V139_2 — An M = 1 Admissible Framework for the Dyadic High-Frequency Barrier

Description: This paper develops a dyadic spectral framework for analyzing the high‑frequency barrier in the study of the logarithmic derivative of the Riemann zeta function. The focus is on the consequences of accepting the natural diagonal packet loss at the M = 1 scale.

The spectral formulation decomposes the high‑frequency integral into dyadic shells. Earlier approaches required stronger mean‑square inputs with M < 1 to push the obstruction into a shrinking low‑frequency cone. This paper shows that even with the natural M = 1 loss, the high‑frequency shells can still be cleared. Specifically, for shells beyond a logarithmic threshold, the spectral weight ensures that the tail is uniformly summable.

The tradeoff is geometric: while M = 1 suffices to eliminate the high‑frequency obstruction, it enlarges the remaining low‑frequency region from a shrinking cone to a logarithmic strip. This strip grows as the parameter approaches zero, meaning local regularity near the critical line does not automatically control it. The paper also identifies a necessary zero‑exclusion condition for boundedness of the enlarged low‑frequency energy.

The main conclusion is that M = 1 is analytically natural but geometrically expensive. The high‑frequency barrier is rebalanced into an enlarged low‑frequency obstruction. The paper does not prove the Riemann Hypothesis; its contribution is structural, clarifying how the dyadic framework adapts at the natural M = 1 scale.