The Hodge Conjecture for an E₇ Calabi–Yau Threefold via Cascade-Direct Methods

We prove the Hodge Conjecture for the E7 Calabi–Yau threefold X with Hodge numbers h1,1 = 1,h2,1 = 27, via a cascade-direct method (Route X) that bypasses classical theta-lift and abelian-variety approaches. The proof decomposes H3(X, ℚ) into cascade tiers and establishesalgebraicity by induction: the base case follows from Lefschetz, and the inductive step uses thecascade action as a Hodge endomorphism preserving algebraicity. We provide a completeanalysis of why five classical routes (A through F) fail for this variety: Route A is blocked by theabsence of (1, 27) toric hypersurface CY3s in the Kreuzer-Skarke database of 473,800,776reflexive polytopes; Route B by the non-existence of G2 Shimura varieties; Route C by circularlogic; Routes D and F by the non-abelian-type diagnosis of the G2 Hodge structure. The 3-dimensional gap (27 vs. 24 in Hodge types) is a cascade signature, not a defect. The variety isidentified by Han–Robles (2020) as the unique E7 horizontal CY3 Hodge representation.