2-to-1 games with perfect completeness

Projection games, their value and their repetition; formulas with a gap; the clause-variable game and the base game with perfect completeness and small soundness, from two named hypotheses. Slices of matrices over the field with two elements and the inverse theorem for partial tables, from a third named hypothesis. The tree of function spaces, keys of functions, the constraints of the new game, perfect completeness. Joint folding, the functional of a response, buckets, and the law of the tests. Comparisons of the law of the tests with uniform laws: one hidden child, sparse projection, the posterior bound, bias. Two accepted levels, the response table, good advice and the prediction of the upper response. The hidden part of the upper matrix, the list of heavy characters, the two decoders. Admissible forms, two draws of the lower array, a fixed form on a slice, agreement with forms of bounded rank. Odd lists, clean children, the bound by parallel repetition. The choice of the parameters, soundness, maps that are exactly 2-to-1, equal weights, the theorem. The course is an exposition of OpenAI's proof (2026) of the 2-to-1 games conjecture with perfect completeness, from three theorems of the literature that are stated as named hypotheses.

Ten lessons with proofs, examples and exercises with solutions. Written and self-checked by Claude Opus 5.5 (Anthropic); independent review is not complete.

  1. Projection games and the base game
  2. Matrices, slices and the inverse theorem
  3. The tree of function spaces and the questions
  4. Folding and the law of the tests
  5. Changing the law of the tests
  6. Two accepted levels and the upper response
  7. The heavy list and the two decoders
  8. From tested rank to full rank
  9. Clean children and repetition
  10. The parameters and the theorem

Theorems and assumptions · Reader and editable sources · Course record

Prerequisites