What was already known
Ruzsa’s inequality gave the upper limit 2. Earlier integer-set constructions and finite searches reached values far below 2.
AI involvement
Hyra, guided by a GPT-5.6 Sol judge in a human-designed search, found the retained construction. Haowei Lin and Shanda Li checked it and wrote the proof.
Hyra supplied the retained central set construction within a human-designed search and judging workflow; the authors supplied the final checked exposition.
- Supporting source (arxiv.org) — Submission history: v1 through review
Importance
Determines the optimal exponent in a standard additive-combinatorics inequality.
Limitations
Lin and Li checked and explain the base-12 construction. A later Lean proof establishes the same exponent through a different base-39 construction, so it does not check each step of the paper’s proof.
Technical verification
Lin and Li expressly report an independent complete check of their construction and proof. Separately, the Lean repository reports a complete check of the same sharp-exponent conclusion by a base-39 variant, with three small native_decide finite certificates; that does not certify every step of the paper’s base-12 proof.
Independent checking
independent author check of construction and proof
- Checker
- Haowei Lin and Shanda Li
- Scope
- Theorem 1.1 base-12 family and sharp exponent
- Basis
- Reported by the linked checker
Independent checking
pinned Lean build and axiom audit; native_decide finite certificates
- Checker
- Lean build in Lin–Li sum-diff-proof project
- Scope
- supremum C(A)=2 and convergence of an explicit base-39 family
- Basis
- Reported by the linked checker
Human understanding
The authors explain the digit gadget, carry automaton, symmetric cyclic basis and CRT combination and manually prepared the proof.
Correction and publication history
Determines the optimal exponent in a standard additive-combinatorics inequality.
Read this published revision