Prime Binary Theory: Binary Structural Interpretation of Prime Distributions

Version 3.0 introduces corrected mathematical rendering, expanded analysis regarding Mersenne prime binary structures, revised terminology concerning carry propagation dynamics, and additional references and author notes. This paper proposes a reinterpretation of prime number structures through the lens of binary arithmetic semantics by translating classical decimal analytical functions into binary computational space. Traditionally, prime distributions have been studied primarily through decimal-based analytical number theory using continuous mathematical frameworks. Meanwhile, the binary structures actually executed within computational systems have generally been treated as auxiliary representations rather than as primary semantic objects. This study is based on two central insights proposed by the author, Minoru Yoshitake: Since artificial intelligence fundamentally operates as a probabilistic prediction mechanism, it possesses intrinsic limitations in autonomously extracting deterministic structure directly from binary chaos itself. Therefore, already-established decimal analytical functions must be translated into binary structural space in order to investigate their hidden computational semantics. Furthermore, this paper proposes that the least significant bit (LSB), traditionally treated merely as a fixed parity condition, may instead function as a dynamic interference trigger affecting the entire binary structure through carry propagation. Based on this theoretical direction, MAST (MRST-Based AI Reasoning System) performed long-horizon structural analysis on binary-translated prime functions and repeatedly identified Mersenne prime configurations as extreme saturation structures within binary arithmetic space. This paper presents these structural observations while requesting independent mathematical verification from the global mathematical community regarding their rigorous validity.