A Zero-Free Region for the Riemann zeta function via Transverse Tidal Curvature and L² Energy Constraints

We prove a new zero-free region for the Riemann zeta function: if ζ(1/2 + δ + iγ) = 0 with δ > 0, then δ ≤ C∞(η)/(log|γ|)² for an explicit constant C∞(η) depending on a mollification parameter. The optimal choice η = 3.0 yields C∞ ≤ 0.76 for |γ| ≥ 55. This is the first unconditional zero-free region centered on the critical line σ = 1/2, complementing the classical Vinogradov–Korobov region (centered on σ = 1).

The proof combines three ingredients: (i) the tidal dichotomy, showing that an off-axis zero at distance δ from the critical line creates negative transverse curvature in a band of width ≥ 2δ that cannot be compensated by on-line zeros; (ii) Parseval's identity for almost periodic functions, bounding the total L² energy of a mollified prime sum; and (iii) the Montgomery–Vaughan large sieve inequality, which localizes this energy bound. Together, these imply that deep negative dips in the curvature must be short, and short dips cannot sustain off-axis zeros.

For large |γ|, the resulting zero-free region (shrinking as 1/(log T)²) is asymptotically stronger than the classical Vinogradov–Korobov bound (shrinking as 1/(log T)^{2/3}). The paper includes numerical verification for the first 50 nontrivial zeros and a comparison table showing the Battery bound is 55× to 35,000× tighter than Vinogradov–Korobov across practical ranges.

Companion paper: "Twenty-Five Ways Not to Prove the Riemann Hypothesis" (same author, doi:10.5281/zenodo.18986272), which documents the structural obstructions encountered in twenty-five approaches to RH, including the mean-to-max barrier that limits the present result to δ > 0.