Multicategory group homophily: constructive feasibility and sharp limits

Unrefereed research candidate. In one sentence: with three or more categories, every category can look homophilous in groups at once, and an exact closed-form ceiling says how strongly, and which budget blocks it. Group homophily measures compare how often a category's members find themselves in groups (teams, meetings, classroom contact groups) with one, two or more others of their own category, against a baseline; the ratio at each count is the category's affinity ratio. Veldt, Benson and Kleinberg (2023) proved that with two categories natural forms of simultaneous group homophily are combinatorially impossible, and asked what changes with more categories. This paper answers the question for a single group size. Theorem 1 (constructive): with three or more categories and any full-support baseline whose category participation matches the observed participation, arbitrarily small participation-preserving changes make every category's affinity ratios strictly increase with its number of members; connected, simple, regular finite hypergraphs realise this at every rational composition (Proposition 3). This proves an earlier informal remark by Sarker, Northrup and Jadbabaie. Theorem 2 gives an exact finite list of inequalities deciding which strict-majority group frequencies can coexist at fixed participation. Theorem 3 gives a closed-form, attained ceiling on how far all majority affinity ratios can rise together, as the minimum of a group-count budget, each category's own participation budget and its complementary participation budget. Consequences: a three-line certificate for the two-category odd-size impossibility of Veldt et al. (Corollary 2), and the fact that matched participation requires the baseline to allow groups without a strict majority (Corollary 3). Exploratory example (Section 7): in the same 4,600 distinct primary-school proximity triples (SocioPatterns), 10 of 10 pupil classrooms meet strict-majority homophily but 0 of 5 merged school years do. The years are not less segregated: merging roughly doubles the two-of-three baseline while most same-year participation sits in all-same-year triples; the criterion is not monotone in segregation. Classroom 1A passes by a margin of two triples. Ceilings are 3.92 (classrooms) and 2.07 (years); a binary pupil/teacher coding gives 0.641, below one as Corollary 2 requires. The archive contains the paper (PDF and Markdown source), the bibliography, exact-rational and LP verification scripts (fail-closed under python -O), aggregate school tables and figures, a rating-free internal review summary, the research-gate record, CI and SHA-256 sums. Raw third-party archives are not redistributed. Assurance: written proofs; the scripts are finite consistency checks; internal AI-assisted review only (not peer review, independent reproduction or formal verification). The results are descriptive measurement feasibility for a relaxed composition model; they do not identify preferences or causal mechanisms. Creator: Anonymous. Licences (mixed). Text and mathematical content: CC0-1.0. Code (*.py and the CI workflow): MIT. Exception: the school-data-derived files in empirical_primary_school/ (CSV and JSON tables, PNG and PDF figures), the Section 7 figure and Appendix B are adaptations of SocioPatterns primary-school data and are licensed CC BY-NC-SA 3.0, not CC0 (attribution: SocioPatterns collaboration, sociopatterns.org; Stehlé et al. 2011; Gemmetto et al. 2014; Chodrow et al. 2021; Benson et al. 2018). The record licence field shows CC BY-NC-SA 3.0 as the most restrictive component licence; see LICENSE.md for the component map.