Hierarchical Constraint Processing: An Axiomatic Theory of Adaptive Failure Modes

We present a formal axiomatic theory of failure modes in hierarchically organized adaptive systems. From five axioms — finite capacity, energetic dependence, hierarchical constraint, threat sensitivity, and structural incompleteness — we derive three fundamental failure modes: collapse (progressive loss of hierarchical constraint under overload), freeze (coherence-preserving lock against revision), and lag failure (temporal misalignment of correction). We prove that collapse and freeze are mechanistically dual, characterized by opposite signatures in revision entropy H(E) with equivalent loss of revision validity V(E), and that these duals require opposite interventions — a result not derivable from either failure mode alone. We derive the freeze threshold as a function of available capacity, showing that the dangerous regime (simultaneous collapse and freeze) arises from a single formal mechanism. We connect the incompleteness axiom to Bennett and Suzuki's Autopoietic Theorem and to Shannon's source coding theorem, grounding the structural necessity of remainder. The theory generates eight falsifiable predictions distinguishing it from existing frameworks.