V129 — Dyadic Mean-Square Thresholds for the Logarithmic Derivative near the Critical Line

Description: This paper investigates dyadic mean‑square thresholds for the regularized logarithmic derivative of the Riemann zeta function near the critical line. The spectral Sobolev gap is decomposed into dyadic shells, revealing a critical high‑frequency threshold of order T2/a. Abstract mean‑square estimates with logarithmic or a-dependent losses are analyzed, showing how such losses enlarge the remaining low‑frequency cone. A precise formulation of cone energy is introduced, retaining the full spectral multiplier to capture contributions from low ordinates. The main localization result states that high‑frequency mean‑square control removes all shells beyond a transition scale R(a), reducing the problem to bounding the exact low‑cone energy. Uniform boundedness of this cone energy excludes off‑line zeros with small ordinates relative to their horizontal displacement. Although unconditional proofs are not claimed, the framework sharpens the spectral barrier: after high‑frequency input, the remaining obstruction is localized to the low‑cone region. This diagnostic perspective clarifies why controlling cone energy is the next barrier toward proving the Riemann Hypothesis.