# Numerical evidence for T692

This build computes the normalized Taylor coefficients of the Bettin–Conrey
reciprocity function for every integer order 0 through 100. The first two
values are exactly -1 and 1/2. The other 99 values are written to 100
significant digits, with rigorous last-place intervals.

## Instructions and sources

The run read both archived skill versions listed in
`../20261109T150707Z-build-skills/manifest.json`. Both are SHA-256
`886390324dd1971e381a43cf3b46b822ab0bc67d62f31d8481e29ec79556746b`.
The same manifest is attached as `skill-manifest.json`.

Conventions and the finite formula are Akatsuka–Murakami (AM),
https://arxiv.org/html/2402.14216v1, equations (1.1)–(1.3).
The independent integral uses equations (3.1)–(3.5), including AM's sign
correction. Euler's even-zeta identity is DLMF 25.6.2. The publication
bibliography was checked against Akatsuka's publication page; the equation
numbers refer to the archived arXiv v1 HTML, whose hash is in the JSON evidence.

## Full-precision computation and comparison

`generate.py` forms each coefficient as an exact rational polynomial in pi^2
and evaluates it with Arb at 717 bits. The worst sum-of-absolute-terms
condition number corresponds to 82.356 decimal digits lost, at order 100.
The smallest measured retained accuracy is 441 bits, at order 100.
A nonconstant rational polynomial in transcendental pi^2 cannot be rational,
so no decimal row is an exactly known rational disguised by rounding.

`checks.py` separately generates Bernoulli numbers from their exact
triangular recurrence and evaluates the direct Bernoulli–zeta expression at
1000 bits. It uses neither the generator's Bernoulli evaluator nor its
polynomial construction. The smallest retained accuracy is 723 bits. For
all 99 inexact rows, both independently evaluated enclosures lie strictly
inside the interval represented by the written 100 digits. Exact initial
values agree as rationals. Deliberate ten-last-place perturbations fail in
all 99 inexact rows.

## Independent analytic and arithmetic checks

The contour integral was evaluated with certified Arb quadrature at 180 bits
for all 101 orders. Both tails beyond |t|=60 are bounded by
(12*60+14)*exp(-3*60/2), included in the returned balls. Every coefficient
agrees, and every resulting radius is below 6.02e-37. The complete derivation
is in `contour-bound.md`. The integral of t on [0,1] returns 1/2; the contour
also recovers the independently known values g_0=-1 and g_1=1/2. A shift of
10^(-30) fails for all 101 contour comparisons. These checks establish an
independent 30-digit comparison; the 100-digit certification comes from the
finite-expression enclosures, not from extrapolating the integral check.

All 99 reciprocal coefficient identities through order 100 hold exactly as
rational polynomial identities. Applying them to the written intervals gives
zero-containing residuals with maximum radius less than 3.17e-72. The first
four closed forms also agree. Six reconstructions at x=31/32,33/32,7/8,9/8,
3/4,5/4 agree with direct finite cotangent sums, including the proved Taylor
tail bound 16*|x-1|^101/(1-|x-1|). These are validation inputs, not table rows.

The ingredient fixtures checked against the live corpus are 101 Bernoulli
numbers from T13, 20 even positive zeta values from T14, and 31 cotangent
sums from T639. Fixtures retain exact spellings and the source identifiers.

The official dry-run checks pass. The written values have maximum length
104 characters and an estimated flat block size of 13.6 KiB. The displayed
Programs snippet was executed at all 101 orders. Sage is 10.9 and numberdb
is 0.1.10. All raw per-order contour enclosures, measurements and outputs
are retained in `numerical-evidence.json`.

## Review boundary

This is build evidence, not an independent critique or argument-selection
acceptance. The finite selection is justified separately in
`argument-selection.md`. No asymptotic-coefficient table or continuous-argument
table was built. No published table was edited. The controller must obtain
its separate critique and repair results before offering the draft.

## Database readback

After filling T692, `verify(sample=None)` reports 101/101 matched, zero
differing, missing or extra entries. The full-precision independent comparison,
strict written-interval containment, all 99 reciprocal identities, six finite
cotangent reconstructions and all 101 saved contour comparisons pass again on
the values read from the database. The JSON evidence retains these reports.