Consumption Types and Dynamic Regimes in Constrained Generative Systems: Refinement of the Irreversibility Axiom and SCC Classification of Asymptotic Behavior

We refine the Irreversibility Axiom (Axiom 2) of the Constrained Generative Systems framework by distinguishing three consumption types: node-consumptive (T1), arc-consumptive (T2), and trajectory-consumptive (T3), according to what structural element is removed or accumulated by the consumption function. We then classify all possible asymptotic behaviors of trajectories in a CGS with finite state space into exactly four dynamic regimes — deterministic cycle (R1), branched attractor (R2), fixed point (R3), and finite acyclic trajectory (R4) — via the strongly connected component (SCC) decomposition of the transition graph. The classification is proved complete and minimal: no fifth regime exists, and no regime is reducible to another.

We establish that regime R2 (branched attractor) is accessible only in trajectory-consumptive systems, where the consumption function records history without removing nodes or arcs from the transition graph. This result identifies a structural boundary between systems that must terminate or cycle deterministically (node- and arc-consumptive) and systems that admit persistent internal choice (trajectory-consumptive).

We prove a forced finiteness theorem: in any CGS whose action alphabet is derived from a continuous group under a distinguishability threshold L > 0, the number of operationally distinct classes is bounded by ⌈1/L⌉, independently of the algebraic structure of the generating group. This extends Axiom 1 from systems with intrinsically finite alphabets to systems where finiteness is forced by the limit.

We verify that the four structural properties of sub-limit dynamics hold under all four regimes, with regime-specific manifestations. In particular, saturation in R2 is asymptotic (convergence of the visit distribution) rather than terminal (exhaustion of capacity), and the efficiency paradox manifests as topological confinement rather than premature termination.