Source
Czarnecki, Gidel, Tracey, Tuyls, Omidshafiei, Balduzzi & Jaderberg, LIT-1504, read in NOTE-1146 from arXiv:2004.09468v2: Definition 1 and Theorem 1 (p. 5, proof p. 13), Propositions 1–2 (p. 5, proofs p. 13), supplement B (pp. 14–16).
What was actually shown
- The construction. Fix the pattern P. Strategy π_i spends its first n bits of moves encoding i and reads the opponent’s code. Both then follow a fixed, lexicographically first line of play reaching the outcome P_ij. That outcome is still attainable, because by definition neither a win nor a loss has yet been ruled out. Each π_i is a deterministic pure strategy, and sign f(π_i, π_j) = P_ij. A cycle of length ⌊2ⁿ⌋ is the special case in which each beats its predecessor and the first beats the last.
- Instances. Tic-Tac-Toe is 5.58-bit communicative by the paper’s linear-time recursion (supplement B), so every 47 × 47 pattern occurs. Go: alternately filling 180 points on each half of the board encodes log₂ 180! bits. The paper says ≈ 1000; the reader computed 1093.9. StarCraft, DOTA or Quake over 10 minutes at 60 Hz: at least 36,000 bits.
The reader re-derived the construction. It needs nothing beyond the definition.
What this does not say
- Anything about draws, in general. The paper states the theorem for P ∈ {−1, 0, 1}. A 0 entry needs a draw to remain attainable after the signalling, which the definition does not provide. The title is limited to wins and losses.
- That strong play cycles. The constructed strategies spend the game signalling and then deliberately lose or win. They are weak, and the theorem places long cycles among weak strategies only.
- That real games have the spinning-top shape. The authors say so themselves (p. 6). That is THEORY-930’s empirical question.
- Anything cardinal. Margins play no part.
Connections
For CLAIM-451 and TERM-077 it is the condition under which cyclic records cannot be excluded on structural grounds. Wherever contestants can recognize one another before the outcome is settled, any pattern of wins can arise. Whether an ordinal potential exists is then a fact about the strategies in play, not about the game (NOTE-1146).
Note of 2026-10-30: the second check
The second check at the status review (ADR-065 §5) followed each step.
Fix an antisymmetric pattern P of wins and losses over m = ⌊2ⁿ⌋ indices. Strategy π_i spends its first n bits of moves encoding i, which the definition lets it do whatever the opponent plays, and decodes the opponent’s index j from the opponent’s moves. Both players now know i, j and P. By the definition, neither a win nor a loss has yet been ruled out, so some continuation reaches the outcome P_ij for π_i. Both follow the lexicographically first such continuation, which each can compute alone. Because P_ji = −P_ij, it is the same line from either side, so the outcome agrees whichever seat each strategy holds, and sign f(π_i, π_j) = P_ij. Each π_i is deterministic, so it is pure. A cycle through all m strategies is the pattern in which each beats its predecessor and the first beats the last.
The instances’ arithmetic: log₂ 180! = 1093.92, computed from the log-gamma function; ⌊2^5.58⌋ = 47; and 10 minutes × 60 s × 60 Hz = 36,000. As a check on the construction, a script (in the session’s scratchpad) built it in a toy game with two bits of signalling followed by a free resignation, and all 64 antisymmetric patterns of wins and losses on 4 strategies were realised.