# Numerical evidence for T680

Skill authority: `agents/runs/20261102T145807Z-build-skills/manifest.json`.
Both archived local and live copies were read and matched SHA-256
`a2b28d9953264d7b0ca62663785c2952190c5be6d6d3597fd695d9a54d606278`. The attached skill-evidence.json records both.

## Methods and accuracy claim

The generator uses mpmath 1.4.1 under Sage 10.9. It computes F and its first
logarithmic solution by Taylor continuation at 140 and 190 decimal working
digits. At the conifold it matches a full Frobenius basis at z=3/4, including
the w log(w) solution for w=1-z. At z=2 the path is explicitly in the upper
half-plane. For negative z, log(-z) replaces log(z) during real transport;
this only subtracts a constant multiple of F and recovers the same real G.
The final values use z exp(G/F), and division by the exact integer C_A gives
the integral convention. The origin's normalizations follow from c_0=1,h_0=0.

All 56 fibers were independently recomputed at 140 decimal digits. Smooth
fibers use a parameter derivative of a generalized hypergeometric function;
the conifold uses a finite sum of 1400 coefficients plus 130 terms of the
gamma-ratio asymptotic expansion, summed by Hurwitz zeta, together with its
parameter derivative. This method shares neither Taylor continuation nor
Frobenius matching with the producer. The all-half conifold also passed a
hypergeometric parameter derivative in the retained first-attempt evidence.

The minimum agreement among F, G and Q is
**133.612 decimal digits**.
The minimum Q agreement between 140 and 190 working digits is
**139.237 digits**.
The smallest computed |F| is **0.95115791574722**;
all independent denominators are numerically nonzero. This is a numerical
separation, not a certified bound. Each stored nonzero component has 100
significant digits. Exact zero imaginary components on the real fibers
follow from real continuation, not from rounding.

The rigour is **heuristic (agreement-checked)**. The interval objects encode
agreement and rounding; they do not turn truncation estimates or asymptotic
tails into certified enclosures.

## Controls, exact checks and size

Before trusting the main comparison, the same continuation algorithm returned
the known rank-two Q values 16 exp(-pi), -16 exp(-pi), and
16 i exp(-pi/2), respectively at z=1/2,-1,2, to more than 150 digits. These
are elliptic-nome identities with the parameter convention of T25. The tail
method's telescoping control returned F=2 and G=1 to more than 153 digits.

Exact rational checks passed for every template: agreement of the cyclotomic
and explicit parameter lists, both differential-equation recurrences through
n=40, all fourteen integer scales, and integrality of q(x) through x^41.
For the quintic and all-half cases, independent factorial/harmonic formulas
also agree; the quintic begins x+770x^2+1014275x^3.

`agents/table-build/dry_run.py` passed on all 112 entries. The measured largest
value representation is 299 characters, and the entries block is 45.6 KB.
The independent check and multiprecision computations took
294.7 seconds. Full records are in checks.jsonl;
exact-checks.json, dry-run.log and prepared-entries.json preserve the remaining
evidence. A prior check exited with status 137 during a mixed-parameter
conifold derivative; it produced no conflicting numerical result. Its output
is retained in the run artifacts, and the replacement method has a separate
nontrivial control.

## Stored-value verification

A fresh `verify(sample=None)` recomputation returned 112/112 matched, with
0 differing, 0 missing and 0 extra. The separate Decimal readback checker
confirmed that each stored component contains the independent value. The
largest displacement was 0.49593824035814644152248933126665019625951594 units
in the last place. It checked all 56 exact scaling identities Q=C_A q against
the stored intervals and all 56 denominators against live T679. Exact-zero
imaginary intervals were retained on every real fiber. Its nonzero positive
control and deliberately incorrect negative control both behaved as expected.
See readback_checks.py, readback-results.json and verify.log.

The final site audit is clean. The audit initially asked whether the two
normalizations were separate quantities; the definition now states explicitly
that these are one number in two invertible conventions. No numbers were
changed in that clarification.

This document is build evidence, not an independent critique or argument
selection acceptance. The draft remains unpublished and unoffered for the
controller's separate critique and repair stages.