Restoring Perceptual Separability After Coherence Shocks: A μ > λ Threshold Inequality in Brand Perception

Brands facing coherence shocks – repositioning campaigns or cultural shifts – confront an asymmetric recovery problem: some regain cohort separability while others enter absorbing cross-cohort interference. This paper derives the first sharp sufficient condition for separability survival: the corrective coherence emission rate μ must exceed the spectral leakage rate λ at the observer cohort's detection scale (μ > λ at scale δ). Cohort perception clouds are modeled as almost-invariant sets in a stochastic flow on an eight-dimensional SBT perception manifold, with separability governed by the spectral gap of the perception operator. The threshold follows from Kato–Rellich perturbation theory and Diaconis–Stroock spectral-gap bounds for reversible Markov chains. A semi-structural re-analysis of Dove's 2003–2023 Real Beauty trajectory, using VECM persistence modeling and impulse-response functions on longitudinal SBT scores, estimates λ ≈ .10 per year from passive Cultural-dimension drift and μ ≈ 4.50 dimension-units per year from Ideological activation. The resulting μ/λ ratio of 45 satisfies the inequality with large margin for the Purpose-Aligned cohort but is sign-inverted for the Skeptic-Critic cohort – precisely predicting the documented divergence in conviction trajectories. Spectral-gap collapse is shown to precede conviction reorientation by 6–18 months, offering a leading indicator unavailable from traditional perceptual maps or aggregate VECMs. The framework bridges Spectral Brand Theory to stochastic-process and operator literatures while delivering an actionable "brand health dashboard" for managers. Monte Carlo simulations and robustness checks confirm the threshold's predictive power. Implications extend to AI-mediated perception where scale-dependent leakage accelerates fine-grained separability loss. Includes paper.yaml (Paper Spec v0.1.0) – a machine-readable specification of the paper's claims, assumptions, and dependencies. See https://github.com/spectralbranding/paper-spec for the standard.