Coherence Type Over Coherence Score: A Stochastic Differential Equation Derivation of Brand Resilience

Spectral Brand Theory 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. No formal derivation has connected this prediction to dynamics. This paper provides that derivation by analyzing the drift structures that different coherence types produce within the stochastic differential equation (SDE) framework of Zharnikov (2026). Two characterizations of drift geometry – isotropy and k-anisotropy – yield three main results. Theorem 1 establishes that absorption probability decreases monotonically in drift isotropy: isotropic drift fields (ecosystem coherence) produce lower absorption risk than anisotropic or stochastic drift. Theorem 2 shows that recovery probability depends on dimension-specific drift, creating asymmetric vulnerability profiles – crises on passive dimensions (e.g., the Economic dimension for Hermès) are disproportionately dangerous. Theorem 3 derives the full resilience ordering – ecosystem > signal > identity > experiential asymmetry > incoherent – from SDE geometry. The type-over-score principle is the central result: crisis survival depends on the minimum drift across dimensions, not the average, so two brands with identical total emission but different distributions have different resilience. The framework is applied to four documented crises and generates three falsifiable predictions. Includes zharnikov-2026s-r12-coherence-resilience.yaml (Paper Spec v0.1.0) – a machine-readable specification of the paper's claims, assumptions, and dependencies. The paper's full machine-first bundle (the SPINE claim/dependency graph and the ONTOLOGY term module) lives in the public repository; see https://github.com/spectralbranding/paper-spec for the standard. This PDF is generated programmatically from that machine-first source under a research-as-repository model.