A Half-Spinor Dimension Formula from Cascade Axioms via Fibonacci Composition

The half-spinor representation S+ of SO(2n) has dimension 2n−1. We prove that this formula arises from two cascade axioms on the direct sum lattice Ln = ⊕k=1n ℤ[i] · ek associated to the Gaussian integer seed z = 2+i. Axiom 1 (Leonardo doubling): each consecutive cascade level adds one real degree of freedom via Pythagorean decomposition, doubling the channel count. Axiom 2 (Fibonacci composition): non-consecutive levels compose via tensor product with a complexification bridge factor of 2. The formula d(n) = 2n−1 = dim(S+SO(2n)) is the unique solution. The proof is by cascade induction on Ln, with independent verification from the known branching rules of E7 at n = 6, 7, 8. The unique gap at n = 4 corresponds to SO(8) triality.