{ "schema_version": "1.0.0", "source": "vibemath", "problem_id": "vibemathed:optimal-exponent-relating-sumsets-and-difference-sets", "title": "Optimal Exponent Relating Sumsets and Difference Sets", "canonical_statement": "For every finite set $A\\subset\\mathbb Z$ with $|A|\\ge 2$, define\n\n$$C(A)=\\frac{\\log\\left(|A+A|/|A|\\right)}\n{\\log\\left(|A-A|/|A|\\right)}.$$\n\nDetermine the largest possible value of $C(A)$, equivalently the least universal exponent $c$ such that\n\n$$\\frac{|A+A|}{|A|}\n\\le\n\\left(\\frac{|A-A|}{|A|}\\right)^c$$\n\nfor every such set $A$. The result proves that the supremum is exactly $2$, although no individual admissible set attains it.", "plain_summary": "VibeMathed reports this item as resolved. VibeMath preserves that report as a source assertion and has not independently authored a plain-language mathematical explanation.", "current_status": "resolved", "solution_events": [ { "id": "vibemathed:optimal-exponent-relating-sumsets-and-difference-sets:event", "problem_id": "vibemathed:optimal-exponent-relating-sumsets-and-difference-sets", "type": "proved", "status": "resolved", "occurred_at": "2026-07-29T00:00:00.000Z", "title": "Lin and Li, “Settling the Optimal Exponent Relating Sumsets and Difference Sets,” arXiv:2607.27199", "summary": "New arXiv preprint with an author-provided Lean formalization; not yet peer-reviewed.", "ai_contribution": "ai_discovered", "attempt_ids": [ "vibemathed:optimal-exponent-relating-sumsets-and-difference-sets:attempt" ], "method_family_ids": [ "vibemathed:optimal-exponent-relating-sumsets-and-difference-sets:method" ], "source_assertion_ids": [ "vibemathed:optimal-exponent-relating-sumsets-and-difference-sets:assertion" ] } ], "attempts": [ { "id": "vibemathed:optimal-exponent-relating-sumsets-and-difference-sets:attempt", "problem_id": "vibemathed:optimal-exponent-relating-sumsets-and-difference-sets", "model": "Hy3", "provider": "Tencent Hunyuan", "attempted_at": "2026-07-29T00:00:00.000Z", "human_collaborators": [ "Haowei Lin", "Shanda Li" ], "exposure": "unknown", "prompt_public": null, "outcome": "Tencent Hunyuan’s Hyra research agent, powered by the Hy3 model, was used to explore and optimize finite-set constructions. During an approximately 24-hour run, Hyra produced the construction underlying the paper after moving from finite numerical searches toward natural-language proposals of general constructions and supporting arguments.\n\nThe human authors independently checked the construction, corrected and rewrote the exposition, and prepared the final mathematical proof manually. GPT-5.6 Sol was used as an exploration judge and later helped translate the natural-language argument into a Lean 4 formalization. The language-model judgments were not used as proof certificates.", "failed_routes": [], "artifacts": [], "sources": [ { "label": "Lin and Li, “Settling the Optimal Exponent Relating Sumsets and Difference Sets,” arXiv:2607.27199", "url": "https://arxiv.org/abs/2607.27199", "kind": "primary-mathematical-source", "license": null }, { "label": "VibeMathed: Optimal Exponent Relating Sumsets and Difference Sets", "url": "https://vibemathed.com/problem/optimal-exponent-relating-sumsets-and-difference-sets", "kind": "source-assertion", "license": "https://vibemathed.com/data-license" } ], "cost_usd": null, "independence": "unknown" } ], "verifications": [ { "id": "vibemathed:optimal-exponent-relating-sumsets-and-difference-sets:verification", "problem_id": "vibemathed:optimal-exponent-relating-sumsets-and-difference-sets", "solution_event_id": "vibemathed:optimal-exponent-relating-sumsets-and-difference-sets:event", "level": "lean_verified_statement_audited", "mathematical_correctness": "supported", "statement_fidelity": "audited", "peer_review": "none", "verifier": "VibeMathed (source-reported label)", "verified_at": null, "note": "This is a newly released arXiv v1 preprint and has not yet been peer-reviewed. It contains an explicit, self-contained mathematical construction and proof.\n\nThe authors also provide a Lean 4/mathlib formalization. The repository reports that `lake build` completes successfully with no `sorry` declarations or warnings. The principal asymptotic and supremum results use three `native_decide` certificates for elementary finite computations concerning a 12-element base-39 digit block. Consequently, those parts additionally trust Lean’s compiler and native execution, rather than relying exclusively on kernel reduction.\n\nThe formalization is strong supporting evidence, but it is author-provided, and no independent expert review was located as of 2026-07-30.", "sources": [ { "label": "VibeMathed: Optimal Exponent Relating Sumsets and Difference Sets", "url": "https://vibemathed.com/problem/optimal-exponent-relating-sumsets-and-difference-sets", "kind": "source-assertion", "license": "https://vibemathed.com/data-license" }, { "label": "Lin and Li, “Settling the Optimal Exponent Relating Sumsets and Difference Sets,” arXiv:2607.27199", "url": "https://arxiv.org/abs/2607.27199", "kind": "primary-mathematical-source", "license": null } ] } ], "sources": [ { "label": "Lin and Li, “Settling the Optimal Exponent Relating Sumsets and Difference Sets,” arXiv:2607.27199", "url": "https://arxiv.org/abs/2607.27199", "kind": "primary-mathematical-source", "license": null }, { "label": "VibeMathed: Optimal Exponent Relating Sumsets and Difference Sets", "url": "https://vibemathed.com/problem/optimal-exponent-relating-sumsets-and-difference-sets", "kind": "source-assertion", "license": "https://vibemathed.com/data-license" } ], "recommended_mode": "replay", "recommended_exposure": "method_aware", "opportunity_signals": [ { "id": "vibemathed:optimal-exponent-relating-sumsets-and-difference-sets:signal:replay_ready", "problem_id": "vibemathed:optimal-exponent-relating-sumsets-and-difference-sets", "kind": "replay_ready", "reason": "A public source and a non-unreviewed verification label support replay triage.", "generated_at": "2026-09-13T16:28:40.178Z" } ], "uncertainties": [ "VibeMath has not independently verified the mathematical claim.", "AI-attempt independence and training-data exposure are unknown unless explicitly documented.", "VibeMath has not independently audited the mathematical statement, proof, or novelty claim." ], "generated_at": "2026-09-13T16:28:40.178Z" }