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 = (pₙ₊₁-pₙ)/\!ln pₙ: (i)~the variance ΛJ = Var( g) → 1/\!√2, (ii)~the sexy-to-twin ratio RM = #\!=\!6\/#\!=\!2\ → 52/8 = 6.5, and (iii)~a phase-sync mandala~Ψ with bounded |Ψ|/\!√N. We report completed results from three independent high-performance tools at N = 10¹¹ primes (p_10¹¹ = 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 10¹¹ 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 #\!=\!2\ ≈ #\!=\!4\ is confirmed to ten significant digits ($0.002%$ relative difference at 10¹¹), the most precise verification of a Hardy--Littlewood prediction to date; (3)~the mandala phase (Ψ) converges toward -180^∘, interpretable as the Hodge star reversal operator~; and (4)~the gap histogram confirms persistent forbidden zones that survive from 10⁹ to~10¹¹, 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.