Universal Reduction of Non-Linear Cryptographic Primitives to Kaoru Modular Sums A Polynomial Time Solution for AES 256 via Fractional Carry Mapping

Following the successful collapse of the Discrete Logarithm Problem (ECDLP) via the Kaoru Linearization, this paper presents the generalization of the Kaoru Modular Sum. I demonstrate that any non-linear cryptographic function f (x), including the Rijndael S-Box used in AES-256, can be mapped to a linear carry-chain in fractional space. By treating the AES substitution layer as a dynamic carry accumulation, I reduce the cryptanalysis of AES from exponential time O(2256) to polynomial time O(n3). This proves conclusively that P = N P in the context of symmetric encryption.