What was already known
The zero-sum rearrangement bound O(d) holds in arbitrary norms. The Euclidean square-root bound was the Steinitz–Bergström conjecture. For prescribed-order signs, an earlier bound was O(sqrt(d) + log N), whose length term grows with the number of vectors (manuscript introduction, Historical context, pp. 2–3).
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
Resolves the Euclidean Steinitz–Bergström conjecture up to the optimal square-root order and removes dependence on sequence length in prescribed-order signing.
Limitations
The claim is existential for finite Euclidean sequences. It does not provide an online signing algorithm or an exact best constant.
Technical verification
OpenAI reports a Lean proof matching the selected claim.
Formal checking
The pinned Lean note includes the O(sqrt(d)) prescribed-order signing theorem and the zero-sum permutation consequence, including repeated and zero vectors.
- Checker
- OpenAI-reported Lean 4 / Comparator
- Scope
- Every finite sequence of vectors in the Euclidean unit ball of R^d has one signing with all prescribed-order prefixes at most C sqrt(d); every indexed zero-sum family has a permutation with the same bound.
- 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