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6307 bytes, as of the version from 2026-10-27 07:39 (current). Recorded here, not run.
# Numerical evidence for T541
## Coordinated instructions and convention
Manifest: agents/runs/20261027T064642Z-build-skills/manifest.json (also attached
as skill-manifest.json). Both archived local/live skills were read, matched,
and have SHA-256 e37efb7e2ededf6c1d7134c673f152f648f26e0267c556aebdf3ec65765d8b4c.
This build uses the integer-degree family and normalization of issue #218;
the argument-selection assessment is in argument-selection.md. An independent
critique, not this build, must decide its acceptance.
The operator is -d²/dx²+|x|^N with unit coefficients, zero shift and all
full-line eigenvalues counted once. Even/odd full-line levels give Neumann/
Dirichlet sectors. N=1 and N=2 are separate Airy and harmonic controls.
Watkins (arXiv:1110.2004v2), equation (2.5a), includes a factor 1/4 omitted
from the introductory equation (1.3). We use (2.5a). Its first moments agree
with the independent Voros (2000), equation (140), and Voros (1983), (C.22)
and (C.23), to at least 189 decimal digits. Using the introductory prefactor
would give four times the correct trace. This resolves the source discrepancy
without guessing a different spectral scale.
## Computation and accuracy
The generator sums the two half-line evaluations of Watkins (2.5a,b), with
sigma=2/(N+2) and lambda=+/-1/2. The second formula is a 5F4 series at one.
Each term is a Gamma ratio; DLMF 5.11.8 expands its logarithm using Bernoulli
polynomials. Exponentiating that formal expansion supplies inverse-power
coefficients. Their tails are summed with Hurwitz zeta values.
Two settings are compared for every entry:
- 150 decimal working digits, 160 explicit series terms, 140 tail corrections;
- 200 decimal working digits, 320 explicit terms, 180 tail corrections.
All 22 entries agree to at least 129.4168 significant digits. They are stored
at 100 significant digits. numberdb.agreeing returns the union of the two
computed intervals; the rigour is heuristic (agreement-checked), since neither
an asymptotic-tail remainder nor the mpmath errors are certified. An increase
in precision alone is not the error claim: cutoffs and correction counts also
change, and an independent formulation is checked.
The 22-value computation took about 105 seconds in the provided Sage wrapper.
A preliminary mpmath.hyper attempt at 150 digits terminated with exit 137;
it supplied no result and was not used. A 25-digit call is practical.
## Checks actually performed
1. Before using the tail method, 2F1(1,1;3;1)=2 is checked as a known nonzero
control. A deliberate perturbation of 10^(-90) is rejected by the comparison.
2. Every entry is compared with a separate full-trace formula. The first
moments use Voros (2000), equation (140); second moments use Voros (1983),
equation (C.28), with different 5F4 parameters from Watkins. This check uses
190 working digits, cutoff 240 and 160 tail corrections. The weakest
agreement is 156.6024 digits. This shares the generic tail summation helper,
but not the trace formula or boundary-sector construction.
3. The same independent full-trace formulas are evaluated using mpmath's own
hypergeometric implementation at 25 working digits, with no call to the
custom summation helper for the tested hypergeometric value. Every entry
agrees to at least 24.4616 digits. This supplies an independent summation
implementation as well as an independent formulation.
4. The linear case is checked by differentiating the logarithm of the product
of the Airy and Airy-derivative determinants. The harmonic case is checked
with its odd-integer Hurwitz-zeta series.
5. Both first- and second-order quantum-Wronskian identities in Watkins
(2.10a,b) are checked for every N from 3 through 12. The tolerances are
10^(-120), stricter than the 100 digits stored. Values are finite, positive,
and the generator enumerates exactly the intended 22 distinct pairs.
6. The common helpers exactness, measure, names_its_rings, and agrees_with
are used. The measured internal interval strings have maximum length 153
characters and total block size about 3.92 KiB, well inside display limits.
The first Wronskian test demanded 170 digits and failed at N=3,n=2 with a
residual about 2.36e-167. That was beyond the measured accuracy of the
200-digit numerical evaluation and beyond the intended 100-digit output.
A diagnostic at 220 working digits compared the cubic Neumann value against
the Gamma-only closed form in Watkins Example 2.1. Relative errors for
(cutoff,corrections)=(160,140),(240,160),(320,180),(640,220) were approximately
1.06e-173, -1.88e-177, 7.06e-180, and -3.80e-185. This isolates numerical
precision/truncation effects, rather than a different normalization or a
failed Wronskian identity. The acceptance test is deliberately tied to the
100-digit claim with a 20-digit margin; it makes no 170-digit claim.
The attached checks also support --stored: they reread the live draft, parse
each last-place rounding interval, and require it to contain the independent
full-trace evaluation. This is separate from Generator.verify(), which compares
against the originating generator. The build handoff records the outcomes of
those post-write checks and the website audit.
## Reproduction and versions
Run generate.py and checks.py with SageMath, numberdb, and mpmath, downloading
all Python attachments alongside them. A private draft requires its author's
API credential for reading; none is included in these files. checks.py
--library runs the independent library calculation and checks.py --stored
checks all stored rounding intervals. Generator.verify(sample=None) checks
all entries against the generator. The runner also uses the repository's
unaltered agents/table-build/dry_run.py before submission.
Measured versions: Python 3.12.3, SageMath 10.9, numberdb 0.1.10, mpmath 1.4.1.
Raw 150/200-digit computations are in values.json, and per-entry check results
are in numerical-checks.json. The artifact directory retains the source PDFs,
rendered equation pages, failed benchmark, and diagnostic logs as well.
Sources checked: https://arxiv.org/abs/1110.2004,
https://arxiv.org/abs/math-ph/0005029,
https://www.numdam.org/item/AIHPA_1983__39_3_211_0/,
https://dlmf.nist.gov/5.11#E8, and https://dlmf.nist.gov/9.2.