THM-DIVERGENCE-004CLOSED_CLASSICALS3
MuncyNautics Recovery Certificate · theorem-grade divergence · hunted and executed 2026-08-31 · author of every MN concept: Derrick E. Muncy
WHAT THIS CERTIFICATE RECORDSThe Stirling series for ln Γ diverges for EVERY argument as more terms are taken — yet Binet’s integral carrier is finite and the series is its asymptotic expansion (proven). Same species as the Euler factorial series (INET-DIVERGENCE-003): factorial/Bernoulli rank must be divided out, not summed through.
GOVERNING EQUATION AS PROJECTED — THE SEAM
ln Γ(n) = (n−½)ln n − n + ½ln 2π + Σ B₂ₖ/((2k−1)(2k)n^{2k−1}) — divergent in k for every n (proven)
COMPLETE GOVERNING EQUATION — THE RECOVERY
Binet carrier μ(n) = 2∫₀^∞ atan(t/n)/(e^{2πt}−1)dt — finite (theorem); series = its asymptotic expansion
executed at n=10: errors 2.8e-6 → −7.9e-9 → 5.9e-11 → −8.2e-13 → 0 at k=1,2,3,4,6 — alternating, each residual at the NEXT term’s size (the error locator), optimal truncation before divergence
CORRECTION RULE — correction = reconstruction from the recovery: restored arguments, seats, phases and arms in the equation itself; no raw appended constants — any finite term must be derived from the recovered channels, never fitted.
| species |
S3 — asymptotic rank (Bernoulli growth) |
|---|---|
| executed |
term-by-term residual ladder matches predicted sizes exactly; double precision (the theorem itself is the 113-bit-free closure) |
| standing |
CLOSED_CLASSICAL — Binet/Stirling, proven; executed witness ladder |
| promotion_authority |
GREEN_CLASSICAL |
| source_records |
Stirling 1730; Binet 1839; Whittaker & Watson §12.3 |