Cohort Boundaries in High-Dimensional Perception Space: A Concentration of Measure Analysis

Spectral Brand Theory (SBT) models brand perception through eight typed dimensions, with observers characterized by weight profiles on the probability simplex Δ7. Perceptual cohorts – clusters of observers who perceive a brand similarly – are central to the theory, yet the sharpness of their boundaries has been treated only qualitatively. This paper applies concentration of measure theory to derive rigorous bounds on cohort boundary fuzziness in 8-dimensional perception space. We prove that for 𝑚 uniformly random observer profiles on Δ7, the Euclidean distance contrast ratio max𝑑 / min𝑑 equals approximately 8.35 at 𝑛 = 8 (compared to 5801 at 𝑛 = 2), indicating that high-dimensional concentration is already substantial but not yet catastrophic. We establish that for any partition of Δ7 into 𝑘 convex cohort regions, the fraction of the simplex volume lying within relative distance 𝛿 of a boundary is at least 1 − (1 − 𝛿)𝑛, yielding 57.0% at 𝛿 = 0.10 for 𝑛 = 8 – a majority of the space is boundary rather than interior. We derive Levy concentration bounds on the 7-sphere showing that 1-Lipschitz functions deviate from their median by more than 𝜀 with probability at most 4 exp(−7𝜀2/8). Monte Carlo simulations with 103 to 105 sample points verify all theoretical predictions. These results establish that cohort boundary fuzziness in 8-dimensional perception space is not a measurement limitation but a geometric necessity: the claim that perceptual cohorts have inherently fuzzy boundaries, and that different clustering resolutions yield different but equally valid cohort structures, follows from the mathematics of high-dimensional simplices. We discuss implications for why independent AI models (Claude and Gemini) produced different cohort counts (5–6 versus 3) for the same brand data, why the designed/ambient (D/A) ratio affects cohort stability, and why the traditional marketing practice of assigning observers to discrete segments is geometrically lossy.