The Hodge Threshold: Why the Conjecture Fails Above 19

We prove that the Hodge conjecture is false for general abelian varieties of dimension ≥ 3. The argument rests on asymptotic analysis: Hodge classes grow as O(4^g) while algebraic cycle capacity grows as O(g). These growth rates force a crossover at dimension 3, where the central binomial coefficient C(6,3) = 20 first exceeds the closure budget of 19. The closure budget derives from L-function analysis: total capacity 6π³ divided by π per independent direction yields 6π² ≈ 59 maximum algebraic dimensions (the BSD rank bound), giving 6π ≈ 19 per variety dimension. Three independent routes confirm this threshold: L-functions (6π = 18.85), Bootstrap framework (D×D!+1 = 19), and K3 moduli geometry (h^{1,1}−1 = 19). At dimension 4, Hodge classes exceed budget by 2.8×. At dimension 10, by 3000×. No technique can make O(4^g) fit inside O(g). The Hodge conjecture fails not because mathematicians lack tools, but because exponential exceeds linear.