# Numerical evidence for T662

This is build evidence, not the independent critique required by the campaign.

## Policy and sources

The controlling manifest is
`agents/runs/20261101T231203Z-build-skills/manifest.json`, attached as
`skill-manifest.json`. Both archived local and live skill versions were read;
each and the canonical snapshot have SHA-256
`a2b28d9953264d7b0ca62663785c2952190c5be6d6d3597fd695d9a54d606278`.
The family issue #292 and its coverage map were read first. Its shared
conventions are followed: H_m=(C_m)^3, q=m^3, n=2, and all complex characters
counted with multiplicity. The attached argument-selection.md reviews m
and the finite range 5 through 40.

Checked primary sources: CKSU, arXiv:math/0511460v1, Theorem 1.8, Section 2,
and Lemma 2.1; Gao–Jing, doi:10.5802/jolt.1399, Theorem 4.1, equation (4.4),
page 605. CKSU's coarse largest-degree estimate is distinguished from our
exact-spectrum root. The latter is obtained by substituting the full spectrum
into their theorem; the source does not print this 100-digit root table.

## Analytical checks

Let q=m^3 and A=2m(m-1). Characters fixed by swapping an ordered pair of
base characters yield 2q degree-one extensions. The distinct pairs form
binomial(q,2) orbits and yield degree-two induced irreducibles. Their squared
dimensions sum to 2q^2=|H_m wr S_2|, exhausting the degree spectrum.
Thus D(s)=2q+binomial(q,2)2^s, including every linear character.

The function g(s)=D(s)/A^s is strictly decreasing, since 1/A and 2/A are
positive and less than one. For m>=5, g(2)>1 and g(3)<1:
    g(2)=m^4/(2(m-1)^2),
    g(3)=(2m^3-1)/(4(m-1)^3).
For the second inequality, (m-1)^3/m^3 >= (4/5)^3 > 1/2 suffices.
This proves a unique beta in (2,3). The CKSU theorem gives g(omega)>=1,
therefore omega<=beta. No algebraicity claim is made.

The coarser bound gamma=(6 log m-log 2)/log(m(m-1)) uses
D(s)<=2^(s-1)m^6. The difference between this upper estimate and D(s)
is q*(2^(s-1)-2)>0 for s>2, proving beta<gamma. The same cubic
inequality proves gamma<3. All these comparisons were additionally checked
on every listed value. The minimum claim is restricted to the listed range;
no global minimum theorem is asserted.

## Computation and independent verification

Generator: exact-rational bisection of a monotone equation, with Arb sign
certification at 429 bits, 357 iterations, and an enclosing finite ball.
Every value retains at least 358 conservative relative accuracy bits.
`agents/table-build/dry_run.py` passes for all 36 values: no floats or
unbounded results, no suspicious exact decimal zeros, named imports and
Sage initialization. Its ball representations have maximum length 125
characters and occupy 5.8 KiB including labels. The stored 100-significant-
digit decimals have maximum length 101 and a 3991-byte JSON numbers mapping.

`checks.py` passed with these independently implemented checks:

- MPFI at 512 bits evaluates the original cost equation at the exact
  rational endpoints of every displayed decimal interval. All 36 left
  endpoints have negative residual and all right endpoints positive.
- mpmath secant solves at 150 decimal digits fall strictly inside all
  displayed intervals. This agreement is additional evidence, not the
  source of certification.
- Both arithmetic/root methods pass a square-root-of-two control before
  measurement. Every generated ball is finite with radius less than 10^-105.
- Every selected m agrees with live T661's exact cubic moment and its
  interval for the 9/4 moment: 72 companion checks in all.
- Direct multiplication in the semidirect product tests every relevant
  quotient triple for m=2,3,4,5. Subset sizes are respectively 4,12,24,40;
  quotient-set sizes are 6,45,160,425. Every case has zero violations.
  Allowing zero in the excluded coordinate at m=2 gives 48 violations,
  confirming that the check can reject a false construction.
- GAP character tables independently give {1:16, 2:28} for m=2 and
  {1:54, 2:351} for m=3, after the S3 degrees [1,1,2] control.
  Larger complete character tables were not computed. The general
  multiplicities follow from the exact character-orbit argument.
- Among m=5,...,40, the smallest root occurs at m=17. The numerical
  approximation is 2.9087787186100874; m=5 and m=40 give approximately
  2.9907539377077031 and 2.9160549411204630.

Software: Python 3.12.3; SageMath 10.9 (its Arb and MPFI implementations);
numberdb 0.1.10; mpmath 1.4.1; GAP 4.15.1. All Sage work runs through
agents/sage.sh. The checks are attached and may be repeated with
`sage -python checks.py --stored` after obtaining access to the draft.

## Corpus checks and limitations

The screen helpers found no duplicate construction-bound family and no
matching issue title. Numerical lookup probes at m=5,17,40 found no hits.
The existing groups, algebra and matrix multiplication tags were checked;
the last has four tables on its API tag page. T661 was read through the API,
but omitted from public title search and unavailable anonymously, so its
permanent identifier is used for the companion link rather than inventing
a slug. T5's address was read from the search results. No existing table was
edited. A new representation theory tag remains a family proposal for
T661, T255 and T607; this run creates no tag.

All raw logs, source downloads, table snapshots and failed checker attempts
are retained in the run workspace. The early checker failures concerned Sage
API availability and coercions; no contradictory numerical answers occurred.
A short lesson artifact records the compatibility findings. The later
controller stages must independently critique arguments and numerics before
offering this private draft. This build does not offer, review or publish it.

## Stored-value verification and audit handoff

After filling, Generator.verify(sample=None) matched 36 of 36 entries, with
zero differing, missing or extra entries. The independent checks were repeated
on the decimal strings read back from T662: all 36 endpoint sign brackets and
secant comparisons passed, along with the companion and small group checks.
The Programs snippet was executed at every selected m and at m=41; all
selected outputs fall in the corresponding stored intervals.

The site audit initially requested an installation command in the generator's
docstring. The generator now includes `sage -pip install numberdb mpmath`.
That documentation-only change leaves the enumeration and arithmetic intact.
The final audit response and read-only selection fingerprint are retained
with the controller handoff. No independent critique sidecar was authored
by this build.