We document twenty-five approaches to the Riemann Hypothesis that the author personally explored and that fail, organized by the structural reason for their failure. These are not historical approaches surveyed from the literature but paths developed, computed, tested, and ultimately killed over several hundred AI-assisted research sessions. The approaches span analytic number theory, spectral geometry, operator algebras, Diophantine approximation, dynamical systems, and statistical mechanics. The central finding is that the dominant obstruction is what we term the mean-to-max barrier: the impossibility of passing from average (L²) control to pointwise (L∞) control of oscillating arithmetic sums. Fifteen of twenty-five approaches encounter this barrier or a structural variant (including three Gel'fond-wall failures and one self-energy reduction). The remaining ten failures fall into four other categories: circularity (3), distinct structural errors (3), orbit independence (2), non-rigorous heuristic (1), and unjustified ansatz (1). This cartography serves three purposes: (1) a negative-results compendium identifying precisely where and why each approach breaks, (2) formalization of the meta-observation that the mean-to-max barrier is a structural feature of the problem rather than a technical difficulty, and (3) motivation for approaches that bypass this barrier entirely. The paper includes a diagnostic checklist enabling researchers to quickly identify whether a new proof strategy encounters a known obstruction before investing significant effort. Companion paper: "A Zero-Free Region for ζ(s) via Transverse Tidal Curvature and L² Energy Constraints" (same author doi:10.5281/zenodo.18986423), which proves the unconditional Battery bound δ ≤ C/(log T)² referenced throughout.
Thierry Marechal (2026) studied this question.