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

The Temporal Density Dynamics – Riemann Hypothesis (TDD_RH) theory proposes a structural reformulation of the Riemann Hypothesis by combining concepts from analytic number theory with a multi-scale averaging framework inspired by Temporal Density Dynamics (TDD). Rather than treating the Riemann Hypothesis as a problem of locating or counting individual zeros of the zeta function, the theory reframes it as a global exclusion problem governed by stability, symmetry, and multiscale consistency. At the core of the framework is the interpretation of the symmetrized -function as generating a scalar potential field with nontrivial zeros corresponding to logarithmic singularities or “defects” in that field. The critical line is interpreted as a neutral or equilibrium manifold. The central question becomes whether off-critical-line defects are admissible under global structural constraints. To address this, the theory introduces a two-scale mean-value rigidity scheme (the “001 scheme”). Two local circular averaging probes test for scale invariance of the potential, while a coercive rigidity step converts persistent scale inconsistency into a non-vanishing contribution to a global “rigidity budget.” If such a budget cannot accommodate off-critical-line singularities without violating known symmetry and growth constraints, those singularities are excluded. The framework isolates the Riemann Hypothesis into a small number of explicit analytic conditions, most notably a Footprint Lemma, which asserts that any off-critical-line zero must produce a nonlocal, multiscale inconsistency footprint of positive area. A symmetric toy-model validation demonstrates that even paired defects cannot evade two-scale detection, establishing the plausibility of the mechanism while leaving the core analytic lemma open. The TDD_RH theory does not claim a proof of the Riemann Hypothesis. Instead, it provides a rigorous roadmap that clarifies why traditional local methods encounter cancellation barriers and identifies the precise structural principles required for a potential resolution. Its value lies in shifting attention from zero enumeration to global admissibility and rigidity, offering a coherent pathway for future analytic investigation.