A Mean-Value Rigidity Framework for the Riemann Hypothesis A Two-Scale Solvability Roadmap

This project investigates the Riemann Hypothesis through a geometric and stability-based framework centered on the Riemann ξ-function. By modeling \Phi(s) = \log|\xi(s)| as a harmonic potential field with logarithmic singularities at its non-trivial zeros, the work reframes the hypothesis as a problem of global structural admissibility rather than zero enumeration. The study introduces the 001 Rigidity Scheme, a multi-scale analytic mechanism that detects failures of scale invariance and converts them into unavoidable global contributions through a coercive rigidity floor. To enhance sensitivity to off-critical-line perturbations, the framework develops anisotropic averaging geometries—most notably triangular and hourglass domains—aligned with the intrinsic flow of the critical strip. Within this setting, two structural principles emerge: a Footprint Lemma, asserting that any zero off the critical line generates a persistent region of scale inconsistency, and a Budget Lemma, establishing an upper bound on the total admissible rigidity compatible with the analytic growth of the ξ-function. Their interaction produces a confinement mechanism in which only zeros on the critical line remain compatible with global stability. The project is presented as a modular research program combining analytic formulation, geometric construction, numerical validation, and asymptotic analysis. Together, these components outline a stability-driven pathway toward resolving the Riemann Hypothesis through global confinement rather than direct spectral counting.