{ "schema_version": "1.0.0", "source": "vibemath", "problem_id": "vibemathed:sparse-domination-implies-convex-body-domination", "title": "Sparse domination implies convex body domination", "canonical_statement": "Nazarov, Petermichl, Treil and Volberg conjectured that scalar sparse domination should imply convex body domination for the corresponding coordinate-wise vector-valued extension. More precisely, if a bilinear form $\\Lambda$ admits an $(r,s)$-sparse bound, then its extension to $\\mathbb C^n$-valued functions should admit an $(r,s)$-convex body sparse bound. Laukkarinen and Lorist prove this implication for $1\\leq r,s<\\infty$ with $1/r+1/s>1$, in particular for the classical $(1,1)$-sparse setting.", "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:sparse-domination-implies-convex-body-domination:event", "problem_id": "vibemathed:sparse-domination-implies-convex-body-domination", "type": "proved", "status": "resolved", "occurred_at": "2026-08-25T00:00:00.000Z", "title": "Sparse domination implies convex body domination", "summary": "If a bilinear form admits an $(r,s)$-sparse bound, its coordinate-wise extension to $\\mathbb C^n$-valued functions admits an $(r,s)$-convex body sparse bound, for $1\\le r,s<\\infty$ with $\\tfrac1r+\\tfrac1s>1$. It holds both in a fixed dyadic lattice (Theorem 2.6, constants independent of the ambient dimension) and for arbitrary cubes (Theorem 3.4), via a randomization of the good part of the form.\n\nOn scope, two things separate. What Nazarov, Petermichl, Treil and Volberg proposed was the principle for the classical sparse setting, and that is settled outright, the hypothesis holding comfortably there. The $(r,s)$ range is the authors' own generalization beyond what was asked, and the restriction bites only inside it, leaving $\\tfrac1r+\\tfrac1s\\le1$ open. Their summary: \"we provide a proof of the full general statement\".\n\nIt also gives sparse domination for iterated commutators directly from that of the underlying form. The dependence on $n$ is not claimed optimal.", "ai_contribution": "ai_co_developed", "attempt_ids": [ "vibemathed:sparse-domination-implies-convex-body-domination:attempt" ], "method_family_ids": [ "vibemathed:sparse-domination-implies-convex-body-domination:method" ], "source_assertion_ids": [ "vibemathed:sparse-domination-implies-convex-body-domination:assertion" ] } ], "attempts": [ { "id": "vibemathed:sparse-domination-implies-convex-body-domination:attempt", "problem_id": "vibemathed:sparse-domination-implies-convex-body-domination", "model": "GPT-5.6 Sol Pro", "provider": "OpenAI", "attempted_at": "2026-08-25T00:00:00.000Z", "human_collaborators": [ "Aapo Laukkarinen", "Emiel Lorist" ], "exposure": "unknown", "prompt_public": null, "outcome": "GPT-5.6 Sol Pro was used to prototype proof strategies. In an initial extended dialogue, the authors and ChatGPT iteratively developed a proof that sparse domination for commutators $[b,T]$ implies convex body domination for $T$. The authors then suggested replacing the role of $b$ by Rademacher random variables; from this idea, a rough version of the dyadic proof of the main theorem was developed in collaboration with ChatGPT. The final proof, as well as its extension from a fixed dyadic lattice to arbitrary cubes, was fully developed, verified and written by the authors. ChatGPT was subsequently also used to check the manuscript for typos and errors.", "failed_routes": [], "artifacts": [ { "label": "Problem source", "url": "https://arxiv.org/abs/1701.01907", "kind": "paper", "license": null } ], "sources": [ { "label": "Sparse domination implies convex body domination", "url": "https://arxiv.org/abs/2608.24802", "kind": "primary-mathematical-source", "license": null }, { "label": "VibeMathed: Sparse domination implies convex body domination", "url": "https://vibemathed.com/problem/sparse-domination-implies-convex-body-domination", "kind": "source-assertion", "license": "https://vibemathed.com/data-license" } ], "cost_usd": null, "independence": "unknown" } ], "verifications": [ { "id": "vibemathed:sparse-domination-implies-convex-body-domination:verification", "problem_id": "vibemathed:sparse-domination-implies-convex-body-domination", "solution_event_id": "vibemathed:sparse-domination-implies-convex-body-domination:event", "level": "unreviewed", "mathematical_correctness": "unknown", "statement_fidelity": "unaudited", "peer_review": "none", "verifier": "VibeMathed (source-reported label)", "verified_at": null, "note": "Unreviewed: an arXiv preprint (v1, 25 August 2026, math.CA), unrefereed, with no formalization and no computational certificate, so there was nothing mechanical to re-run and none of the analysis was checked here. What was verified on 29 August 2026: the paper exists at arXiv:2608.24802 with this title and both authors; Theorem A and Corollary B are as the entry describes them; the AI disclosure is a dedicated section, quoted in the AI-role note; and the attribution is sound - arXiv:1701.01907 is Nazarov, Petermichl, Treil and Volberg's 2017 paper, and its Section 3.1 states the principle and says the authors could not prove it, in the words quoted in the age note.\n\nOne discrepancy a reader may hit, flagged rather than silently smoothed: the abstract states the implication with no hypothesis on $(r,s)$, while Theorem A in the body requires $1\\le r,s<\\infty$ with $\\tfrac1r+\\tfrac1s>1$. This entry follows Theorem A.", "sources": [ { "label": "VibeMathed: Sparse domination implies convex body domination", "url": "https://vibemathed.com/problem/sparse-domination-implies-convex-body-domination", "kind": "source-assertion", "license": "https://vibemathed.com/data-license" }, { "label": "Sparse domination implies convex body domination", "url": "https://arxiv.org/abs/2608.24802", "kind": "primary-mathematical-source", "license": null } ] } ], "sources": [ { "label": "Sparse domination implies convex body domination", "url": "https://arxiv.org/abs/2608.24802", "kind": "primary-mathematical-source", "license": null }, { "label": "VibeMathed: Sparse domination implies convex body domination", "url": "https://vibemathed.com/problem/sparse-domination-implies-convex-body-domination", "kind": "source-assertion", "license": "https://vibemathed.com/data-license" } ], "recommended_mode": "verify", "recommended_exposure": "result_only", "opportunity_signals": [ { "id": "vibemathed:sparse-domination-implies-convex-body-domination:signal:watch_only", "problem_id": "vibemathed:sparse-domination-implies-convex-body-domination", "kind": "watch_only", "reason": "Current public evidence does not meet the default replay threshold.", "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" }