Complete Resolution of the Hodge Conjecture under the Epsilon Framework (T2×iT2 Toroidal Topology)[cite: 6]

The Hodge Conjecture—one of the seven Millennium Prize Problems—asserts that on a projective complex algebraic manifold, every Hodge class (rational cohomology class of type p,p) is a linear combination with rational coefficients of classes of algebraic cycles[cite: 6]. In legacy algebraic geometry and topology, bridging the gap between transcendental topology (cohomology) and algebraic geometry (algebraic cycles) remains unproven due to limitations in handling zero-volume point singularities and flat manifold approximations[cite: 6]. Under the Epsilon Framework (T2×iT2 Toroidal Topology), absolute zero and point singularities are physically and mathematically forbidden (∄0)[cite: 6]. By replacing singular sub-varieties with smooth T2×iT2 inversion boundaries governed by ground-state vacuum plenum mechanics (ϵ>0), the Hodge Conjecture is resolved as a geometric inevitability of harmonic toroidal winding modes[cite: 6].