Unified Determinant-Lock Theory for Class Number One Quadratic Rings: From Gauss to Half-Integers

This work is about the 'Hydra Method,' a novel and efficient mathematical approach for the factorization analysis of Fermat numbers. Instead of relying on classical division or GCD algorithms, it utilizes a mandatory 'determinant lock' mechanism within the number's geometric structure to directly construct the cofactor via a closed-form formula