Prime-Phase Entropy and Fractal Scaling in Random Euler-Product Models: A Two-Regime Diagnostic Framework Motivated by GUE Zero Statistics

This note introduces a regularized prime-phase entropy observable built from the phases tlog p 2π, p≤ P, and studies its scaling behavior in two distinct regimes. The first regime is the microscopic height-increment regime. Since the regularized entropy is smooth in the height variable \(t\), its natural structure-function baseline is differentiable: ζₕₜ(q)=q. The second regime is the prime-scale, or renormalization-scale, regime. In the independent-prime random model, prime blocks are added and measured using the variance clock V(P)=σ_ε²∑p≤ Pwₚ². For the default weight wₚ=(log p)⁻¹, the clock satisfies V(P)~ σ_ε²P/(log P)³. In this random prime-block model, a triangular-array central limit theorem gives the Brownian scaling law ζsc(q)= q2. For long blocks satisfying P₂-P₁/log P₁→∞, and under the stated PNT-regularity condition, the default-weight block variance obeys Δ V(P₁,P₂)~σ_ε²∫P₁P₂dx/(log x)³. The note also defines height and scale anomaly scores, Aₕₜ(T;Q), Asc(T;Q), for measuring deviations from the smooth height baseline and the Brownian prime-scale baseline. The Riemann Hypothesis is not proved or reformulated here. RH and GUE zero statistics serve only as motivation for the observable and for the Brownian prime-scale baseline. Arithmetic links, such as those involving pair correlation or subconvexity, are recorded only as conditional diagnostic bridges requiring explicit transfer hypotheses. Main components: - regularized prime-phase entropy;- separation of height-increment and prime-scale regimes;- smooth height baseline \(ζₕₜ(q)=q\);- independent-prime block Brownian baseline \(ζsc(q)=q/2\);- variance-clock asymptotics for \(w_p=(log p)⁻¹\);- default-weight long-block Brownian law;- height and scale anomaly scores;- conditional arithmetic bridge under explicit transfer assumptions;- optional local Wasserstein stability interpretation.