Logarithmic Gate Geometry for Centered Riemann Zeros: Positive Leakage Energy and a Conditional Spectral Nullity Program

This preprint develops an exact logarithmic coordinate system for a point ρ = 1/2 + ε + iγ in the upper half-plane. The centered principal logarithm separates a real radial coordinate from a real gate phase α = −arctan(ε/γ). The associated leakage energy L(α) = −log(cos α) is nonnegative, reflection-invariant, and vanishes exactly when Re ρ = 1/2. The release also gives a finite-dimensional self-adjoint realization and a conditional spectral closure theorem. The accompanying Lean 4 package formalizes the scoped algebraic and analytic core. It was built with Lean 4.19.0 and checked with --trust=0; the axiom audit reports no sorryAx. Version 1.1 revises the presentation while preserving the proved, verified, computed, conditional, open, and rejected classifications recorded in the manuscript.