What was already known
Donaldson posed the tamed-to-compatible existence question. Prior affirmative cases included integrable complex surfaces (Buchdahl, Lamari, Li–Zhang), CP², residual sets in the b₂⁺ = 1 setting, and selected rational surfaces under curve hypotheses; none covered every smooth tamed almost complex structure on a closed four-manifold (manuscript introduction, pp. 1–2).
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
Settles Donaldson’s general tamed-to-compatible existence question in real dimension four.
Limitations
The almost complex structure is fixed, but the compatible form need not represent the original taming cohomology class. This record does not assert general equality of tame and compatible cohomology cones; the October 6 revision restricts that stronger equality.
Technical verification
OpenAI reports a Lean proof matching the selected claim.
Formal checking
Pinned OpenAI Lean solution module contains a proved matching headline theorem. This is source-reported code inspection; local build and imported-module audit were not performed.
- Checker
- OpenAI-reported Lean 4
- Scope
- Existence of a compatible symplectic form for the given tamed almost complex structure on a closed connected four-manifold.
- 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 verification-scope review for non-notifying local inclusion.
Read this published revision