Towards the Riemann Hypothesis: Zero-Free Regions via Mesoscopic Energy Separation

We develop a mesoscopic energy separation framework for the Riemann ξ-function using
Gaussian energy functionals, annular Fej´er bridges, and Schur operator bounds. The key
innovation is a mesoscopic scaling h(T) ∼ 1/ log T and M(T) ∼ log T that respects the
logarithmic growth of zero density. We prove: (i) an explicit annular Fej´er lower bound with
constant c2(c) = e−πcπ2 log 2; (ii) Schur-Young operator bounds scaling as O(1/ log T); (iii)
a Poisson-Jensen bridge converting local energy gaps to zero-free regions; (iv) that for any
fixed σ∗ > 0 and sufficiently large T, there are no zeros with |ℜ(s) − 1/2| ≥ σ∗.
Critical limitation: This establishes a zero-free strip away from the critical line at high
heights, but does not prove that all zeros lie exactly on ℜ(s) = 1/2. The transition σ∗ → 0
requires additional analysis beyond the current framework. We present this as a partial
result towards the Riemann Hypothesis, not a complete proof.