Write a PREreview

**Abstract**We present a domain-agnostic stability criterion that detects impending structural collapse insymbolic complex systems prior to failure. By extracting the sorted absolute eigenvalues ofthe n-gram transition probability matrix as a topological invariant and subjecting thesequence to controlled stochastic perturbations, we compute a composite Stability IndexSI_final = SI_perturb × (1 − C(S)). Collapse manifests as one of two nonlinear phasetransitions: pathological rigidity (low entropy, infinite repetition) or structural entropy (highentropy, uniform noise). A second-derivative trigger based on perturbation intensity identifiesthe critical acceleration toward either mode before SI_final reaches zero. The criterion istheoretically grounded in spectral graph theory and information theory, and is independent ofsemantic content. Preliminary computational demonstrations on text, code, and genomicsequences show reliable early detection. This work establishes a foundational invariant forproactive stability monitoring in artificial intelligence, legal frameworks, financial systems,.and biological sequences