Abstract

This paper establishes Operatiology (Japanese: Sousa Genron, 操作原論) as the foundational executive layer of Cognitional Mechanics (CM; Japanese: Chinou Rikigaku, 知能力学), derived from Noology (Japanese: Chigaku, 知学; Version 2, 2026). Operatiology answers a fundamental question: what is the minimal structure any system must have in order to make distinctions at all? Starting from three primitive notions of Noology — Ordo (order-sensitivity), Consensus (internal consistency), and Arbitrium (configurational freedom) — the paper derives a unique minimal operational structure C_min governed by three axioms: non-commutativity (A1), finite generator rank with saturation closure (A2), and redundancy exclusion (A4). The key result is that commutativity of all operations would collapse the history space O^n/S_n, destroying the configurational trace required by Arbitrium. Non-commutativity is therefore not assumed but derived as the minimal structural response. Version 3 adds Lemma 3.19 (Word Relation Describability Constraint): the word relations of C_min cannot contain negative-sign expressions such as g^n(s) = -g(s), because the axiom system imposes no additive structure on the state space. This closes a logical gap between the Tier-2 operational structure and its algebraic realisation. Version 3 supersedes Version 2 as the authoritative reference. Note: ×＝，^＝**，．＝.，：＝:，＿＝_，−＝-，！＝!=．The following substitutions are used solely for plain-text compatibility in this description. No algorithmic or semantic modification has been made. ◆◆◆ ［PYTHON ILLUSTRATION CODE — note：×＝，^＝**，．＝.，：＝:，＿＝_，−＝-，！＝!=］ ◆◆◆ from collections import defaultdict def compose（f，g）： return {s： f［g［s］］ for s in g} states ＝ ［0，1，2，3］ A ＝ {0：1，1：2，2：3，3：0} B ＝ {0：1，1：0，2：2，3：3} AB ＝ compose（A，B） BA ＝ compose（B，A） witnesses ＝ ［s for s in states if AB［s］ ！＝ BA［s］］ s0 ＝ witnesses［0］ print（f"A and B commute? {len（witnesses）＝＝0}"） print（f"Witness s＝{s0}： A o B（{s0}）＝{AB［s0］}，B o A（{s0}）＝{BA［s0］}"） C ＝ D ＝ {0：1，1：2，2：3，3：0} print（f"C o D ＝＝ D o C： {compose（C，D）＝＝compose（D，C）}"） print（"All orderings collapse −− Arbitrium violated．"） ◆◆◆ ［END］ ◆◆◆ Published on July 21, 2026. DOI: 10.5281/zenodo.21471088