This deposit contains the complete Lean 4 formalization of the HQIV (Horizon-Quantum Information Vacuum) framework together with its computational payoff: the OSH (Octonionic Sparse Horizon-causal) oracle. The core contribution is a discrete null-lattice model on the 3-D light cone. Nonnegative integer triples (x, y, z) with x+y+z=m are counted via stars-and-bars, lifted by an exact octonion factor of 8, and shown to force the curvature-imprint exponent α = 3/5 identically (proved via the hockey-stick identity). Octonions are realized as ℝ⁸ vectors with left-multiplication matrices that preserve the Euclidean inner product, yielding a fully norm-preserving digital quantum state space on the angular ladder ℋL of cardinality (L+1) ². All gates (phase flips, ℓ-preserving swaps, octonion left actions, rational Hadamard shells) are proved unitary by construction. The OSH oracle implements a sparse, horizon-causal register: each causal expansion doubles support while wrapping indices modulo |ℋL|; after any digital gate the flipped-support prune guarantees |prune (flip (R, UR) ) | ≤ 2|R| under the finite predicate plo (L, n) ≡ (n ≤ L). The bound is fully verified in Lean and supplies an exact digital hook for period-finding oracles (including Shor-style circuits) and for any energy-landscape evolution All combinatorial identities, norm-preservation lemmas, pruning bounds, and rational Fourier kernels are proved in the companion Lean modules (OctonionicLightCone. lean, OctonionBasics. lean, DigitalGates. lean, DiscreteSchrodinger. lean, OSHoracle. lean). This is version 1. 0. Future versions will add the polished AFM manuscript and further applications.
Steven Jr Ettinger (Wed,) studied this question.