Towards the Riemann Hypothesis: Zero-Free Regions via Matched-Scale Mesoscopic Framework

We develop a matched-scale mesoscopic framework for analyzing zeros of the Riemann
ξ-function. The key innovation is a structural coupling between the tent scale h(T) ∼ 1/ log T
and the distance σ∗ of a hypothetical off-line zero, which automatically produces constants
independent of σ∗ and forces zeros into a shrinking strip.
We prove: (i) sharp Poincar´e-Wirtinger bounds for the detrending projector; (ii) explicit
high-frequency capture with constant η(K) = 2K3e−2K; (iii) Fej´er mass preservation at
matched scale; (iv) rigorous Schur bounds for all operator components including drift; (v) a
Poisson-Jensen bridge yielding zero-free regions; (vi) that for any fixed σ∗ > 0, there are no
zeros with |ℜ(s) − 1/2| ≥ σ∗ at sufficiently large height T.
Main Result: All zeros at height T satisfy |ℜ(s) − 1/2| = O((log T)−1).
Critical Limitation: This establishes asymptotic approach to the critical line but does
not prove exact localization ℜ(s) = 1/2. The transition σ∗ → 0 uniformly remains open. We
present this as a significant partial result towards the Riemann Hypothesis, not a
complete proof.