What was already known
Friedgut and Kalai proved order 1/log n and identified 1/(log n)² as the conjectured sharp graph scale; later bounds remained short of the exact exponent.
AI involvement
OpenAI is credited as author and says an internal model produced the mathematical manuscripts and proof artifacts; item-specific autonomy is not documented.
OpenAI is credited as manuscript author and the repository directly attributes the mathematical outputs to its internal model; it does not document the little-or-no-human-input needed for AI-autonomous.
Importance
Attains the conjectured graph threshold-width exponent.
Limitations
A published Lean theorem proves the graph statement as scoped. The later hypergraph companion is a separate result; its AI-model review does not establish human checking of this graph theorem.
Technical verification
The published Lean source states and proves the graph quantile-width bound under vertex invariance, increasingness, nontriviality, n>=2 and 0<epsilon<1/2. This is a source-reported formal artifact. The later hypergraph theorem has a distinct checking scope.
Formal checking
Problem313.sharp_threshold_width for monotone relabeling-invariant graph properties; Lean theorem proof in OAI/Combinatorics/SharpThreshold/Main.lean against the comparator statement
- Checker
- Lean 4 artifact published by OpenAI
- Scope
- Problem313.sharp_threshold_width for monotone relabeling-invariant graph properties
- Basis
- Reported by the linked checker
Checking a formal statement does not establish that it matches the original problem.
Human understanding
No human account explains the graph proof's central argument; this is a default for undocumented human understanding, not a finding that people cannot understand it.
Correction and publication history
Attains the conjectured graph threshold-width exponent.
Read this published revision