nx_bigfloat120_gamma.nx

nx_bigfloat120_gamma.nx -- SOVEREIGN gamma oracle on the 120-bit bigfloat (ME2 rung 1: DLMF ch.5 enters the engine). Composition of blessed organs: ln Gamma(z) = (z - 1/2) ln z - z + ln(2 pi)/2 + sum_{n=1..17} B_2n / (2n (2n-1) z^(2n-1)) [z >= 30] with the INTEGER Bernoulli rationals B_2..B_34 (DLMF 24.2 -- mathematical constants like k!; B_36's numerator exceeds i64 and is not needed: at z = 30 the n=17 tail is < 2^-138, 28 bits past the oracle's 2^-110 contract). x < 30 shifts up by the recurrence Gamma(x) = Gamma(x+k)/(x(x+1)..(x+k-1)); negative non-integers reflect through pi/(sin(pi x) Gamma(1-x)) over the blessed trig organ. exp from the pow organ; ln rebuilt bigfloat-in (the atanh form; exponent split is exact). C99 tgamma specials: +-0 -> +-inf, negative integers and -inf -> NaN, +inf -> +inf, x > 172 -> +inf (true overflow 171.624), x < -184 -> +-0 (|Gamma| below the subnormal floor; sign by interval parity). Self-anchor (see _bf_gamma_gate_authored): EXACT factorials to 18!, the recurrence identity across different shift depths, Gamma(1/2)^2 = pi, and the reflection product r(x) r(1-x) sin(pi x) = pi on (0,1). license_tier: ORIGINAL