HUNT-005 — Stirling series — door 2 proof-out: carrier (Binet) reconstructs ln Γ at 4e-10 where the series bests 2.9e-4

WHAT THIS CERTIFICATE RECORDSDoor-2 proof-out, both legs. DETECTION: the Stirling correction series Σ B₂ₖ/(2k(2k−1)x^{2k−1}) at x = 1 has its optimal truncation at k = 3 with error 2.9e-4, then diverges factorially (B₂ₖ growth) — executed term-by-term, the divergence is visible by k = 7. RECOVERY: the carrier is the function itself, written as Binet’s second integral ln Γ(x) = (x−½)ln x − x + ½ln 2π + 2∫₀^∞ arctan(t/x)/(e^{2πt}−1)dt — the same asymptotic-carrier lane as INET-DIVERGENCE-003 and HUNT-002. RECONSTRUCTION LEG: the carrier integral reconstructs the EXACT function at 4e-10 everywhere tested (x = 1, 3/2, 2) — including x = 1, where the series’ own best effort was 2.9e-4. The carrier does what the divergent freight never could, and the freight remains its lawful asymptotic expansion.

Σ B₂ₖ/(2k(2k−1)x^{2k−1}) read as a convergent sum — diverges factorially; at x = 1 best error only 2.9e-4 (k = 3)

carrier: Binet integral — reconstructs ln Γ at 4e-10 at x = 1, 3/2, 2 ✓
the series re-derived from the carrier as its asymptotic expansion — the ledger returns, the lane is named

2 — detection + reconstruction

S3 — lane

CLOSED_CLASSICAL — Stirling 1730; Binet 1839; Euler–Maclaurin remainder theory

GREEN_CLASSICAL

Binet 1839; Whittaker & Watson; the mp113 engine’s own Γ route (golden vectors)