What was already known
Earlier discrete metric-space examples lacked a uniform positive lower bound on pairwise distances. Kalton asked whether uniform discreteness forces bounded approximation; countable proper spaces were known to have the metric approximation property.
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
Answers Kalton’s uniform-discreteness question negatively.
Limitations
The witness metric space is unbounded and not proper; distinct points are at distance at least one.
Technical verification
OpenAI reports a Lean proof matching the selected claim.
Formal checking
The pinned Lean note constructs the uniformly discrete space and proves approximation without any bounded approximation constant.
- Checker
- OpenAI-reported Lean 4 / Comparator
- Scope
- There is a countable uniformly discrete metric space whose real Lipschitz-free space has the approximation property but fails every bounded approximation property.
- 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