What was already known
The manuscript says the characteristic-zero affine-space cancellation problem in dimensions at least three was still listed as open in July 2026; low-dimensional and positive-characteristic results did not answer the complex fourfold case.
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
Gives a negative answer to complex affine-space cancellation in dimension four.
Limitations
The formalized result concerns the fourfold and its cylinder. Separate stable-coordinate and affine-fibration consequences are outside this check.
Technical verification
OpenAI reports a Lean proof matching the selected claim.
Formal checking
The pinned Lean note gives a finite-type complex domain of Krull dimension four whose polynomial extension is the five-variable polynomial ring but which is not the four-variable polynomial ring.
- Checker
- OpenAI-reported Lean 4 / Comparator
- Scope
- There is an explicit integral complex affine fourfold X such that X × A¹ is isomorphic to A⁵ but X is not isomorphic to A⁴.
- 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 scoped checking review for non-notifying local inclusion.
Read this published revision