A Universal Stability Criterion for Symbolic Complex Systems: Detecting Structural Deviation Before Catastrophic Collapse (USG)

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