Angular Hurwitz Equivalence: General Relativity and Quantum Mechanics as Adjacent Grades of a Single Octonionic Structure

We propose that general relativity (GR) and quantum mechanics (QM) represent physicallycomplete but angularly inequivalent descriptions of the same underlying octonionic structure,separated by exactly one 90-degree turn. We formalize the turn as a family of operators {Tu},one for each imaginary unit u in the Cayley–Dickson algebra, each recording the discrepancybetween two mutually admissible projection modes on the pair (u, −u): a unary truth-valueprojection (πAND) and a signed-polarity projection (πOR). The residue Tu(u2 + (−u)) = 2 is derivedfrom first principles as the failure of the Boolean idempotent equation A = A at the first non-Boolean integer. The seed z = 2 + i of the underlying Gaussian integer cascade is the uniqueminimal Gaussian integer whose real part is this residue and whose imaginary part is theperpendicular storing the sign ambiguity. GR operates at exterior algebra grade 2 (the F4 level,πAND projection) while QM operates at grade 3 (the E6 level, πOR projection). The number ofindependent turn operators at each Cayley–Dickson grade — 1, 3, 7 — coincides with thedimension of the imaginary subspace and with the rank structure forcing the exceptional Liealgebra chain. We show that e7 = 14 + 21 + 98 = 133 is the unique simple Lie algebracontaining both grades simultaneously.