Dynamics of Operatiology: Operational Reconfiguration, Structural Control, and the Necessity of Logical Leaps

This paper establishes Version 5 of the dynamic extension of the axiomatic framework of Operatiology (Japanese: Sousa Genron, 操作原論) and its Tier-3 realization, Cognitional Mechanics (CM; Japanese: Chinou Rikigaku, 知能力学). Version 5 updates the foundational reference from Operatiology Version 4 to Operatiology Version 5, incorporating two fixes that bear directly on the Dynamics paper. Fix3 directly requires g_new ∈ Maps(S,S) \ O as a type-level constraint in the definition of Structural Leap (Operatiology Version 5, Definition 8.5), rather than treating this as a downstream consequence of other conditions. Fix6 adds Π_O ≠ Π_disc as an explicit axiom condition O2(iii), ensuring that Mode I describes operational equilibrium within a genuinely non-trivial partition structure. The present version supersedes Version 4, which was grounded in Operatiology Version 4. The direction of dependence is Operatiology → Dynamics, not Cognitional Mechanics → Dynamics. The type constraint g_new ∈ Maps(S,S) \ O is now incorporated at the definition level of the Dynamical Structural Leap, the SC* scope condition, and the Theorem hypothesis. The Mode II exclusion argument is reformulated accordingly: since g_new ∈ Maps(S,S) \ O, the augmented generator set G' = G_ref ∪ {g_new} satisfies G' ⊄ O, which directly violates the domain condition G ⊆ O of Mode II. This type-level argument is logically prior to any reachability computation and supersedes the prior reachability-based exclusion used in Version 4. The DynV5 extension of the operational-history relation ∼_G and partition Π_G to generator sets G ⊆ Maps(S,S) is given by explicit definition: s ∼_G t iff ∀w ∈ ⟨G⟩: w(s) = w(t), Π_G := S/∼_G. When G ⊆ O, this coincides with Operatiology Version 5, Definition 3.2. Three dynamic modes are formulated entirely in the vocabulary of Operatiology Version 5: Mode I (Π_G-maximality, within a non-trivial partition structure guaranteed by O2(iii)), Mode II (partition change without inaccessible domain entry), and Mode III (Dynamical Structural Leap, via inaccessible domain entry). On SC*-systems, Logical Leaps are structurally necessary occurrences of Dynamical Structural Leaps. Representative code (Mode III / W-Event: Structural Leap into Inaccessible Domain): from collections import defaultdict def operational_partition(states, ops): cls_id = {s: 0 for s in states} while True: key = {s: (cls_id[s], tuple(cls_id[op[s]] for op in ops)) for s in states} sig_to_cls = {}; new_id = {}; ctr = 0 for s in states: k = key[s] if k not in sig_to_cls: sig_to_cls[k] = ctr; ctr += 1 new_id[s] = sig_to_cls[k] if new_id == cls_id: break cls_id = new_id part = defaultdict(list) for s, c in cls_id.items(): part[c].append(s) return frozenset(frozenset(v) for v in part.values()) def is_strictly_finer(pi_fine, pi_coarse): if pi_fine == pi_coarse: return False for cls_fine in pi_fine: if not any(cls_fine