Can every group look homophilous at once? An Evidence Press briefing. Researchers often ask whether people end up in groups, such as teams, meetings or clusters of children in a playground, with others of their own category more often than chance would predict. For each category they compare how often its members are in groups with one, two or more others like them against a baseline. That comparison is called an affinity ratio. In twenty twenty-three, Veldt, Benson and Kleinberg proved a surprising limit. With only two categories, the bookkeeping makes it impossible for both to look homophilous at once in certain natural senses. They asked what happens with more categories. This paper answers that question for groups of one size. With three or more categories, the obstruction disappears: small adjustments can make every category's ratios rise together. The paper also gives an exact test of which combinations are possible, and a simple formula for the highest level all categories could reach together, naming the budget that blocks them. That formula gives a three-line proof of the two-category impossibility. A school example shows why this matters. In the same four thousand six hundred three-person contact groups, all ten classrooms pass a strict majority homophily test, but none of the five school years formed by merging them does. The years are not less segregated. Merging roughly doubles how often two members of three share a year by chance, while most same-year contact happens in groups that are entirely from one year. So a failed test is not proof of weak homophily. The results describe measurement, not preferences or causes, and the school example is exploratory: one school, over two days. This is an anonymous, AI-assisted, unrefereed candidate, published on the third of November twenty twenty-six; its paper and evidence are linked from the Evidence Press release page. This briefing uses an AI-generated voice and is not additional scientific evidence.