What was already known
Shapiro–Rudin constructions gave real-sign maxima of order sqrt(N), with a larger constant; complex unimodular coefficients admitted ultraflat constructions. A fixed relative gap above the Parseval lower bound was conjectured for real signs (companion manuscript introduction, pp. 1–2).
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
Proves that the minimum normalized maximum tends to one through all integer lengths, refuting the real-sign fixed-gap conjecture.
Limitations
This is a one-sided maximum bound. It supplies no uniform positive lower bound for the modulus on the circle and does not establish two-sided real-sign ultraflatness. It is existential without an efficient signing algorithm.
Technical verification
OpenAI reports a Lean proof matching the selected claim.
Formal checking
The pinned Lean note gives a real Littlewood polynomial of every sufficiently large length with maximum at most (1+eta) sqrt(N), for each fixed eta > 0.
- Checker
- OpenAI-reported Lean 4 / Comparator
- Scope
- For every eta > 0 and all sufficiently large lengths N, a real-sign Littlewood polynomial of length N has circle maximum modulus at most (1+eta) sqrt(N).
- 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 companion manuscript, attribution, prior-work, and exact Lean-scope review for non-notifying local inclusion.
Read this published revision