Boundary Volume and Cohort Reassignment on the 7-Simplex: A Concentration Analysis Under the Uniform Null

This paper applies concentration-of-measure techniques to the probability simplex Δ7 under the uniform Dirichlet(1, …, 1) null, which is treated throughout as a worst-case geometry rather than as a description of observed populations. 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 Δ7, at least 52.2% of the volume lies within a tenth of a cell’s volume radius of that cell’s boundary. Third, Lévy’s lemma on the 7-sphere yields P(|f − Mf| ≥ ε) ≤ 4exp (−7ε2/8) for 1-Lipschitz functions. Monte Carlo simulations with 105 draws confirm the predictions within sampling error. Because that boundary volume is dominated by proximity to the faces of the simplex, reassignment risk is measured separately: approximately two-thirds of observers lie within one component standard deviation of an inter-cohort bisector, a fraction stable across k ≥ 3 and, under partitions re-fit to the population as clustering re-fits them, across α ∈ [1, 30]. Concentration sharpens boundaries only against a zone held fixed. The results give a geometric foundation for preferring continuous observer profiles over discrete cohort labels once perceptual dimensionality exceeds five. 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 github.com/spectralbranding/paper-spec for the standard. This PDF is generated programmatically from that machine-first source under a research-as-repository model.