[SMT-VOL8 & STCT-VOL4] HODGE THEORETIC PHASE TRANSITIONS VIA ROUGH OPERATOR ALGEBRA: TOPOLOGICAL CONFINEMENT AND ALGEBRAIC CYCLES

Abstract

The Hodge Conjecture seeks to identify the algebraic structure within the topological framework of complex projective manifolds. Traditional approaches have failed to construct algebraic cycles directly from harmonic forms due to the limitations of absolute smooth analysis. In this paper, we propose a rigorous resolution via Rough Operator Algebra (ROA) and theSeonggil Theory of Composite Torsion (STCT). We redefine Hodge classes not as smooth differential forms, but as elements of a Rough Current Space D′_α(X) parameterized by a geometric roughness index α. Introducing the Universal Arithmetic Friction constant η ≈ 10^−22, we define algebraic cycles as the unique ’topological condensates’ formed during the phase transition toabsolute smoothness (α → 1). By extending Siu’s Theorem into the non-commutative domain, we prove that the integrality of the Lelong number emerges as a strict quantization effect. This quantization is enforced by the Dolbeault non-commutative residual scaled by η. We establish that any non-algebraic cycle, including non-(p,p) forms, induces an infinite entropy divergence due to chiral imbalance and non-commutative friction, forcing a ”Topological Confinement” into rectifiable, positive algebraic skeletons. Thus, the algebraic structure of Hodge classes is proven as an inevitable geometric consequence of Roughness Symmetry Breaking.