What was already known
Carbery had proposed the c=2 sharpened Lp triangle inequality. The human authors already suspected a counterexample and had found one by random brute-force search, but it gave neither a structural explanation nor the optimal exponent p′.
AI involvement
Grok supplied the structural counterexample and the proof of the key kernel lemma. Chen, de Dios Pont, Ivanisvili, Madrid and Wang corrected and generalized the example, checked the lemma and completed the argument.
Grok originated the retained structural counterexample and the key optimality lemma, while the authors supplied the reduction, corrections and broader paper.
Importance
Refutes the proposed exponent and identifies the sharp necessary exponent constraint.
Limitations
The original model example had numerical errors, corrected by the authors. Their separate three-function inequality and root-counting proof are human-led results.
Technical verification
The authors explicitly report verifying the correctness of Grok's Lemma 2.3 proof and carefully checking the details of its use. They present a corrected counterexample with a direct limiting calculation. The check is source-reported human proof review, not independently reproduced.
Formal checking
The authors explicitly report verifying the correctness of Grok's Lemma 2.3 proof and carefully checking the details of its use. They present a corrected counterexample with a direct limiting calculation. The check is source-reported human proof review, not independently reproduced.
- Checker
- Chen, de Dios Pont, Ivanisvili, Madrid and Wang, the human paper authors
- Scope
- Grok's Lemma 2.3 and its use in the optimal exponent argument
- Basis
- Reported by the linked checker
Checking a formal statement does not establish that it matches the original problem.
Human understanding
The authors explain the shared set plus private sets and the limiting calculation that forces c≤p′; they also explain how the Grok lemma enters their reduction.
Correction and publication history
Grok supplied the structural counterexample and key lemma; the authors corrected, checked and explained them.
Read this published revision