One clue word, a 25-card board, and a partner who has to recover exactly the cards you mean. This paper measures how much a single legal word can communicate under a fixed common-English vocabulary, and finds a sharp limit: exact recovery of all nine of your cards is available on 0.286% of boards, the wall is the vocabulary rather than the geometry, and the clues a similarity decoder certifies as perfect mostly fail on a real reader. For the accessible version, read the write-up or try the interactive explorer.

Abstract

Cooperative word-association games such as Codenames pose a problem of constrained reference: a speaker indicates a hidden subset of a 25-word board to a partner using one legal word and a count. We ask whether a single clue can reliably indicate an arbitrary nine-word subset, and decompose the question with a six-rung model ladder separating channel capacity, embedding geometry, vocabulary projection, and pragmatic interpretation. Under a frozen common-English vocabulary and a deterministic embedding-rank decoder, exact recovery of a nine-word subset is available for only 0.286% of board-relative target sets, by exact enumeration over 2.0×10⁸ assignments, with a practical frontier near six. The limit is not geometric: synthetic vectors separate essentially every subset in 300 dimensions, so the wall is the projection from an ideal separating direction onto a legal word. In controlled pilot audits, the separations the decoder certifies do not transfer to pragmatic receivers: language-model and blinded-human guessers recover ordinary semantic clues at 23 of 24 but decoder-certified challenge clues at 0 of 44, a dissociation that survives stronger receivers and reproduces on natural game distributions. The dominant variable for natural recoverability is the availability of a clean separating concept, not the target count. A symbolic WordNet replication reproduces the collapse with no embedding. More broadly, the result bounds when embedding cosine similarity is a safe proxy for communication: least so at high cardinality and low margin, the regime where it is most tempting to trust.

Try it

The interactive explorer runs the decoder in your browser: pick a target subset and watch a single word separate it, or hit the wall.

Cite

@misc{liou2026oneclue,
  title  = {Decoder-Recoverable Is Not Communicable: One-Clue Reference Under Constrained Vocabularies},
  author = {Liou, Brian H.},
  year   = {2026},
  note   = {https://brianhliou.com/publications/one-clue-reference/}
}