# Numerical evidence for T552

The stored 163 reduced rational coefficients agree exactly with all independent
checks in numerical-evidence.json. No finite-precision claim is made.

The generator uses the source's character expansion, with Murnaghan-Nakayama
characters and Jacobi-Trudi Schur specializations. The independent checker uses
no character or Schur routines. It enumerates S_k, groups unknown coefficients
by cycle type, and solves G_N * w = T_ab using rational Gaussian elimination,
where G_N(sigma)=N^(number of cycles) and
T_ab(sigma)=product over cycles of (N for even length, a-b for odd length).
For k<=N this Gram matrix is invertible, so these equations uniquely determine
the coefficients. All 474 full permutation equations were also checked.

For an independent matrix-entry check, write a unitary column as a normalized
vector of independent complex Gaussian variables. Its squared moduli have the
Dirichlet(1,...,1) law. The sum Q over the first a coordinates is Beta(a,b),
and W_11=2Q-1. Expanding (2Q-1)^k and using E[Q^j]=(a)_j/(a+b)_j gives
all 61 moments without the character expansion or Gram solver. Summing the
stored coefficient over every permutation gives the same moment. At a=b=1,
W_11 is uniform on [-1,1], providing nonzero controls 1/3 and 1/5 for its
second and fourth moments. A deliberately corrupted coefficient fails the
moment check. Gaussian elimination has the independent control
[[2,1],[1,3]] x = [1,2], whose solution is [1/5,3/5].

The 48 low-degree cases agree with Matsumoto's Example 4.1. All 163 values
also satisfy signature reversal and the convolution with the corresponding
coefficients independently read from T549. Thirteen balanced odd-degree
coefficients are exactly zero. These facts are verified again on stored
values, not merely on a second invocation of the generator.

The initial checks encountered missing Sage coercion initialization, rather
than a disagreement between computations. The logs retain those failures.
Constructing every rational from integer numerator and denominator, with QQ
on both sides of division, resolved them. No failed check was waived.

The archived local and live skill versions match; policy evidence is
agents/runs/20261027T144347Z-build-skills/manifest.json. Both SHA-256 digests are
e37efb7e2ededf6c1d7134c673f152f648f26e0267c556aebdf3ec65765d8b4c.
The exact source PDF and source-file checksums are in numerical-evidence.json.

These are build checks. Independent critique, including argument selection,
remains required. The builder has not offered, reviewed or published T552.