What was already known
The Ryser circulant Hadamard conjecture had ruled out many orders but remained open in full; its even Barker-sequence consequence likewise remained unresolved.
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
Settles the circulant Hadamard conjecture and the remaining even-length obstruction for Barker sequences.
Limitations
The Lean scope checks the exact circulant classification and even Barker lengths. The manuscript’s full list of odd Barker lengths relies on separate earlier work and is outside this selected claim.
Technical verification
OpenAI reports a Lean proof matching the selected claim.
Formal checking
The pinned OpenAI Lean scope note identifies this selected claim.
- Checker
- OpenAI-reported Lean 4 / Comparator
- Scope
- A real circulant Hadamard matrix of positive order n exists exactly when n=1 or n=4.
- 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 precise partial or mismatched Lean-scope review for non-notifying local inclusion.
Read this published revision