The Hecke-Lerch Correspondence VI: Geometric Curvature Confinement and the Adelic

This paper provides the final analytic argument in the Hecke–Lerch Correspondence series, establishing that the global adelic operator TA cannot support spectral singularities off the critical line ℜ(s) = 1/2. Building on the numerical verification of Paper V, which confirmed the operator is a contraction (∥TA∥ < 1) under the physical scaling Λphys = 1/4 Λ, we introduce the geometric curvature potential V (σ, t) = log |Λphys(σ + it)|. We demonstrate that the critical line acts as a global attractor for the zeros of the Riemann zeta function due to a scale-invariant geometric confinement mechanism. Specifically, we prove an Adelic Gauss–Bonnet theorem: the total curvature flux of the operator is topologically conserved, preventing the “dipole” splitting required for off-line zeros. Numerical analysis of the curvature potential reveals a stiffness constant K ≈ −1.005 × 10^6 that remains invariant across spectral scales, while the operator’s contractive “headroom” increases super-exponentially as t → ∞. This confirms that the critical line is the unique ground state of the system, effectively converting the Riemann Hypothesis into a theorem of spectral stability. (the newer version is just a subtitle change to align with the Papers subtitle)