Dark-State Invariants: A Deterministic Framework for Detecting Computational Redundancy in Belief Propagation Dynamics

We introduce Dark-State Invariants, a deterministic framework for identifying temporally stable regions in belief propagation (BP) dynamics. Rather than optimizing update rules or convergence speed directly, this approach measures where iterative message-passing has effectively ceased to produce meaningful changes. A node is classified as dark-stable when both its sign and magnitude remain invariant within a fixed tolerance across consecutive iterations. This yields a per-iteration mask and derived metrics quantifying the fraction of computationally inactive nodes. Dark-state invariants provide a new observable complementary to existing measures such as cosine fidelity and sign agreement. Empirical evaluation across deterministic stress scenarios demonstrates that convergent systems rapidly accumulate high dark-state fractions, while oscillatory and chaotic systems do not. This enables a principled shift from acceleration-based optimization toward invariantdriven computation elimination, where redundant updates can be safely avoided.