This paper applies concentration-of-measure techniques to the probability simplex \ (⁷\) under the uniform Dirichlet\ ( (1, , 1) \) null. Three results follow. First, the Euclidean distance contrast ratio degrades to approximately 7. 5 at \ (n = 8\), placing the space in a transitional regime where clustering remains informative yet noisy. Second, for any convex \ (k\) -partition of \ (⁷\), the volume fraction within relative Euclidean distance \ (=. 10\) of any boundary is at least 52. 2%. Third, Lévy’s lemma on the 7-sphere yields \ (P (|f - Mf|) 4 (-7²/8) \) for 1-Lipschitz functions. Monte Carlo simulations with \ (10⁵\) draws confirm the predictions within sampling error. These bounds imply that a majority of observer profiles lie near at least one cohort boundary under the uniform null, making discrete assignment inherently unstable. The uniform distribution maximizes boundary volume; symmetric Dirichlet\ ( (, , ) \) with \ (3\) reduces the fraction by \ (^-7/2\), producing operationally crisp boundaries (below 2%) for empirically plausible \ (\). The results give a geometric foundation for preferring continuous observer profiles over discrete cohort labels once perceptual dimensionality exceeds five. Fisher-Rao recalculation via the sphere isometry confirms the qualitative conclusions. Includes zharnikov-2026f-r3-cohort-boundaries. 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.
Dmitry Zharnikov (Sat,) studied this question.