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

The $E_8$ Diamond framework predicts three ``physical constants of arithmetic'' governing normalized prime gaps $\tilde g_n = (p_{n+1}-p_n)/\!\ln p_n$: (i)~the variance $\Lambda_J = \operatorname{Var}(\tilde g) \to 1/\!\sqrt{2}$, (ii)~the sexy-to-twin ratio $R_M = \#\{g\!=\!6\}/\#\{g\!=\!2\} \to 52/8 = 6.5$, and (iii)~a phase-sync mandala~$\Psi$ with bounded $|\Psi|/\!\sqrt{N}$. We report completed results from three independent high-performance tools at $N = 10^{11}$ primes ($p_{10^{11}} = 2{,}760{,}727{,}302{,}517$): 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 ($E_8$ root triplet coherence analysis). The original $E_8$ 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 $\dim(F_4)/64 = 52/64 = 13/16 = 0.8125$, encoding the $F_4$~Jordan core within the $E_8$~spinor sector; (2)~the twin--cousin degeneracy $\#\{g\!=\!2\} \approx \#\{g\!=\!4\}$ is confirmed to \emph{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 $\arg(\Psi)$ converges toward $-180^\circ$, interpretable as the Hodge star reversal operator~$\star$; and (4)~the gap histogram confirms persistent forbidden zones that survive from $10^9$ to~$10^{11}$, revealing quantized exclusion in the arithmetic vacuum. We interpret the singular series as the ``classical limit'' and identify the $F_4/E_8$ ratio as the first correction: the ``viscosity'' of the arithmetic vacuum determined by the exceptional Jordan algebra $J_3(\mathbb{O})$. We outline a Lean~4 formalization path and propose a new attack on the Riemann Hypothesis via the phase-locked mandala.