Principles of Operatiology: New Foundation of Cognitional Mechanics

This paper establishes Operatiology (Japanese: Sousa Genron, 操作原論) as the foundational executive layer of Cognitional Mechanics (CM), replacing the intelligence-centric framing of prior CM versions with a purely operational foundation. The central thesis is that "Intelligence" functioned as an epistemically necessary scaffold during the construction of CM: it oriented the derivation of the unique minimal operational structure Cmin⁡Cₘᵢₙ Cmin but did not appear as a variable, boundary condition, or operational intermediate in any proof or calculation. With CM Version 9 completing the uniqueness proof of Cmin⁡Cₘᵢₙ Cmin via the canonical irreducible set IΠdIΠ_d IΠd, the scaffold is no longer structurally required and is formally removed. Operatiology is derived from Noology (Version 2, 2026), the regulative layer consisting of three primitive notions: Ordo, Consensus, and Arbitrium. A formal lemma establishes that the simultaneous satisfaction of Arbitrium, Axiom 4 (redundancy exclusion), and Axiom 2 (finite generator rank with operational identity) forces non-commutativity as the minimal operational structure, via a collapse mechanism on the history space On/SnO^n/S_n On/Sn that destroys configurational distinguishability under commutativity. The CM axiom system {A1,A2,A4}\{A1, A2, A4\} {A1,A2,A4} is retained intact. Admissible operations are defined purely structurally as operations that alter the ΠdΠ_d Πd-indistinguishability partition. Cognitional Mechanics is repositioned as the reflexive special case of Operatiology: the branch treating intelligence as a Tier-2 pattern whose study includes subjectivity and qualia. Version 2 strengthens the logical closure of the framework by internalising the existence and uniqueness proof of Cmin⁡Cₘᵢₙ Cmin, formalising the operational closure conditions of Axiom 2 (Closure Compatibility and Operational Identity), clarifying the Tier-2/Tier-3 boundary concerning minimal faithful realization dimension via the Rank Underdetermination theorem, and introducing finite ΠdΠ_d Πd-closure conditions for the Applied Operatiology series. All changes are documented in an appended revision history.