What was already known
Dimension-free metric-entropy duality was known in special geometric regimes and with dimension-dependent losses. Pietsch’s unrestricted constant-factor comparison 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
Disproves the universal metric-entropy duality conjecture even with a cube as the covering body.
Limitations
The counterexample dimension depends on the proposed constants; the claim does not fix one dimension for all a and b.
Technical verification
OpenAI reports a Lean proof matching the selected claim.
Formal checking
The pinned Lean note proves the failure for every a,b≥1 and also records a vanishing dual-to-primal entropy ratio family.
- Checker
- OpenAI-reported Lean 4 / Comparator
- Scope
- For every proposed constants a,b≥1, some origin-symmetric convex body K in a finite dimension violates Pietsch’s proposed dimension-free covering-entropy comparison with the cube: log N(K,B∞)>b log N(B∞°,a⁻¹K°).
- 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