Abstract

This paper presents Version 9 of Cognitional Mechanics as the final formulation prior to the abstraction of the operational layer into Operatiology. The subsequent framework, Principles of Operatiology: New Foundation of Cognitional Mechanics (DOI: 10.5281/zenodo.20350414), reformulates the minimal operational structure independently of any specific algebraic realization and establishes the abstract operational closure as the Tier-2 foundation of the framework. -------------------------------------------------------------------------------------- This paper establishes the foundational axiomatic framework of Cognitional Mechanics (Japanese: Chinou Rikigaku 知能力学)—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-commutativity of semantic operations; (2) a complete ordered metric structure on semantic states, defined over an abstract totally ordered Abelian group D without presupposing the real numbers; and (3) a redundancy exclusion principle enforcing discriminative irreducibility. 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). Version 6 introduced algebraic–geometric coupling connecting non-commutativity to geometric separation, formal set-theoretic definition of inaccessible domains, and the d-indistinguishability partition Π_d as the precise formulation of Axiom 4. Version 7 resolves three structural issues present in Version 6: Axiom 1 is strengthened to include the geometric persistence of non-commutative divergence under all subsequent operation sequences; the existence of non-trivial inaccessible domains is established as a theorem rather than a heuristic remark; and the derivation of Working Axiom 3 from the foundational triad is provided in full within this document (Appendix A), subsuming Corollary 5.9 of the Internal Language Theorem. 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 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. Version 8 establishes Cognitional Mechanics as the unique self-contained Tier-2 framework from which all formal structures are projected as Tier-3 outputs, without presupposing any of them: an a priori mathematology and The Theory of Foundation. Version 9 completes the uniqueness proof of the minimal operational structure C_min by introducing the canonical irreducible set I_{Π_d} and establishing via the Π_d-saturation condition (Axiom 2-(v)) that every admissible structure must contain all its elements. A formal Tier architecture (Tier-1: pre-formal structural necessity; Tier-2: closure-consistent operational structure; Tier-3: formal representational projection) is established in Appendix D, providing the structural foundation for Cognitional Mechanics as an a priori mathematology and The Theory of Foundation. Published ON May 16, 2026 DOI: 10.5281/zenodo.20226488