What was already known
The general midpoint-convexity versus asymptotic-uniform-convexifiability question had already been answered negatively by Baudier. The manuscript supplies a reflexive example with an explicit modulus and answers a separate reflexive diamond-distortion question.
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
Strengthens the earlier negative result to a quantitative reflexive example and separates the corresponding metric diamond criterion.
Limitations
This is not the first negative answer to the general renormability question. The numerical bounds refer to the constructed norm and countably branching diamonds.
Technical verification
OpenAI reports a Lean proof matching the selected claim.
Formal checking
The pinned Lean note proves reflexivity, the averaged midpoint modulus, failure of every equivalent AUC norm and the diamond-distortion lower bound.
- Checker
- OpenAI-reported Lean 4 / Comparator
- Scope
- A separable reflexive real Banach space has an asymptotically midpoint uniformly convex given norm with averaged midpoint modulus at least √(1+t²/12)−1 for t>0, but admits no equivalent asymptotically uniformly convex norm; depth-k countably branching diamonds have distortion at least √(1+k/12) in it.
- 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