What was already known
The sharp degree-one bound and finite dimensionality for every fixed growth degree were known. Real noninteger-degree counterexamples did not settle the integer-degree Euclidean comparison posed by Yau.
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 integer-degree form of Yau’s harmonic-dimension comparison question.
Limitations
The formalized witness uses dimension 16 and degree 50000, is Euclidean near the origin, and has asymptotic volume ratio strictly between zero and one.
Technical verification
OpenAI reports a Lean proof matching the selected claim.
Formal checking
The pinned Lean note constructs the dimension-16 degree-50000 witness and proves its harmonic-function space exceeds the Euclidean count.
- Checker
- OpenAI-reported Lean 4 / Comparator
- Scope
- A complete smooth nonnegatively Ricci-curved metric on Euclidean space in dimension 16 has more harmonic functions of pointwise polynomial growth at most integer degree 50000 than Euclidean space has harmonic polynomials of that degree.
- 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