# Build handoff: T541

Draft: https://numberdb.org/T541
Title: Full spectral zeta values of homogeneous oscillators
Family: numberdb-data issue #218.

The draft was created and filled in this run. It contains 22 entries at 100
significant digits: (N,2) for N=1,...,12, and (N,1) for N=3,...,12. It is
unpublished, and this actor performed no offer or review action. The controller
must run the separate critique and any repair before offering it. No independent
argument-review acceptance sidecar was written by this build.

## Instructions and argument selection

Coordinated manifest:
../20261027T064642Z-build-skills/manifest.json, or the attached
skill-manifest.json. Both valid archived versions were read and match SHA-256
e37efb7e2ededf6c1d7134c673f152f648f26e0267c556aebdf3ec65765d8b4c.

The argument-selection.md attachment assesses both N and n as discrete_index.
It justifies the practical degree prefix, the first-two-moments slice, the
convergence omissions, the absolute-power potential, the unit coefficients,
the zero shift, the parity convention and multiplicities. There are no sampled
continuous couplings or reference grids. The enumeration and completeness note
agree. The independent critic should obtain a fresh fingerprint using at least
generate.py, accelerate.py, checks.py and check.py; none imports an additional
local computational helper. values.json is output evidence, not generator input.
selection-fingerprint.json records the final build fingerprint, not acceptance.

## What was checked

- Current corpus and request screening, sample numerical lookups, and live tags.
  No matching published family was found. The harmonic value also occurs as a
  Favard constant, which is a different construction. Airy component traces
  and oscillator eigenvalues are linked using verified addresses.
- All 22 values at 150/200 working digits with distinct tail cutoffs and
  correction counts: minimum agreement 129.4168 significant digits.
- Separate full-trace formulas: minimum agreement 156.6024 digits.
- Independent mpmath hypergeometric evaluation at 25 working digits: minimum
  agreement 24.4616 digits. This evaluator does not call the custom accelerator.
- Known nonzero control, deliberate perturbation rejection, both first- and
  second-order Wronskian identities across all N=3,...,12, Airy determinant
  derivatives, and the harmonic odd-integer spectrum.
- The unmodified repository dry_run.py passed: 22 entries, no exactness or
  disguised-exact-value findings, maximum internal length 153 characters,
  approximately 3.92 KiB. Stored numbers are at most 102 characters.
- The displayed mpmath program was executed and compared with the generator.
- Generator.verify(sample=None): 22/22 matched, no missing or extra entries.
- All 22 values reread from the website: their last-place rounding intervals
  contain the separate high-precision full-trace computations.
- Website audit: clean, findings=[]. The first audit requested a dependency
  installation command; that was added to the generator docstring and the
  audit rerun. No mathematical source was changed in that fix.
- Anonymous API read cannot see the draft. All nine principal attachments
  were downloaded and their SHA-256 hashes matched the local files.

## Decisions and limits for the critic

The proposal's numerical viability was unmeasured. A direct 150-digit
hypergeometric attempt ended without a value (exit 137); the Gamma-ratio
asymptotic-tail method made the selection practical. No eigenvalue cutoff was
used to approximate the infinite trace. The rigour remains heuristic
(agreement-checked), because the asymptotic remainder is not rigorously enclosed.
The alternative high-precision formula shares the generic tail helper; the
25-digit library check supplies implementation independence at lower precision.

Watkins's introduction (1.3) omits the factor 1/4 present in (2.5a). We use
(2.5a), as independently confirmed by Voros's full-trace formula. This is a
resolved source inconsistency, not a choice of a new operator normalization.

An initial 170-digit Wronskian tolerance exceeded the measured numerical
accuracy. Diagnostics at higher working precision recovered more digits and
agreed with the cubic Neumann Gamma-only value. The acceptance tolerance is
10^(-120), with a margin beyond the 100 stored digits. See numerical-evidence.md
and retained tail-diagnosis.log, rather than inferring certified 200-digit
accuracy from a working-precision setting.

Existing tags used: special values, analysis, physics, special functions.
The live tag list has no spectral zeta tag. The proposed tag would apply to
at least T531 (Airy zeta values), T532 (Airy derivative zeta values), T539 and
T540 (homogeneous-oscillator determinants), and T541, as well as the queued
parity-twisted trace table. No new tag was created here.

## Artifacts

The work directory retains source PDFs and equation images, screening results,
raw values, all successful and failed diagnostic logs, the dry-run log,
fill/verify log, stored-check results, audit results, attachment hashes,
argument-selection rationale, numerical evidence, and a proposed lesson.
The website has the reproducible generator, generic accelerator, numerical
checks and unchanged common check helper, together with the rationale and
numerical evidence. Post-write evidence is attached separately.

This source snapshot was not committed or pushed. No pipeline source, prompt,
skill, published table, controller state or completion receipt was changed.
The sole controller handoff is the requested build.table.json naming T541.
The proposal queue is left for the controller's successful critique/repair
lifecycle; this draft has not been represented as a published answer.