OpenAI has published a preprint dated October 25, 2026 claiming to refute the Exponential Time Hypothesis (ETH). The accompanying scope notes describe a subexponential runtime bound formalized in Lean and a sharper bound in the manuscript. The difference concerns how strong the runtime guarantee is; it does not mean that only the sharper bound could contradict ETH.
ETH rules out algorithms that solve 3-SAT in subexponential time in the number of variables. In 3-SAT, the task is to decide whether a Boolean formula has a satisfying assignment, with each clause containing at most three literals. A literal is a variable or its negation.
OpenAI says the repository contains mathematical manuscripts and supporting proof artifacts produced by an unnamed internal OpenAI model. This article compares the supplied preprint record and scope notes. No independent mathematical verification, Lean verification run or SAT-solver benchmark was performed for this article.
What the formalization covers
The scope notes describe exact counting and SAT on canonical inputs in subexponential time and polynomial space. Counting asks how many assignments satisfy a formula; SAT asks whether there is at least one.
The documented theorem applies to k-CNF formulas for each fixed k of at least two. Each clause contains at most k literals, so the stated scope includes 3-SAT. The input must use the canonical encoding described by the theorem, and the runtime includes a polynomial factor in that encoding's bit length.
How the runtime bounds differ
Let n be the number of variables, m the number of clauses and L the encoding's bit length. According to the scope notes, the Lean bound is:
(L + 1)^{O_k(1)} × 2^{O_k(n(log log n)^2 / sqrt(log n))}
The October 25 manuscript's bound is:
(n + m)^{O_k(1)} × 2^{O_k(n log log n / sqrt(log n))}
These expressions transcribe the published scope notes; they are not executable code or measured timings. The subscript k means the hidden constants may depend on the fixed clause width.
The Lean exponent contains (log log n)^2, while the manuscript exponent contains a single log log n factor. The manuscript therefore states a sharper asymptotic bound. The polynomial overhead also uses different quantities: encoding length L in the Lean statement, and variables plus clauses in the manuscript statement.
Both displayed exponential factors are subexponential in n: their exponents grow more slowly than n because the logarithmic ratios tend to zero. That conclusion keeps k fixed and retains the stated polynomial input overhead. The scope notes themselves describe the formalized result as subexponential. On its stated scope, that is already the kind of guarantee that challenges ETH; the manuscript's sharper analysis is a separate claim to check.
The notes also list a further manuscript improvement when the number of clauses is at most D times the number of variables, for fixed D greater than zero. Under that condition, the displayed exponential factor becomes 2^{O_{k,D}(n / sqrt(log n))}.
What readers can conclude
The published documents distinguish a claimed formalized subexponential result from stronger manuscript bounds. They do not supply an elapsed time for a developer's workload. An asymptotic improvement alone does not establish practical solver performance.
OpenAI describes the repository's manuscripts as raw research data and defers questions of credit and precedence to the mathematical community. Readers assessing the ETH claim should check the formal theorem's canonical-input assumptions and verification configuration, then examine the manuscript's stronger runtime analysis separately. Independent verification remains outside this article's evidence.