A Three-Mode Utility-Hysteretic Hybrid Regulator: A Conditional Practical ISS-to-Set Theorem for the ISRM Skeleton (v.1.4)

A Three-Mode Utility-Hysteretic Hybrid Regulator:A Conditional Practical ISS-to-Set Theorem for the ISRM Skeleton (v1.4) Abstract: This note formulates an Interactionist Self-Regulation Model (ISRM)-motivated three-mode regulator as a projected hybrid/switched system and states conditions under which it satisfies a practical input-to-state stability result relative to a compact residual set. The continuous state x = (E, S, C) ∈ X = [0,1] × [−M_S, M_S] × [−M_C, M_C] represents energy, stress deviation, and cognitive-load deviation. The discrete mode q ∈ {Conserve, Recalibrate, Proceed} is governed by utility hysteresis on U(x) = αE − βS − γC, with an enforced Recalibrate dwell gate. We do not claim a new hybrid-systems theorem and do not claim that utility itself proves stability. The utility variable schedules switching and supplies residence-time information; stability is carried by a quadratic Lyapunov function W, projected mode-wise Lyapunov inequalities, a truncated Lyapunov argument outside a residual set, and an average Lyapunov-exponent dominance condition over actual macro-cycles. The main theorem is intentionally conditional: if the projected hybrid data are well posed, the residual set is guarded so that non-Recalibrate modes are inactive inside it, the disturbance bound satisfies |d(t)| ≤ D ≤ D̄ a.e., mode-wise inequalities hold for the implemented hysteresis controller, and exterior residence times satisfy an average exponent dominance condition, then the system is forward invariant, non-Zeno, forward complete, and practically ISS at macro-cycle times with respect to A₀ := {x ∈ X : W(x) ≤ W₀}. A full hybrid-time ISS estimate follows under explicit bounded-excursion regularity. A worked feasibility certificate establishes that the assumptions are simultaneously satisfiable. The result should be read as a rigorous conditional control-theoretic skeleton for this ISRM-style regulator, not as empirical validation of ISRM. Version history. v1.4 (May 2026): The mathematical claim was scoped to what's actually defensible; practical ISS to a compact residual set rather than to a point, with utility U cleanly separated from the Lyapunov function W and all hypotheses made explicit and checkable. The ISRM model itself (three modes, tri-signal utility, hysteresis switching, dynamics) is unchanged; only the formal claim built on top of it tightened.v1.3 (Oct 2025): Added complete Lyapunov-based hybrid stability proof, explicit ISS constants, constructive dwell-time bounds, and corrected Proceed-mode descent inequality. Integrates all prior derivations into a unified mathematical framework.v1.2 (Oct 2025): Major revision addressing peer review, added axiomatic derivation of multiplicative form, generalized ISS to class-( K_∞), provided empirical cortical circuit example.v1.0 (Sept 2025): First public preprint release of The Inevitability of ISRM.