What was already known
Balls were known to minimize at sufficiently small volumes and coordinate cuts at half volume. The full ball–tube–slab profile and all transition equality cases remained conjectural.
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
Determines the complete isoperimetric profile and classifies all minimizers of the cubic flat three-torus.
Limitations
Classification is up to ambient isometry and null-set changes; the statement concerns the unit cubic flat three-torus.
Technical verification
OpenAI reports a Lean proof matching the selected claim.
Formal checking
The pinned Lean note covers the profile, existence and classification of all minimizers, including transition equality cases.
- Checker
- OpenAI-reported Lean 4 / Comparator
- Scope
- In the unit cubic flat three-torus, every volume-minimizing region is a ball, a circular tube about a shortest closed geodesic, a coordinate slab or the complement of one of these, in the predicted volume ranges, including both types at the transition volumes 4π/81 and 1/π.
- 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