity, decrease, log convexity and all applicable recurrence identities
  were tested again on the values read from T531.
- `verify(sample=None)` returned 19/19 matched, with zero differing,
  missing or extra entries.
- The Programs snippet was executed and all nineteen results agreed.
- Final `api/table/T531/audit`: `findings: []`, `clean: true`.
- An unauthenticated table read does not expose T531.

The reproducible generator, checks, shared checking helper, argument-selection
note, numerical evidence and skill manifest are attached to T531; their live
bytes were checked against the local files. `attachment-hashes.json` records
them. `stored-numerical-evidence.json` records the successful independent
read-back run; `stored-checks-output.txt` contains its output.

## Instructions and next stage

Read and cited instruction manifest:
`agents/runs/20261026T224836Z-build-skills/manifest.json`.
The archived local and live versions, the canonical source and a fresh live
fetch all have SHA-256
`b4d51ded0d52cfe3dcc0859de0845c3879b0c2363273ba7e809a7e06ded40b18`.
The per-file read record is `skills-read.json`; the manifest is also attached
as `skill-manifest.json`.

`selection-fingerprint.json` records a builder fingerprint, not an acceptance.
The critic must independently assess n and obtain its own current fingerprint.
The generator's entire computation and enumeration live in `generate.py`;
there are no imported local generator helpers. The recorded fingerprint also
covers `checks.py`, `check.py` and `argument-selection.md` for reproducibility.
The critic must use its own NUMBERDB_RUN_ID as reviewer_run and write the
required sidecar beside its own report. No changes to sources or selections
have been made after the final fingerprint.

Minor deployment and checker findings are preserved in
`environment-findings.md` and `lessons-proposed.md`. No pipeline source,
skill, Git repository or supervisor state was changed.