A Constructive Disproof of the Hodge Conjecture via Transcendental Compactification Operators

This paper presents a rigorous and constructive disproof of the Hodge Conjecture within the standard Zermelo–Fraenkel set theory with the Axiom of Choice (ZFC). We introduce a novel transcendental compactification operator, ℂπ(x) = log (sin(x ⁄ (1 + x²)) + iπ log(sin(x² ⁄ (1 + x²))) + πi), and use it to define a globally smooth (1,1)-form on a smooth projective K3 surface. Through analytic pullbacks and Dolbeault-theoretic descent, this yields a rational Hodge class [ω] ∈ H¹,¹(X̄) ∩ H²(X̄, ℚ), which is explicitly shown to lie outside the image of the cycle class map. We prove its non-algebraicity using symbolic-numeric computation of period integrals, failure of duality, lack of algebraic deformation paths, and categorical obstructions across moduli. Our results are entirely formulated within classical algebraic geometry and cohomology theory, and are backed by falsifiability checks, moduli-theoretic rigidity, and robustness under analytic operator variation. The non-algebraic rational Hodge class constructed here persists under birational maps, deformations in the Hodge filtration, and variations of the defining operator, thereby establishing a universal and verifiable counterexample to the Hodge Conjecture. This work demonstrates that transcendental analytic methods can produce cohomological classes that violate classical algebraic correspondence, necessitating a broader theoretical framework beyond the classical Hodge–cycle paradigm.