What was already known
Patterson predicted the first-moment bias. Dunn and Radziwiłł obtained a smoothed asymptotic under the Generalized Riemann Hypothesis for Hecke L-functions; an unconditional sharp-cutoff result was not known in the manuscript’s account.
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 Patterson’s first-moment conjecture without the earlier GRH premise.
Limitations
The result is for primary Eisenstein primes and this normalization; no uniformity in growing angular modes is claimed. The Lean check is reported by OpenAI and was not reproduced here.
Technical verification
OpenAI reports a Lean proof matching the selected claim.
Formal checking
The pinned Lean scope note identifies the unconditional sharp-cutoff first moment and fixed-mode angular statements.
- Checker
- OpenAI-reported Lean 4 / Comparator
- Scope
- Over all primary Eisenstein primes of norm at most X, the sum of normalized cubic Gauss sums has the sharp-cutoff asymptotic (6/5)c₍*₎X^(5/6)/log X, where c₍*₎=(2π)^(2/3)/(3Γ(2/3)).
- 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