When a language model must pick one answer from a large space of equally valid options, which does it pick -- and how often is it the same answer every other model picks? Asked to "pick a word -- any word," 44 models chose "serendipity" 41% of the time. We characterize this convergence with a deliberately minimal instrument: 31 single-turn prompts, each naming a category with many valid one-word answers ("Name a tree."), asked four times per model with no system prompt.

Analysis is exact-match on normalized tokens -- no embeddings, no judge -- at about a dollar per model. That models converge is well documented; our contribution is the instrument itself -- the One-Word Census -- and what it reveals about the structure of the convergence.

We score each model by answer-choice surprisal: the average $-\log2$ probability of its answers under the pooled answers of all other models, leave-one-out. Convergence is extreme -- in 7 of 31 categories one answer takes over 80% of all answers -- yet conformity varies more than fourfold across models, and the variation is structured.

Persona- and community-tuned models are the most divergent; the newest mainline flagships are the most conformist, producing almost no answer no other model gave. Within four lineages (Claude, GPT, Qwen, Grok) conformity rises with each generation -- but reverses for the latest flagship Claude and GPT models, a possible early signal of repositioning at the top tier.

Rankings are robust to roster composition (leave-one-family-out rho = 0.985). Against human category-production norms, the field is more concentrated than people in 18 of 20 shared categories.

All prompts, transcripts, and code are public.