Principles of Operatiology: New Foundation of Cognitional Mechanics

This paper presents Version 5 of Operatiology (Japanese: Sousa Genron, 操作原論), the substrate-independent foundational executive layer of Cognitional Mechanics (CM; Japanese: Chinou Rikigaku, 知能力学). Version 5 is not a full redefinition of Version 4: it preserves the upper structure of Version 4 intact — the transformation monoid (O, ◦, e), the minimal operational closure C_min, and the Tier-2/Tier-3 separation — while correcting five internal defects identified in that version and introducing one controlled local extension required to express Structural Leap within a type-safe boundary. The five defects are: (1) the identity-collapse of Π_G under the original Definition 3.1 (the identity e ∈ ⟨G⟩ forced Π_G = Π_disc); (2) the absence of e ∈ C from the admissibility condition, which broke Lemma 7.8; (3) the structural incompatibility between Definition 9.1 (Π_Gref = Π_O) and Definition 8.4(iii) (Π_G′ ≺_ref Π_Gref) when g_new ∈ O; (4) the degenerate case Π_O = Π_disc making Structural Leap condition (iii) impossible in principle; and (5) the collapse of the proof of Theorem 8.1 under the corrected Definition 3.1. Note: Definition 3.5 of Version 4 already used the correct notation Π_G∪{g}; no correction was required there. The corrected version employs the effective operational history H⁺_G (excluding the identity from the quantification in ∼_G), adds e ∈ C to the admissibility condition, restricts g_new to Maps(S, S) \ O with a type firewall confining the local extension to Definition 8.4, adds O2(iii) excluding Π_O = Π_disc, and strengthens Coupling Persistence from ≁_O to ≠. All structures of Version 4 not affected by these corrections — O1, Cl(G), I_op, G_ref, the Tier-3 projection, EP, and all reading maps — are preserved without modification. A reading map from Version 4 to Version 5 is provided in Appendix C. Note: The following substitutions are used solely for plain-text compatibility and to avoid PhilPapers URL false-positives. No algorithmic or semantic modification has been made. ×＝, ^＝**, ．＝., ：＝:, ＿＝_, −＝-, ！＝!=. Additionally, to avoid URL false-positives, function-call parentheses are replaced with full-width brackets: append［c］, elements［…］, new_f［…］, frontier［…］. ◆◆◆ PYTHON ILLUSTRATION CODE ◆◆◆ def compose（f，g）： """Composition f o g.""" return {s： f［g［s］］ for s in g} def identity(states): return {s: s for s in states} def submonoid(states, generators): """Generate all elements of by iterated composition.""" e = identity(states) elements = [e] frontier = [e] while frontier: new_f = [] for w in frontier: for g in generators: c = compose(g, w) if c not in elements: elements［append］ c new_f［append］ c frontier = new_f return elements states = [0, 1, 2, 3, 4] A = {0:1, 1:2, 2:2, 3:3, 4:4} B = {0:3, 1:1, 2:2, 3:3, 4:4} AB = compose(A, B) BA = compose(B, A) witnesses = [s for s in states if AB[s] ！＝ BA[s]] s_init = witnesses[0] s0 = AB[s_init] s_star = BA[s_init] print(f"O1: A and B commute? {len(witnesses)==0}") print(f"Witness s={s_init}: " f"A o B({s_init})={s0}, B o A({s_init})={s_star}") print(f"=> s0={s0} != s*={s_star} [O1 satisfied]") all_words = submonoid(states, [A, B]) e = identity(states) non_id = [w for w in all_words if w != e] persistence = all(w[s0] != s_star for w in non_id) print(f"\nCoupling Persistence (ii'):") print(f" || = {len(all_words)}, " f"| \ {{e}}| = {len(non_id)}") print(f" All P(s0) != s: {persistence}") print(f" => s is inaccessible from s