What was already known
Friedgut proved a sharp transition around a density that could depend on formula size. Gaia Carenini established the fixed-k limiting threshold first; the OpenAI manuscript credits her priority and gives another proof.
AI involvement
OpenAI is the credited author and attributes the proof artifacts to its internal model. No named person is documented as checking or explaining its concentration-and-comparison proof.
OpenAI authorship and release provenance credit the internal model with the alternative proof; session-level autonomy is not documented.
Importance
The contribution is a distinct formalized proof of the threshold-existence theorem.
Limitations
The result does not compute αₖ, give a convergence rate, or cover the threshold point. Carenini’s separate proof confirms the conclusion but does not check OpenAI’s argument.
Technical verification
The release maps the fixed-k threshold theorem to a Lean 4 proof, with proper clauses sampled independently with replacement and strict-side limits. The separate challenge target is a statement to prove, not the proof file.
Formal checking
Lean proof of OAI.FixedClauseThreshold.main using formalized random-SAT convergence, matched to the fixed-clause comparator statement; Final main theorem and preceding strict-side limits; see lean/docs/235.md, Scope
- Checker
- Lean 4 kernel, as reported by the OpenAI math release
- Scope
- For each fixed k ≥ 3, proper random k-SAT with replacement has a positive finite limiting threshold with strict-side limits
- Basis
- Reported by the linked checker
Checking a formal statement does not establish that it matches the original problem.
Human understanding
Carenini's independent proof establishes the conclusion by another method, but no named person explains OpenAI's capped-index concentration and system-size comparison mechanism. Few is the evidence default.
Correction and publication history
The contribution is a distinct formalized proof of the threshold-existence theorem.
Read this published revision