The Prime Field Architecture of the Arithmetic Modulation Field: Yang-Mills Existence and Mass Gap on R⁴ — Volume 13: Computational Verification Record

Volume 13 of 14. The complete computational verification record for the Prime Field Architecture Yang-Mills proof program: 18 independent verifications, zero failures, five engines. Documents five computational engines: Engine A (PFAE Sieve, O(N log log N), Wheel-30 exact arithmetic), Engine B (PFA_PiN, O(N^{2/3}/ln N), Meissel-Lehmer variant, reaches N = 10¹⁵), Engine C (SJQBL v8, full sieve with classify phase, N up to 3×10¹¹, benchmark 394.7s at N = 3×10¹¹), Engine D (Python/NumPy, 64-bit algebraic verification), Engine E (Direct Arithmetic, exact integer computation). Presents complete verification tables for: the Prime Arithmetic Theorem at 13 scales (N = 10³ to N = 10¹⁵, zero divergence, two independent engines); the trecade verification at T1-T11 (SJQBL v8, zero error); the rotor amplitude verification (Inverse Prime Reaction Law to 7 decimal places); the gate structure; the vacuum sector enumeration; all 15 Serre relations (max error 0.00e+00); all 64 Gell-Mann commutator pairs (max error 0.00e+00); the 4 PSGW structure constants (max error 0.00e+00); the 4-gluon color factors W^{1346} = −1/2 (error 0.00e+00) and W^{1468} = −√3/2 (error 2.22e−16, machine precision); the physical mass gap ratio 2π√48/13 = 3.34855 (0.11% agreement with lattice QCD); and the geometric series ratio r = 0.167620 < 1. Includes step-by-step instructions for independent reproduction of every computation using standard tools.