What was already known
The 1966 conjecture sought 1+o(1). The previous asymptotic upper bound was approximately 1.399+o(1), while known primitive layers approach 1.
AI involvement
An autonomous GPT-5.4 Pro run produced an initial proof and suggested the von Mangoldt-weighted chain method. Eight human authors rewrote and reviewed the final proof; Math Inc.’s Gauss formalized the quantitative result.
The authors explicitly identify an autonomous model run that produced the initial proof of this exact theorem.
Importance
Settles the conjectured sharp leading constant with a quantitative error term.
Limitations
The human proof review and Lean formalization are source-reported. The relationship between the released Lean statement and the formal conjecture has not been independently checked.
Technical verification
The authors report a completed review of the final proof, and the cited Lean project formalizes the quantitative estimate. Both checks are source-reported; neither was rerun here.
Formal checking
Generation and review of the final proof
- Checker
- Eight human authors of the primitive-sets paper
- Scope
- Theorem 1.1 quantitative primitive-set bound
- Basis
- Reported by the linked checker
Formal checking
Lean formalization in Main.lean and FormalConjecturesErdos1196.lean
- Checker
- Lean 4, in Math Inc.'s Erdos1196 project generated by Gauss
- Scope
- Every primitive A above x satisfies f(A) <= 1+O(1/log x), with a bridge to the formal #1196 statement
- Basis
- Reported by the linked checker
Checking a formal statement does not establish that it matches the original problem.
Human understanding
The authors explain the chain/antichain duality, von Mangoldt transition and sub-invariant weight behind the bound.
Correction and publication history
Settles the conjectured sharp leading constant with a quantitative error term.
Read this published revision