Hodge Classes on Self-Powers of Split Weil Eightfolds A Conditional Dimension-Eight Extension

Recent work establishes three geometric inputs that areunusually well matched in dimension eight: algebraicityof the full Weil plane on every split Weil-type abelianeightfold, the rational Hodge conjecture for complex CMabelian varieties, and algebraic quadratic spin-contractiontensors arising from Kuga–Satake constructions. Assumingthese inputs, we prove a dimension-eight extension:every rational Hodge class on every self-power of a splitWeil-type abelian eightfold is algebraic. The new workis representation-theoretic and special-period in nature.We establish an eight-dimensional determinant-supplylemma, a special-unitary spin-completion mechanism forbalanced four-dimensional determinant words, algebraicclosure of the three exceptional minuscule derived blocksA31, A1C2, and Bspin3 , and a rank-two graph closure forresidual SL2 identifications. An exhaustive Albert Type-IV audit reduces the arithmetic possibilities to fifteengrouped cases; an exact quartic Galois calculation showsthat only the Klein-four case contributes an extra determinantdirection, which is again supplied by the spin-boxmechanism. The theorem is conditional on the statedgeometric inputs and is not a solution of the generalHodge conjecture.