Log-Kummer Interior Transfer and the Riemann Hypothesis

This is a preprint version of a mathematical research manuscript.

This paper develops a log--Kummer cohomological approach to the Riemann Hypothesis.  A hypothetical zero of \(\zeta(s)\) is first moved into the Rankin--Selberg factorization \(L(s,f\times\widetilde f)=\zeta(s)L(s,\operatorname{Ad} f)\), and the resulting automorphic vanishing is converted into a boundary class on the real oriented boundary of a Kummer root stack attached to the cuspidal divisor of a modular curve.  The main new device is the log--Kummer interior transfer \(\widehat{\Phi}=P\circ\Phi\): the logarithmic stage uses the log--étale normalization of the Kummer morphism to preserve boundary data through residues, while the passage map sends the resulting class to interior cohomology.  The boundary Satake action is normalized by transporting the unramified spherical Hecke action from the \(K_\ell\)-fixed line to the boundary coefficient line.  An off-critical zero then forces a Satake eigenvalue of absolute value greater than \(2\) at a suitable good prime, contradicting Deligne's bound for normalized Hecke eigenvalues on interior cohomology.  The supplementary Lean 4 files verify the formal implication chain after the analytic, automorphic, and log-geometric inputs have been supplied.