Breakthrough Tracker record
A ChatGPT-generated construction is claimed to refute Yau's scalar-curvature integral bound
Tianze Hao and Jintian Zhu construct a complete one-ended three-manifold, diffeomorphic to R3 and with strictly positive Ricci curvature, for which the scalar-curvature integral over radius-R balls divided by R has infinite limsup. This would refute Yau's unrestricted asymptotic scalar-curvature bound.
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- Stable ID
math-yau-scalar-curvature-integral-counterexample-2026- Revision
math-yau-scalar-curvature-integral-counterexample-2026.v1- Field
- Mathematics · Differential geometry, Ricci curvature and geometric analysis
- Evidence
- Tier 1 · Peer reviewed: No
- Record state
- Provisional · Provisional AI-generated counterexample
- Last checked
AI role
The authors state that GPT-5.6 Pro generated the main theorems, proofs and entire construction after they supplied the target and suggested an iterative method from earlier work.
Record details
- Problem or result
- Yau's asymptotic scalar-curvature integral bound under nonnegative Ricci curvature
- Authors
- Tianze Hao and Jintian Zhu
- Institutions
- Peking University; Westlake University
- Result date
- Submitted September 6 and announced September 9, 2026
Why it matters
The construction targets a remaining unrestricted scalar-curvature case after positive results under stronger geometric assumptions and would also defeat a related one-ended conjecture.
Limits
This is a long first-version preprint with no external review or formal proof artifact. The authors say they checked the mathematics and take responsibility, but that is not independent validation.
Sources
Correction and revision history
- 2026-09-10 — Added after primary-source, scope, status, AI-role and limitation review.
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