Foundations of Cognitional Mechanics

This paper establishes the foundational axiomatic framework of Cognitional Mechanics (Japanese: Chinou Rikigaku) —a pure theoretical system that formalizes the structural mechanisms inherent to intelligence through mathematical abstraction. Unlike conventional approaches rooted in physical dynamics or engineering implementations, Cognitional Mechanics constructs a self-contained theory based on three independent foundational axioms: (1) non-commutative semantic operations, where the order of operations determines meaning; (2) a complete ordered metric structure on semantic states, defined over an abstract totally ordered Abelian group DD D without presupposing the real numbers; and (3) a redundancy exclusion principle enforcing minimal description. A fourth principle, the operational bound, is retained as Working Axiom 3 for notational continuity with the established corpus; it is a derived corollary of the three foundational axioms, as established by the Internal Language Theorem (DOI: 10.5281/zenodo.19350315). By introducing a rigorous axiomatic structure—including a metric space of semantic states with defined distance functions satisfying non-negativity, symmetry, and triangle inequality—this framework transcends subjective interpretations of intelligence and establishes it as an object of formal, mathematical inquiry. The theory absorbs conventional criticisms regarding measurability by positioning the operational limit c as an abstract logical boundary rather than a numerical constant, thereby maintaining theoretical self-containment without dependence on external observation or physical instantiation. Cognitional Mechanics provides a foundation for understanding intelligence-specific phenomena such as logical conflicts arising from non-commutative operation sequences, convergence and divergence in meaning formation, and the emergence of fundamentally unreachable semantic states. This work serves as a foundational reference for subsequent investigations into concrete distance metrics, structural correspondences with quantum mechanical formalisms, and applications to artificial intelligence architectures. The theory is presented as a complete, closed deductive system, independent of external frameworks, and axiomatizing the operational structure of intelligence as a subject of pure mathematical mechanics.