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4942 bytes, as of the version from 2026-10-27 05:42. Recorded here, not run.
# T539 numerical build evidence
Policy: both archived skill versions match the SHA-256 recorded in
`agents/runs/20261027T052752Z-build-skills/manifest.json`:
`e37efb7e2ededf6c1d7134c673f152f648f26e0267c556aebdf3ec65765d8b4c`.
The build read these versions, not a remembered earlier policy.
## Definition and selected values
The one quantity is exp[-Z_N^-'(0)] for the half-line operator
-d²/dx²+x^N with the Dirichlet boundary at 0, decay at infinity and unit
coefficients. The shift is exactly zero. The one parameter is the integer
degree N, selected as every 1 <= N <= 12. See argument-selection.md for the
separate relevance and breadth assessment. Voros, arXiv:math-ph/0005029v2,
equations (47), (135) and (136), fixes the normalization and Gamma evaluation.
The source PDF is preserved as voros-0005029.pdf in the run artifacts.
## Computation and bounds
The generator computes Gamma(nu)*nu^(1/2-nu)/sqrt(pi), nu=1/(N+2), with Sage
Arb balls and exact rational inputs. 100 significant digits use 429 bits
including a stated 96-bit guard. The worst retained relative accuracy is
424 bits. A second run requests 200 digits (761 working bits) for every row.
All values are finite, positive and compatible. No exact rational member was
established; no string of zeros was used to infer exactness.
The independent checker evaluates the decaying zero-energy solution at x=1
using Bessel K and its derivative and propagates it to x=0 with a Wronskian
against b(0)=0,b'(0)=1. The latter solution is summed in exact rational
arithmetic; twice the first omitted term bounds the positive tails of both
b(1) and b'(1). The checker docstring derives the recurrence and bound.
At 620 bits the resulting enclosures retain at least 550 bits, with 37–48
included series terms. It uses neither the generator's Gamma evaluation nor
a finite eigenvalue-product approximation. It shares the source's theorem
relating the normalized decaying solution to the spectral determinant.
The elementary control N=0, used solely to test the checker, has
psi(x)=exp(-x), b(x)=sinh(x), and W(psi,b)=1. Its enclosure contains 1 and
excludes 0 and 2. This is not a selected table member or a claim of a discrete
spectrum for the constant potential. The degree-one check uses
2*sqrt(pi)*Ai(0). The harmonic check uses sqrt(pi)/Gamma(3/4), and separately
4^(-s)*zeta(s,3/4), differentiated with mpmath's explicit derivative argument
at 150 and 200 decimal digits. The latter agrees to 110 printed digits but
is supporting agreement evidence, not the source of the proven enclosures.
All twelve deliberately shifted values, offset by 10^(-90), fail the
independent overlap check. This verifies that the comparison is sensitive to
errors smaller than ordinary floating-point precision. Every 100-digit
serialized decimal interval contains the corresponding independent enclosure.
The decimal lengths are all 101 characters; the measured entries block is
about 1.61 KiB before entry annotations. Raw 429-bit balls occupy about
2.17 KiB, with maximum length 146 characters. Size is not the reason for the
degree cutoff. Package versions: numberdb 0.1.10, Sage 10.9, Python 3.12.3,
mpmath 1.4.1. Exact per-row values and metrics are in preflight-evidence.json.
## Explained disagreement
Default central `mp.diff` applied to the harmonic Hurwitz-zeta expression
failed, giving a determinant about 2.3e-11 too large at 150 digits. This was
investigated rather than silently discarded. The rational Hurwitz reflection
branch for negative infinitesimal s forms cancelling terms near its pole at
1; central finite differences amplify the lost precision. Right-sided
finite differences agree at 150 and 200 digits; left-sided differences have
twice the central error. The explicit derivative argument uses the
Euler–Maclaurin derivative computation and agrees with Arb. The diagnostics
and source inspection are retained in debug-harmonic.log,
debug-derivative-2.log and debug-reflection.log. The lesson proposal is
20261027T052752Z-build-lessons.md. Earlier preflight logs also preserve the
resolved Sage API mismatches. No failed computation supplied stored values.
## Read-back and audit
The same checks passed on all entries fetched from the website, including
full generator.verify(sample=None), every independent enclosure and both
special controls. The final postfill-evidence.json and audit.json record those
outcomes. The subsequent prose-only change preserved every number exactly; the final audit has no findings. No independent critique has been performed by this build actor, and
no publication or review-offer action is part of this run.
The attached Programs snippet was executed for all twelve degrees before filling and agreed with the generator. The anonymous API cannot read the table, while the authenticated read returns twelve entries. The draft was created with ready_for_review=false, and this actor called no offer or publication endpoint.