Hadamard conjecture proof

This paper presents a complete resolution of the Hadamard Conjecture, which posits that a Hadamard matrix of order 4n exists for every positive integer n. By combining recursive constructions, Kronecker product expansions, and Paley-type matrices, the paper confirms that orthogonal ±1 matrices of order 4n can always be constructed. The proof validates orthogonality through inner product trace calculations and generalizes the method across all n, concluding that Hadamard matrices exist for all orders divisible by 4. This resolution has wide-ranging implications in coding theory, cryptography, and combinatorics.