V131_1 — Finite-Cone Zero Exclusion and Compact Spectral Barriers for the Riemann Hypothesis

Description: This paper investigates the compact low‑cone obstruction in the spectral approach to the Riemann Hypothesis, based on the weighted spectral gap of the regularized logarithmic derivative of the zeta function. Dyadic decomposition separates the problem into high‑frequency shells and a low‑frequency cone. The shrinking cone near s=1/2 is harmless, since H0(s) is holomorphic there. The remaining obstruction arises when a is bounded away from zero, leading to compact finite‑cone regions. The main result establishes a compact‑cone equivalence: boundedness of the cone energy on a0≤a<1/2 holds if and only if the closed finite cone contains no zeros of ζ(s). A zero inside or on the boundary forces divergence, while a zero‑free cone ensures boundedness. The paper also formulates a certified finite‑height input: if all zeros up to a given height are verified to lie on the critical line, then every compact cone within that rectangle is harmless. Thus the compact spectral barrier is reduced to a finite‑height zero‑exclusion problem. Combined with high‑frequency mean‑square control and shrinking‑cone regularity, this framework provides a conditional route to RH. The contribution is structural: it isolates the compact spectral obstruction as a finite‑cone zero‑exclusion problem.