V130 — Low-Cone Regularity and the Remaining Spectral Barrier for the Riemann Hypothesis

Description: This paper analyzes the low‑cone contribution to the spectral gap associated with the regularized logarithmic derivative of the Riemann zeta function near the critical line. Previous dyadic decompositions reduced the spectral problem, under suitable high‑frequency mean‑square input, to bounding the exact low‑cone energy. The main result is a shrinking‑cone regularity theorem: if the transition scale satisfies aR(a)→0 as a→0+, then the low‑cone energy is automatically bounded, since H0(s) is holomorphic in a neighborhood of s=1/2. This shows that purely logarithmic high‑frequency losses lead to a shrinking cone and remove the low‑cone obstruction. More generally, if high‑frequency estimates involve a power loss a−M, the cone shrinks only when M<1−ε. Thus, the remaining spectral barrier is sharpened: the critical task is to establish high‑frequency mean‑square estimates with sufficiently small a-loss. The paper does not prove RH but removes the shrinking low‑cone obstruction unconditionally, reducing the problem to the high‑frequency regime.