Geometric Necessity of Fuzzy Cohort Boundaries: A Concentration Analysis of the 7-Simplex

This paper applies concentration-of-measure techniques to the probability simplex \(\Delta^7\) under the uniform Dirichlet\((1,\ldots,1)\) 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 \(\Delta^7\), the volume fraction within relative Euclidean distance \(\delta = .10\) of any boundary is at least 52.2%. Third, Lévy’s lemma on the 7-sphere yields \(P(|f - M_f| \geq \varepsilon) \leq 4\exp(-7\varepsilon^2/8)\) for 1-Lipschitz functions. Monte Carlo simulations with \(10^5\) 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\((\alpha,\ldots,\alpha)\) with \(\alpha \geq 3\) reduces the fraction by \(\alpha^{-7/2}\), producing operationally crisp boundaries (below 2%) for empirically plausible \(\alpha\). 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.