The Hodge Conjecture as an Admissibility and Rigidity Problem

The Hodge Conjecture concerns the characterisation of those rational cohomologyclasses of type (p,p)(p,p)(p,p) on a smooth projective complex variety that arise fromalgebraic cycles. While traditionally approached through explicit constructionsin algebraic geometry, the conjecture may instead be viewed as a question ofglobal compatibility between analytic, topological, and algebraic structures. This paper develops an admissibility–rigidity reformulation of the HodgeConjecture. Building on earlier work that introduced admissibility and globalcoherence probes in the context of the Riemann Hypothesis, and subsequent work oninjectivity and rigidity of admissible configurations, a similar framework wasrecently applied to the Navier–Stokes regularity problem. The present workextends that programme to Hodge theory. Within this framework, a notion of admissible Hodge class is defined, capturingthe absence of persistent global incompatibilities between harmonicrepresentatives, intersection theory, and period pairings. To detectinadmissibility, the paper introduces abstract global coherence probes,designed to identify global obstructions that are invisible to local or purelyformal criteria. A rigidity principle is then established: admissible Hodge classes admit nopersistent global obstruction and are therefore forced to arise from algebraiccycles. Admissibility is shown not to be an external assumption, but a structuralcondition naturally enforced by the coherence inherent in Hodge theory itself. The contribution of this work is structural rather than constructive. No explicitcycle constructions or new analytic estimates are introduced. Instead, the paperclarifies the logical architecture underlying the Hodge Conjecture by isolatingglobal admissibility as the decisive condition and rigidity as the mechanism bywhich algebraicity is enforced. This perspective situates the Hodge Conjecture within a broaderadmissibility–rigidity programme spanning number theory, analysis, and geometry,and provides a unified structural pathway consistent with established theory.