What was already known
The previous lower-bound improvement was logarithmic over n^{-2}, and the nearly n^{-2} upper-bound formulation remained open.
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
Disproves the nearly n^{-2} upper-bound formulation of Heilbronn’s triangle problem.
Limitations
The linked Lean scope proves a power improvement only along an unbounded sequence of sizes n, not the manuscript’s every-sufficiently-large-n assertion. That narrower result already refutes the proposed almost-n^{-2} upper bound.
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
- There is an unbounded sequence of n with point sets in the unit square whose smallest triangle area is at least n^{-2+η} for one fixed η>0.
- 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