# T540 build numerical evidence

Policy manifest: `agents/runs/20261027T061240Z-build-skills/manifest.json`.
Both archived skills and the snapshot canonical skill have SHA-256
`e37efb7e2ededf6c1d7134c673f152f648f26e0267c556aebdf3ec65765d8b4c`.
This is build evidence, not an independent critique or an offer for review.

The source is Voros, *Exercises in exact quantization*, corrected preprint
math-ph/0005029v2. Equations (7), (47), (125), (135), (136), (148), (160)
fix the zeta convention, boundary condition, potential scale, Bessel
reconstruction, Gamma formula, Airy control and harmonic control respectively.
The later survey 2206.14482v3, sections 1.1 and 3.1, independently confirms
integer degrees including odd absolute-power wells and parity indexing.
Downloaded PDFs and their hashes are retained in the run work directory.
DLMF 25.11.13 and 25.11.18 give the harmonic Hurwitz-zeta values at zero.

The generator evaluates all N=1,...,12 as real balls at 429 bits, with at
least 425 measured relative bits. A 200-digit recomputation agrees for every
entry. `checks.py` reconstructs -psi'(0) by a constant Wronskian at x=1,
using Bessel K and a rational Frobenius series with geometrically bounded
positive tails. At 720 bits and 100 terms, its smallest measured relative
accuracy is 708 bits. This is independent of evaluating the Gamma formula,
although both analytic representations come from the same source.

The control uses N=0 solely in the verifier: psi=exp(-x), F=cosh(x), giving
exactly 1. Its enclosure contains 1 and excludes 1+10^(-90). Every selected
degree agrees with the reconstruction and rejects a 10^(-90) perturbation.
The Airy derivative at zero checks N=1. The harmonic Gamma identity and
spectral-zeta derivative check N=2. The Dirichlet product identity is tested
for all twelve degrees. The stored-data mode checks exact decimal endpoints:
each stored rounding interval must contain the independent enclosure, and
each stored Neumann/Dirichlet product interval must contain the sine identity.
A report of the stored-data mode is retained separately after filling.

A discrepancy in the extra mpmath check was investigated before acceptance.
In mpmath 1.4.1, default numerical differentiation of
4^(-s)*zeta(s,1/4) at zero produced an error 1.56216e-11 in the derivative.
Explicit differentiation of Hurwitz zeta agreed with the analytic control;
controlled central differences at steps 10^(-55), 10^(-60), 10^(-80) converged
quadratically. The determinant differences for the latter two steps were
2.4337715e-121 and 2.4337715e-161, exactly the expected step-squared scaling.
This identifies cancellation from the default extremely small finite-difference
step near s=0, rather than a determinant normalization error. The final check
uses controlled steps, increased internal precision, and a nonzero exp(2s)
derivative control. This extra check is heuristic; the table's proven status
rests on Arb enclosures of the closed formula, not on finite differences.

`dry_run.py` passed: 12 values, exact rational inputs or error-carrying balls,
no disguised exact integer, longest working-ball spelling 146 characters,
2.3 KiB estimated working entries block. Stored values are requested at 100
significant digits. No entry is known to be an exact rational from its definition.

Software recorded in `offline-checks.json`: numberdb 0.1.10, mpmath 1.4.1.
All Sage computations used the campaign's `agents/sage.sh` wrapper. No live
website container was used for computation. Reproduce the offline checks with
`sage -python checks.py` and `sage -python dry_run.py generate.py`; use
`sage -python checks.py --stored` with a NumberDB API key to read the private draft.