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# Numerical evidence for T542
Run: 20261027T075514Z-build. This is build evidence, not an independent critique.
The coordinated policy manifest is
`agents/runs/20261027T075514Z-build-skills/manifest.json`, attached as
`skill-manifest.json`. Both archived versions were read and compared, and the
source snapshot canonical copy has the same SHA-256:
`e37efb7e2ededf6c1d7134c673f152f648f26e0267c556aebdf3ec65765d8b4c`.
## Definition and selection
The operator is -d²/dx²+|x|^N, with both coefficients one, zero shift, and
full-line levels starting at the even ground state. Each value is the Neumann
inverse-power trace minus the Dirichlet trace. The 22 argument pairs are
(N,2) for 1<=N<=12 and (N,1) for 3<=N<=12. There is no spectral cutoff,
finite-part value, radial-coupling grid, or absolute value of the output.
`argument-selection.md` assesses both free parameters and all fixed conventions.
## Primary computation
Watkins (2012), DOI 10.1142/S140292511250026X, (2.5a,b), with sigma=2/(N+2)
and radial lambda=-1/2 for Neumann and +1/2 for Dirichlet. The first formula
includes 1/4. Its omission from introductory (1.3) is a source discrepancy,
resolved here by (2.5a), the independently derived Voros first-moment formula,
and the spectral identities. It is not a choice between untested answers.
The scanned equations (2.5a,b), (2.10a,b), and Examples 2.1-2.2 were inspected.
At N=1 the answer is 1/kappa-kappa², with
kappa=3^(1/3) Gamma(2/3)/Gamma(1/3). At N=2 the answer is Catalan's constant.
The generalized hypergeometric series is summed through a finite prefix and
an asymptotic Gamma-ratio tail. The log-Gamma expansion uses Bernoulli
polynomials (DLMF 5.11.8), its exponential is expanded coefficient by
coefficient, and the tail powers are summed with Hurwitz zeta values.
The helper `accelerate.py` is reused from the website's T541 attachment,
SHA-256 `c0a1143175711f966663c5c9fef7bf06edffe104f612204b649844d6b9d4baf0`.
Its source was inspected. The two settings are:
- 150 decimal working digits, cutoff 160, 140 correction terms;
- 200 decimal working digits, cutoff 320, 180 correction terms.
Every one of the 22 results agreed to at least 128.957 significant digits.
`values.json` records both results, the two boundary traces, and the 100-digit
written values. No asymptotic remainder bound is claimed: the table's rigour
is **heuristic (agreement-checked)**. The agreement varies the tail cutoff and
truncation as well as floating-point precision.
The N=3 Neumann second-moment hypergeometric benchmark took approximately
2.17 seconds and 3.39 seconds at the respective settings. All 22 values took
116.86 seconds. The wrapper returned 143 after printing the complete result
and summary; the full dry run and separate checks subsequently completed
successfully. No incomplete result is used. Runs use agents/sage.sh only.
## Independent checks actually performed before filling
1. A fresh transcription from Voros (1983), Appendix C, (C.22), (C.28),
(C.33), evaluated at 190 digits with cutoff 240 and 160 corrections.
The scanned printed pages 327-329 were inspected, including the Gamma
factors. The parity second moment uses two different 5F4 series rather
than the generator's two boundary formulas. All 22 answers agree to at
least **157.166 significant digits**. This check shares the generic
asymptotic-tail helper, so it is independent formula checking rather
than a completely independent numerical algorithm.
2. The same alternate formulas with mpmath's own hyper() at 25 working
digits, without the shared tail helper. Every entry agreed to at least
**24.857 significant digits**. This is a lower-precision algorithm check,
not an independent certification of all 100 digits.
3. At N=1, numerical differentiation of the logarithm of the Airy determinant
ratio, independently of the primary Gamma expression. At N=2, the two
Hurwitz zeta arithmetic progressions of the harmonic spectrum. These are
part of both whole-table comparisons.
4. The first two quantum-Wronskian identities from Watkins (2.10a,b), for
every degree 3 through 12, with absolute residuals below 10^(-120).
5. The sextic parity second-moment Gamma closed form (Watkins Example 2.2),
and the cubic Neumann second-moment closed form (Example 2.1).
6. Before interpreting results: the nonzero control 2F1(1,1;3;1)=2 for the
tail algorithm (error below 10^(-150)); the same control for the separate
library evaluation; and the harmonic Catalan control. A deliberate
perturbation of 10^(-90) fails the 100-digit written-interval test.
7. The repository dry_run.py computed all 22 entries without sending any,
passed exactness and source-import checks, and found no exact-in-disguise
warning. Its longest returned representation was 154 characters and block
3.9 KB. Every stored decimal is 100 significant digits (102 characters,
including `0.`), well inside readability and storage limits.
`numerical-checks.json` preserves every alternate value, agreement count,
Wronskian residual and control outcome. The checked papers' SHA-256 values
are recorded there; copies and scanned-page images remain in the run work
artifacts. Sources: Voros (1983), Numdam AIHPA_1983__39_3_211_0; Voros (2022),
arXiv:2206.14482v3; Watkins (2012), arXiv:1110.2004; NIST DLMF 9.2 and 5.11.
## Reproduction
Download all attached files. With SageMath and numberdb/mpmath installed:
```
sage -python generate.py
sage -python checks.py
sage -python checks.py --library
sage -python checks.py --stored
```
Reading a private draft requires NUMBERDB_API_KEY. The generator's --fill
writes entries only and neither publishes nor offers the table. `checks.py`
compares the saved generator computation or the live written intervals with
fresh alternate formulas. A separate controller-managed critique must assess
both numerical evidence and arguments. The selection-relevant source files
are `generate.py` and `accelerate.py`; an independent critic should fingerprint
both, plus `argument-selection.md` as the selection rationale.
## Readback verification
The draft was filled once with 22 entries and all nine declared attachments.
Generator.verify(sample=None) matched 22/22, with no missing, extra, or
contradicting entries. checks.py --stored recomputed the alternate formulas
and found all 22 reference values inside the actual written rounding
intervals. It also reconstructed the boundary sectors from the stored full
trace T541 and stored parity trace T542: both quantum-Wronskian identities
contained zero in propagated finite intervals of diameter below 10^(-94),
for every N from 3 through 12. This does not certify the heuristic reference
values themselves. Versions: Python 3.12.3, Sage 10.9, mpmath 1.4.1,
numberdb 0.1.10. Detailed readback results are in numerical-checks.json.