Multi-Scale Exponential Law for Riemann Zero Spacings Outside the GUE Framework

The zeros of the Riemann zeta function on the critical lineencode the oscillatory structure of the primes through theexplicit formula. The Montgomery-Odlyzko law describes theirspacing statistics via the Gaussian Unitary Ensemble (GUE) ofrandom matrix theory -- but GUE is blind to the arithmeticorigin of the zeros. We ask a sharper question: among thespacings that deviate most from the mean, do their positionscarry arithmetic structure that GUE cannot replicate? We introduce a probability measure {p_n = w_n/Z} on zerospacings and establish four numerical results on N=3000 zeros(verified at N=1000 in two independent environments). (1) Multi-scale exponential law: the mean gap k(c) betweenanomalies at threshold c satisfies k(c) ~ 0.834 * exp(1.730 * c), R^2 = 0.998, across five levels (c in {1.0, 1.5, 2.0, 2.5, 3.0}), eachwith CV < 0.53 and bootstrap p < 1e-4. The coefficient B=1.730is preliminary; N >= 1e4 is required for convergence. (2) GUE matrix comparison: 500 independent GUE matrices(500x500) yield CV_GUE = 3.23 versus CV_Riem = 0.40 -- adifference of 20.7 sigma in the OPPOSITE direction. Thisstructure lies strictly outside the GUE framework. (3) Shannon entropy: S_hat(T) = S(T) / log N(T) -> 1 asT -> infinity, consistent with RH. (4) Zero detection: local maxima of |zeta'(1/2+it)/zeta(1/2+it)|^2locate all 29 zeros in [14, 100] with 100% accuracy and zerofalse positives. No proof of the Riemann Hypothesis is claimed. The analyticidentity of B ~= 1.730 is the central open problem. ----------------------------------------------------------------MSC 2020: 11M06 (primary); 11M26, 94A17, 65B99 (secondary)