Hierarchical Emergence, Compression, and Observer-Dependent Structure Formation: A Rigorous Reformulation Grounded in the Universal Balance-Feedback Framework

Abstract We present a rigorously revised dynamical systems framework that formally grounds hierarchical emergence, information compression, and observer formation within the Universal Balance-Feedback Framework (UBFF). The UBFF comprises four universal laws: (I) System Integrity / Karma, (II) Universal Balance in Nature, (III) Universal Feedback Loop Mechanism, and (IV) Universal Interconnected Nodes. We show that these laws map precisely onto the mathematical structure of a coupled nonlinear stochastic system on a complete metric space, and that their interaction produces nested invariant attractors spanning physical, biological, cognitive, social, and symbolic regimes. We strengthen the original proof sketch by invoking the Krylov-Bogoliubov theorem for invariant measures, Tikhonov's theorem for timescale separation, and the Banach fixed-point theorem for observer-compression closure. We introduce a stable bounded memory field with exponential decay, correct the unbounded accumulation flaw in earlier formulations, and provide a concrete worked example in neural population dynamics that demonstrates the attractor hierarchy empirically. We further situate the framework relative to Friston's Free Energy Principle, Hoel's Causal Emergence, and Wolfram's computational irreducibility.