Coherence Type as Crisis Predictor: A Formal Derivation from Non-Ergodic Dynamics

Spectral Brand Theory (SBT) classifies brands into five coherence types – ecosystem, signal, identity, experiential asymmetry, and incoherent –and predicts that coherence type determines crisis resilience better than coherence score. This paper provides a formal derivation by analyzing the drift structures that different coherence types produce within the stochastic differential equation (SDE) framework of Zharnikov (2026j) on the positive octant of the 7-sphere. We introduce drift isotropy and k-anisotropy as characterizations of drift geometry and derive three main results. Theorem 1 establishes that absorption probability decreases monotonically with drift isotropy, so isotropic drift (ecosystem coherence) produces lower absorption risk than anisotropic drift (signal, identity) or stochastic drift (incoherent). Theorem 2 shows that recovery probability following a crisis shock depends on the drift available on the affected dimension, creating asymmetric vulnerability profiles for anisotropic brands. Theorem 3 combines drift isotropy with dimension-specific volatility to derive the full resilience ordering – ecosystem > signal > identity > experiential asymmetry > incoherent – as a mathematical consequence of the SDE geometry. The gap between types increases with crisis severity. We apply the framework to four well-documented brand crises (Tylenol 1982, BP 2010, Volkswagen 2015, Tesla 2019) and state three falsifiable predictions for empirical testing.