Non-Ergodic Brand Perception: Diffusion Dynamics on Multi-Dimensional Perceptual Manifolds

Brand perception is inherently dynamic, yet formal continuous-time manifold models of multi-dimensional brand perception remain scarce. This paper models an observer’s brand perception as a stochastic process on \(S^7_+\), the positive octant of the 7-sphere in Spectral Brand Theory’s eight-dimensional perception space. Signal encounters drive Brownian motion on the manifold, signal decay introduces drift toward a neutral prior, and negative conviction creates absorbing boundaries. A Stratonovich SDE on \(S^7_+\) yields four results: well-posedness up to absorption (Theorem 1); exponential survival probability decay at rate \(\lambda_{D,1} \sigma_0^2/2\) with eigenvalue \(\lambda_{D,1} = 112\) (Theorem 2); Dirichlet spectral gap \(\lambda_{D,2} - \lambda_{D,1} = 48\) with faster mixing to quasi-stationary distribution than on the full sphere (Theorem 3); and time-ensemble divergence with probability 1, establishing non-ergodicity via absorbing-boundary dynamics independent of Peters’ (2019) multiplicative framing (Theorem 4). Application to five brands shows absorption risk nearly matches SBT coherence grading with one inversion explained by coherence-type-dependent drift (Proposition 7). Recovery time scales as \(1/\delta^2\) in residual dimensional perception (Proposition 5), consistent with Tylenol–BP divergence. A sensitivity analysis around a representative D/A ratio recovers the empirically observed Goldilocks zone \(r^* \in [.55, .65]\) (Proposition 6). Results formalize the dynamic distinction between vectorized and rasterized brand management. Includes zharnikov-2026j-r6-diffusion-dynamics.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.