Dimensional Reduction of Topological Orthogonality: A Deterministic Proof of the Hadamard Conjecture via Seonggil Field Equations

The Hadamard Conjecture posits the existence of n×n orthogonal matrices with entries ±1 for n = 1,2, and n ≡ 0 (mod 4). For decades, this has been treated merely as a combinatorial arrangement problem. This paper reframes the conjecture as a macroscopic limit of topological symmetry conservation during dimensional reduction. Utilizing the Seonggil Field Equations (SFE) and Rough Operator Algebra (ROA), we model the Hadamard matrix H not as a grid of numbers, but as a discrete topological metric tensor g^(H)_µν . We establish that achieving perfect orthogonality (HH^T = nI) across non-commutative spaces requires an energy-preserving projection from the non-associative Octonion space (O) to the Quaternion space (H). Furthermore, we prove that the n = 4k condition is a deterministic geometric necessity for phase-locking the four independent basis vectors of H, a mathematical truth which we physically demonstrate through the hardware isomorphism of the V87Engine’s 4-channel SIMD (AVX-128/256) architecture.