What was already known
Previously known pseudo-telepathic games admitted perfect maximally entangled strategies. Earlier work showed some games could also use nonmaximally entangled states, but did not show maximally entangled states were insufficient for every perfect strategy.
AI involvement
Lalonde devised inner-product games and the search method. Codex found the retained game matrix, pruned the game and extracted and implemented its exact rational exclusion certificate.
The human author supplied the conceptual framework, but Codex found the retained game matrix and certificate essential to the counterexample.
Importance
Gives a concrete finite-dimensional counterexample to completeness of maximally entangled strategies.
Limitations
The Lean certificate checks the finite-dimensional maximally entangled exclusion. The analytic existence argument has no separately documented completed check, and no infinite-dimensional exclusion is claimed.
Technical verification
The repository reports an exact Lean certificate excluding all finite-dimensional maximally entangled perfect strategies for the concrete game. The existence of a perfect nonmaximally entangled strategy is argued in the paper, without a documented completed check of that analytic side; the repository also does not claim infinite-dimensional tracial exclusion.
Formal checking
The repository reports an exact Lean certificate excluding all finite-dimensional maximally entangled perfect strategies for the concrete game. The existence of a perfect nonmaximally entangled strategy is argued in the paper, without a documented completed check of that analytic side; the repository also does not claim infinite-dimensional tracial exclusion.
- Checker
- Lean 4 checker in Lalonde's repository, as reported by the author
- Scope
- No perfect finite-dimensional maximally entangled strategy with arbitrary six-outcome POVMs exists for the specified 4-by-3 game
- Basis
- Reported by the linked checker
Checking a formal statement does not establish that it matches the original problem.
Human understanding
Lalonde describes his own conceptual game class, heuristic, hierarchy modification and the proof plan for the concrete counterexample.
Correction and publication history
Codex found the retained game matrix and exact exclusion certificate within Lalonde’s design.
Read this published revision