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 a closed-form sufficient condition for separability survival – to the author’s knowledge the first threshold inequality for cohort-level separability in a multi-dimensional perception space: 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. Author-proposed Dove-calibrated parameters – \(λ ≈ .10\)/year and \(μ ≈ 4.50\)/year – motivate a Monte Carlo demonstration of the threshold’s predictive properties. The \(μ/λ\) ratio of 45 satisfies the inequality for the Purpose-Aligned cohort but is sign-inverted for the Skeptic-Critic cohort, correctly predicting the documented conviction divergence. Spectral-gap collapse precedes conviction reorientation by 6–18 months, offering a leading indicator unavailable from perceptual maps or aggregate VECMs. Implications extend to AI-mediated perception where scale-dependent leakage accelerates fine-grained separability loss. Includes zharnikov-2026ad-r22.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.