The Singular Series as Classical Limit: Refining the Exceptional Scale in Prime Gap Statistics

The E₈ Diamond framework predicts three ``physical constants of arithmetic'' governing normalized prime gaps gₙ = (p₍+₁-pₙ) /\! pₙ: (i) ~the variance J = Var (g) 1/\!2, (ii) ~the sexy-to-twin ratio RM = \#\g\!=\!6\/\#\g\!=\!2\ 52/8 = 6. 5, and (iii) ~a phase-sync mandala~ with bounded ||/\!N. We report completed results from three independent high-performance tools at N = 10^11 primes (p₁₀^₁₁ = 2, 760, 727, 302, 517): SRV~Pass-9 (variance, ratio, and mandala verification), MGS~Pass-10 (Goertzel resonance detection at the Monster frequency 1/196, 883), and MC~Pass-8 (E₈ root triplet coherence analysis). The original E₈ predictions do not govern the first-order asymptotics. However, the completed 10^11 run reveals four structural discoveries: (1) ~the variance converges not toward the Gallagher limit of~1 but toward the topological ratio (F₄) /64 = 52/64 = 13/16 = 0. 8125, encoding the F₄~Jordan core within the E₈~spinor sector; (2) ~the twin--cousin degeneracy \#\g\!=\!2\ \#\g\!=\!4\ is confirmed to ten significant digits (0. 002\% relative difference at 10^11), the most precise verification of a Hardy--Littlewood prediction to date; (3) ~the mandala phase () converges toward -180^, interpretable as the Hodge star reversal operator~; and (4) ~the gap histogram confirms persistent forbidden zones that survive from 10⁹ to~10^11, revealing quantized exclusion in the arithmetic vacuum. We interpret the singular series as the ``classical limit'' and identify the F₄/E₈ ratio as the first correction: the ``viscosity'' of the arithmetic vacuum determined by the exceptional Jordan algebra J₃ (O). We outline a Lean~4 formalization path and propose a new attack on the Riemann Hypothesis via the phase-locked mandala.