A Unified Structural Framework for the Millennium Problems

This preprint develops a unified structural framework for the Clay Millennium Problems by treating them not as seven isolated questions, but as manifestations of a small family of recurring obstruction phenomena. Rather than proposing new proofs, the paper argues that each problem sits at a boundary where a natural invariant threatens to lose boundedness, positivity, localisation, representability, or computability. Within this framework, the Riemann Hypothesis is recast as a problem of spectral positivity for a quadratic form à la Weil–Hilbert–Pólya; Birch–Swinnerton–Dyer as a dimensional coherence condition aligning analytic and arithmetic notions of rank; and the Hodge Conjecture as an obstruction to representability of analytic Hodge classes by algebraic cycles. The Navier–Stokes regularity problem and the Yang–Mills mass gap are interpreted as scale-coherence and spectral-coherence obstructions, respectively, in which globally bounded energy fails to guarantee control over small-scale concentration. The P\mathbf{P}P vs. NP\mathbf{NP}NP problem is framed as an irreversibility obstruction between efficient verification and efficient construction in large combinatorial spaces. The resolved Poincaré Conjecture is treated as a template case in which a suitable coherence-restoring flow (Ricci flow with entropy) successfully neutralises the obstruction. Across these case studies, the paper formulates conditional “structural reduction” theorems: assuming the existence of appropriate coherence operators (e.g. positivity, dimensional matching, or scale-localisation operators), the standard conjectural statements follow formally. This shifts the emphasis from ad hoc, problem-specific techniques toward the search for coherence-preserving structures and dynamics that act across domains. The resulting framework is intended as diagnostic and organisational: it clarifies why similar conceptual barriers recur in number theory, arithmetic geometry, Hodge theory, PDEs, gauge theory, and complexity theory, and suggests that future progress may hinge on discovering new mechanisms that prevent coherence collapse rather than new invariants alone.