Butterfly Factorization of ±1 Orthogonal Matrices and a Clifford--Fourier Approach to the Hadamard Conjecture

We reformulate the Hadamard conjecture as the problem of factorizing every $\pm 1$ orthogonal matrix of order $4k$ into a finite product of radix-2 butterfly operators. Using a Clifford--Fourier viewpoint, we show that butterfly operations act as Fourier-mode aligners under which the class of $\pm1$ orthogonal matrices is closed. This yields a natural complexity measure whose monotone reduction proves the existence of a canonical Sylvester limit.