A Solvency Theorem for Adaptive Networks: Energetic Cost of Enforced Decoupling v3

This work presents a formal solvency theorem for adaptive networks governed by an information–thermodynamic coherence functional. The paper defines a dimensionless coherence ratio C(W)C(W)C(W) measuring ordering (Minimum Description Length reduction) relative to maintenance entropy (Landauer-normalized), and establishes a Lyapunov-based solvency threshold C∗C^*C∗. The central result proves that enforcing strict conditional independence between intrinsically coupled observables incurs a minimum hidden-state entropy cost proportional to conditional mutual information. Under sustained enforcement, this entropy cost reduces the coherence functional below its solvency threshold, violating bounded operation. The theorem is model-independent within first-order linearized neighborhoods of the operating manifold and applies to bounded adaptive systems operating under fixed encoding classes. The result connects Shannon mutual information, Landauer’s thermodynamic erasure bound, and Lyapunov stability into a single solvency constraint. No metaphysical or cosmological claims are made. The theorem addresses structural stability properties of adaptive networks under enforced decoupling and establishes energetic limits on sustained independence constraints. A companion empirical illustration (neural-network multi-seed sweep) is available separately.Keywords#adaptive_networks #complex_systems #information_thermodynamics #mutual_information #Landauer_principle #Lyapunov_stability #solvency_threshold #minimum_description_length #conditional_independence #entropy_cost #network_decoupling #bounded_dynamical_systems #coherence_functional #stability_analysis #information_theory #control_theory #energy_information_tradeoff Version: v3License: CC BY 4.0ORCID: 0009-0004-2830-4089