A Unified Framework via Degrees of Freedom and Structural Collapse

We present a unified framework connecting logical correctness, adversarial robustness, and dynamic stability. Using a minimal language based on states, rule closure, break structures, and system dynamics, we show that these notions coincide under precise structural conditions. We prove that a state is canonical if and only if it is unbreakable and dynamically stable, provided the system admits full certificate coverage and contains no unavoidable external inputs. This equivalence is characterized by the vanishing of all degrees of freedom---representation, source, and dynamic---and is not assumed but arises as a structural consequence. We identify the exact boundary at which this collapse fails and develop a quantitative robustness theory for open systems. In such settings, robustness is governed by finite break costs and is often dominated by execution-layer vulnerabilities rather than governance or protocol design. Empirical verification against three DeFi protocols (Aave, Compound, Sky) confirms that all observed instability events are instances of DoFₛrc > 0 materialized without an explicit adversary.