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2985 bytes, as of the version from 2026-10-29 04:58 (current). Recorded here, not run.
# Independent continuation used in checks.py
The skill manifest is `agents/runs/20261029T043345Z-build-skills/manifest.json`;
its two versions have the SHA-256 recorded in `skill-evidence.json`.
The generator evaluates the finite signed-Stirling and central-factorial
formulas of Cunha and Freitas, Theorem 3.3, with Arb balls. The check uses an
independently derived continuation of the defining spectrum and mpmath, not
the generator's combinatorial coefficients or its zeta-derivative code.
Put a=(d-1)/4 and x=k+a, so lambda=4(x^2-a^2). For d>=2 the multiplicity
polynomial is
P(x) = 4x/(d-1)! product(2x+i-(d-1)/2, i=1,...,d-2).
For d=1, P(x)=2. This is Spreafico Section 3's polynomial Q_d(2k).
We also check it exactly against the binomial difference for every d=1,...,20
and k=1,...,50. Expand it independently as sum(c_j x^j).
Remove k=1,...,K-1 explicitly, with K=30, and set q=K+a. The remaining zeta
function is
4^(-s) sum_j c_j sum_(ell>=0) (s)_ell/ell! a^(2ell)
zeta(2s+2ell-j,q).
The expansion is initially valid in the convergent half-plane and continues
meromorphically to a neighborhood of zero. Its ell-tail converges uniformly
there because a/q<1. Its derivative at zero has the following terms:
* ell=0: 2 zeta'(-j,q) - log(4) zeta(-j,q).
* ell>=1, 2ell-j != 1: a^(2ell)/ell zeta(2ell-j,q).
* ell>=1, 2ell-j = 1: a^(2ell)/ell times
[-psi(q) + H_(ell-1)/2 - log(4)/2].
The last expression retains the finite contribution from the pole of Hurwitz
zeta: (s)_ell/ell! = s/ell + H_(ell-1)s^2/ell + O(s^3), and
zeta(1+2s,q)=1/(2s)-psi(q)+O(s). Omitting this pole correction is wrong in
even dimensions. The finite initial spectrum contributes
-sum_(k<K) m_(d,k) log(lambda_(d,k)). Negating the total gives L_d.
For truncation before ell=M, where 2M>j+1 for all polynomial degrees, the
absolute omitted contribution is at most
sum_j |c_j| a^(2M)/(M(1-rho))
* [q^(j-2M) + q^(j-2M+1)/(2M-j-1)], rho=(a/q)^2.
This follows from the geometric-series bound on sum_(ell>=M) r^ell/ell
and the integral bound for the decreasing power sum over k>=K. The check
increases M until this bound is less than 10^(-170). At a=0 the tail is zero.
The continuation is evaluated at 190 decimal digits, with extra precision
for tiny positive-s Hurwitz zeta values. The d=1 control returns the known
nonzero result 2 log(pi) before other dimensions are evaluated. All 20 values
must agree with the generator at 200 working digits within 10^(-160), and all
20 determinants must match Appendix B's six-decimal rounded values. This
cross-check's floating arithmetic is corroboration only. The table's proven
claim is supplied by finite Arb enclosures from the published finite formulas,
not by wrapping mpmath outputs in an interval.
The finite formulas contain log(4(n-1)) in odd dimension d=2n-1 and thus must
not be used literally at d=1. The generator obtains that endpoint directly
from Z_1(s)=2^(1-2s) zeta(2s), giving D_1=pi^2 and L_1=2 log(pi).