# Numerical evidence for T612

Run: 20261029T192200Z-build. The table remains an unpublished draft. This
build evidence is not independent critique approval.

## Instruction record

The coordinated archived manifest is
`agents/runs/20261029T192200Z-build-skills/manifest.json` in this run's source
snapshot. Both `skill-local.md` and `skill-live.md` were read; both SHA-256
hashes are `28ecbeeccb0c5e5d6eb541b5a5b0f88aec47b154374d909085116d34f5629199`.
`skill-evidence.json` records the manifest path and its versions. The
campaign argument-selection gate and `agents/argument_review.py` were read.
`argument-selection.md` explains the only free parameter and its finite extent.

## Conventions and sources

Inoue, Section 4 (arXiv:0809.2111), defines the two pentagonal flowers glued
along their boundaries to produce a trivalent sphere. L_n has n sides per
base, n >= 5. The PDF SHA-256 is
`f694e4f5358d3ec375bdfce1fbd99d6b12333b83c76774e63ba8d22492843550`.

Kellerhals–Kolpakov, *The minimal growth rate of cocompact Coxeter groups in
hyperbolic 3-space*, Canadian Journal of Mathematics 66 (2014), 354–372,
DOI 10.4153/CJM-2012-062-3, Section 2.2 specifies distinct elements by minimum
length in the face reflections. Equation (2.5) gives Steinberg's formula;
Section 4.2, equation (4.3), gives the compact right-angled growth series in
terms of the face count. The author-hosted PDF at
https://homeweb.unifr.ch/kellerha/pub/CJM-KellKolp-J0844-FINAL.pdf has SHA-256
`8e428dc81785be0d649ad65b1e9eb022e4c670a7dd6089bbaa61c093e91c8ca6`.

Using F=2n+2 in that formula yields denominator
(1-t)(1-(2n-2)t+t^2). Its positive poles are tau_n^(-1), 1, tau_n,
where tau_n=n-1+sqrt(n(n-2)); the numerator (1+t)^3 cancels none of them.
This settles the root and word metric. In particular, the values are not
word counts or growth in generators of the orientation-preserving subgroup.

## Computation and checks

`generate.py` uses named Sage rings and Arb at 397 bits, comprising 100
decimal digits and a 64-bit guard. All 26 values n=5,...,30 are finite,
positive, irrational, and greater than 1. Irrationality is checked exactly:
n(n-2) is nonsquare; in general it lies strictly between (n-2)^2 and (n-1)^2.
The measured worst relative radius is less than 2^(-395), and every row
satisfies relative radius < 10^(-117). Every serialized value has exactly
100 significant digits and at most 101 characters including its decimal point.
The dry-run helper reports 26 entries, longest internal ball spelling 137
characters, and an estimated 4.6 KiB block; its exactness, prose and suspicious
trailing-zero checks pass.

`checks.py` constructs the dual adjacency graph from two cyclic rings of
pentagonal faces and two bases. For every n it enumerates all cliques and
finds [1, 2n+2, 6n, 4n], with no larger cliques. It assembles the growth
denominator by Steinberg's formula over those finite elementary abelian
2-subgroups, in exact rational polynomial arithmetic. This agrees exactly
with the displayed growth series for all 26 n.

Independently of the rational-series calculation, it enumerates the distinct
elements at lengths 0,1,2,3, reducing by involution and all commuting adjacent
swaps. Its controls first give [1,3,3,1] for (Z/2)^3 and [1,3,6,12] for three
free involutions. Every resulting coefficient agrees with the series. The
endpoint counts are [1,12,102,812] for n=5 and [1,62,3602,208862] for n=30.
This finite check tests the metric and group combinatorics; the general
rational-series identity is justified by the cited theorem.

The reciprocal cubic assembled from those clique counts is bisected 440
times in exact rational arithmetic on [2n-3,2n-2]. A separate sqrt(2) control
first checks the signs and exact bracket width. Every obtained bracket lies
inside the independently computed Arb ball's neighborhood and strictly inside
the decimal's claimed interval. The cubic has opposite signs at the two
exact endpoints of every stored decimal interval; its relevant root is unique
on the initial bracket. No midpoint residual is treated as an error bound.

The checks also evaluate the quadratic, reciprocal trace identity and
log(tau_n)=arcosh(n-1) in finite, narrow balls. They enumerate small exact
solutions of a^2-d*b^2=+/-4 with the maximal-order parity conditions, find the
smallest unit >1, and verify its power equals tau_n exactly in Q(sqrt(d))
for every n. In particular n=10 gives the sixth power of (1+sqrt(5))/2.
The read-back mode also checks log(tau_n)/m against T131's stored regulator
interval wherever the field discriminant is present. Discriminants 1020,
1292, 1596 and 1932 exceed T131's coverage; their unit-power identities are
checked exactly, without claiming a missing regulator comparison.

All comparison balls are required to be finite and sufficiently narrow. All
100-digit intervals strictly contain the computed balls and exact rational
root brackets. Verification uses the NumberDB convention of one unit of
uncertainty in the final written decimal place.

## Reproduction

Install NumberDB in SageMath and download `generate.py` and `checks.py` into
one directory. `sage -python checks.py` runs the independent computations.
With read access to the draft, set NUMBERDB_API_KEY and run
`sage -python generate.py` and `sage -python checks.py --stored` to recompute
and check the database values. The campaign uses `agents/sage.sh` for every
Sage command; its stdout evidence is retained in the run-work directory.