The Stochastic Internal-Loop Directed-Information Bound with Continuous-Time Companion: The Stochastic Internal-Loop Invariance Entropy Bound (SILEB) Bridging Stochastic Invariance Entropy and Control-Information Lower Bounds in Closed Self-Referential Systems

/The Stochastic Internal-Loop Directed-Information Bound with Continuous-Time Companion: The Stochastic Internal-Loop Invariance Entropy Bound (SILEB) Bridging Stochastic Invariance Entropy and Control-Information Lower Bounds in Closed Self-Referential Systems

Abstract

Bridge theorems closing the final gap in the Ashby–Landauer–Friston Viability Bound for closed self-stabilizing systems. Part I (SIDI) is the primary discrete-time result with full proof. Part II (SILEB) is the continuous-time Itô companion (citation-audit pending). Together they complete the ALF chain for both discrete and continuous regimes

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