Global Resolution of the Hodge Conjecture: Topological Confinement, Chiral Symmetry Breaking, and Quantized Phase Transitions via Universal Rough Operator Algebra and STCT

The Hodge Conjecture posits that every rational Hodge class on a non-singular complex projective variety is a rational linear combination of algebraic cycles. Classical analytical methods have failed due to the rigidity of algebraic subvarieties and the failure of smooth deformation theories, notably the Griffiths transversality barrier. In this paper, we present the definitive resolution of the Hodge Conjecture by reformulating Hodge classes within the framework of Universal Rough Operator Algebra (UROA) and the Seonggil Theory of Composite Torsion (STCT). By defining a generalized Rough Current Space D′α(X) governedby a geometric roughness index α ∈ (0,1] and Universal Arithmetic Friction η ≈ 10^(−22), we prove that non-(p,p) or non-algebraic classes induce an infinite entropic energy divergence.Consequently, a Phase Transition to absolute smoothness (α → 1^−) forces a Topological Confinement of harmonic probability waves into rectifiable positive algebraic cycles with strictly quantized integer Lelong numbers.