What was already known
Chao and Dobriban’s earlier one-dimensional q=r=2 bounds did not match; the uncontaminated n-rate was already known.
AI involvement
GPT-5 Pro suggested the decisive optimal-transport interpolation for the upper-bound contamination bias and the dynamic Benamou–Brenier route for the lower bound. Dobriban says both began from human-provided reasoning; he and Chao supplied the problem, earlier bounds, proof repairs and generalization. This is AI-centered collaboration, not model authorship of the whole theorem.
Dobriban identifies GPT-5 Pro's transport ideas as crucial central steps retained in the sharp result while explicitly documenting the human setup, correction and completion.
Importance
Closes the contamination-rate gap and extends the sharp result across dimensions and transport parameters.
Limitations
The theorem’s q and r ranges, smoothness and loss assumptions are essential. Chao and Dobriban supplied the setup, corrections, final proof and verification.
Technical verification
Dobriban explicitly reports careful human verification of model-generated calculations and proofs, including resolution of the optimal-map, interpolation-density and dynamic-transport gaps before the final revision. This is source-reported author checking of the assessed argument, not an independently reproduced check.
Formal checking
Careful human verification of GPT outputs, literature and regularity assumptions through iterative proof correction
- Checker
- Edgar Dobriban, author-reviewer
- Scope
- The transport-based upper and lower bounds yielding the sharp pointwise density-estimation minimax rate
- Basis
- Reported by the linked checker
Checking a formal statement does not establish that it matches the original problem.
Human understanding
Dobriban independently explains the transport interpolation, dynamic continuity equation and repaired Jacobian-density step, including why initial references failed.
Correction and publication history
Closes the contamination-rate gap and extends the sharp result across dimensions and transport parameters.
Read this published revision