This note introduces a regularized prime-phase entropy observable built from the phases t p 2, 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: ₇ₓ (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) =_² wₚ². For the default weight wₚ= (p) ^-1, the clock satisfies V (P) _²P (P) ³. In this random prime-block model, a triangular-array central limit theorem gives the Brownian scaling law ₒ₂ (q) = q2. For long blocks satisfying P₂-P₁ P₁, and under the stated PNT-regularity condition, the default-weight block variance obeys V (P₁, P₂) _²䃑^P₂dx (x) ³. The note also defines height and scale anomaly scores, A ₇ₓ (T;Q), A ₒ₂ (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 \ (₇ₓ (q) =q\) ;- independent-prime block Brownian baseline \ (ₒ₂ (q) =q/2\) ;- variance-clock asymptotics for \ (wₚ= (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.
Byoungwoo Lee (Tue,) studied this question.