What was already known
The strongest cited upper bound was of stretched-exponential form, N exp(−c√log N); a fixed-power saving remained open.
AI involvement
OpenAI is credited as author and identifies this ordinary manuscript as internal-model output. The release does not establish complete autonomy for this result.
Institutional AI authorship and the collection's model-production account establish central AI work, without proving complete autonomy from human task choice.
Importance
Improves the classical Furstenberg–Sárközy upper bound to a fixed power saving.
Limitations
The Lean theorem and matching comparator cover this upper bound; later lower-bound constructions and companion polynomial-difference results are separate.
Technical verification
The released repository provides an actual Lean theorem with the matching uniform power-saving statement and a Comparator mapping, and its scope note reports a formalization of that result. The challenge template's sorry is a specification placeholder; the solution module contains a proof. This is a source-reported machine check.
Formal checking
The released repository provides an actual Lean theorem with the matching uniform power-saving statement and a Comparator mapping, and its scope note reports a formalization of that result. The challenge template's sorry is a specification placeholder; the solution module contains a proof. This is a source-reported machine check.
- Checker
- Lean kernel through OAI.Combinatorics.SquareDifference.Main, as reported by OpenAI
- Scope
- Existence of absolute c > 0 and C with |A| <= C N^(1-c) for every square-difference-free A subset {1,...,N}
- Basis
- Reported by the linked checker
Checking a formal statement does not establish that it matches the original problem.
Human understanding
No named human explanation of the new finite-field tuple law and contraction proof is documented; the model-produced manuscript alone does not establish human understanding.
Correction and publication history
Improves the classical Furstenberg–Sárközy upper bound to a fixed power saving.
Read this published revision