Major Conjectures and Cosmological Tests L: The Cosmological Falsifiability of the Hodge Conjecture

This paper proposes an innovative interdisciplinary research framework thatconnects the pure mathematical Hodge Conjecture with observations of the largescale structure of the physical universe. By introducing the axioms of informationconservation and computability, we translate the duality between rational cohomology classes and algebraic cycles on complex projective algebraic varieties, asdescribed by the Hodge Conjecture, into observable features of the cosmic webstructure under multi-scale topological analysis. The theoretical prediction indicates that if the geometry of the universe obeys a certain ”Hodge-type” rationalconstraint, then the length distribution of the persistent homology barcodes of itsstructure should exhibit a clustering pattern in integer multiples of a fixed fundamental scale (L0 ≈ 2.1 Mpc/h), and the number distribution of topological featuresacross different dimensions should satisfy specific number-theoretic constraints. Using high-fidelity simulation data from the DESI and Euclid surveys for testing, wedetect a significant discrete clustering signal in the length distribution consistentwith the prediction. The core contribution of this paper lies in constructing aclear, operational, and falsifiable cosmological testing path for this highly abstractmathematical conjecture.