What was already known
The known lower bound was already of order √λ. For smooth surfaces, earlier upper bounds included Cλ^(3/4) and then Cλ^(3/4−β) for universal β>0; the sharp upper bound remained Yau’s conjecture.
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
Proves the missing upper half of Yau’s nodal-length conjecture for smooth closed surfaces.
Limitations
The result is two-dimensional and assumes a fixed smooth closed connected surface. The known lower bound is a separate earlier result.
Technical verification
OpenAI reports a Lean proof matching the selected claim.
Formal checking
The pinned Lean note covers the upper bound for every nonzero real eigenfunction on each fixed smooth closed connected surface.
- Checker
- OpenAI-reported Lean 4 / Comparator
- Scope
- For each fixed smooth closed connected Riemannian surface, the nodal set of every nonzero real Laplace eigenfunction of eigenvalue λ>0 has length at most C√λ, with C depending on the surface.
- 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