What was already known
Earlier directional zero–one results treated independent environments; finite-range dependence allows neighboring transition rows to be correlated and requires control across infinite separated sets.
AI involvement
OpenAI credits itself as manuscript author and states that its internal frontier model produced the results in this release. Individual human input is not detailed enough to establish task-only autonomy.
OpenAI is named as author on the manuscript, and its release attributes the results to an internal frontier model. The individual human input is not detailed enough to establish autonomy.
Importance
Extends directional zero–one behavior beyond independent random environments.
Limitations
Finite-range dependence is required for full transition rows, including infinite deterministic sets. The linked Lean scope proves the iid strictly elliptic case, not correlated finite-range environments.
Technical verification
OpenAI reports a Lean check of a narrower theorem, case, or component; the selected claim extends beyond that scope.
Formal checking
The pinned OpenAI Lean scope note verifies only this narrower statement, not the full selected claim.
- Checker
- OpenAI-reported Lean 4 / Comparator
- Scope
- For iid strictly elliptic nearest-neighbor environments on Z^d, d≥3, the annealed probability of escape in any fixed nonzero direction is zero or one.
- Basis
- Reported by the linked checker
Checking a formal statement does not establish that it matches the original problem.
Human understanding
Few is the default when evidence for a higher level is insufficient. No named person is documented as explaining the central argument for this assessed contribution.
Correction and publication history
Primary manuscript, attribution, prior-work, and precise partial or mismatched Lean-scope review for non-notifying local inclusion.
Read this published revision