{ "schema_version": "1.0.0", "source": "vibemath", "problem_id": "vibemathed:asymptotically-attaining-the-moore-bound", "title": "Asymptotically attaining the Moore bound", "canonical_statement": "For positive integers $d$ and $k$, let $n_k(d)$ be the maximum order of a graph of maximum degree at most $d$ and diameter at most $k$. It is shown that\n$$\\lim_{d \\to \\infty}\\frac{n_k(d)}{d^k} = 1$$\nfor every fixed $k$, thereby resolving the asymptotic degree-diameter problem for fixed diameter. \n\nAlso proved a similar lower bound on the edge-variant of the problem, and a tight asymptotic for the bipartite variant of the edge problem.", "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:asymptotically-attaining-the-moore-bound:event", "problem_id": "vibemathed:asymptotically-attaining-the-moore-bound", "type": "proved", "status": "resolved", "occurred_at": "2026-08-04T00:00:00.000Z", "title": "Asymptotically attaining the Moore bound", "summary": "Settles two conjectures. Theorem 1.1 proves Bollobas's asymptotic degree-diameter conjecture, in the stronger liminf form rather than the conjectured limsup. Corollary 1.2 proves Conjecture 3 of Cambie, Cames van Batenburg, de Joannis de Verclos and Kang on the edge variant, again in the stronger liminf form, and is tight for bipartite graphs.", "ai_contribution": "ai_co_developed", "attempt_ids": [ "vibemathed:asymptotically-attaining-the-moore-bound:attempt" ], "method_family_ids": [ "vibemathed:asymptotically-attaining-the-moore-bound:method" ], "source_assertion_ids": [ "vibemathed:asymptotically-attaining-the-moore-bound:assertion" ] } ], "attempts": [ { "id": "vibemathed:asymptotically-attaining-the-moore-bound:attempt", "problem_id": "vibemathed:asymptotically-attaining-the-moore-bound", "model": "GPT-5.6", "provider": "OpenAI", "attempted_at": "2026-08-04T00:00:00.000Z", "human_collaborators": [ "Wouter Cames van Batenburg", "Samuel Korsky" ], "exposure": "unknown", "prompt_public": null, "outcome": "The paper's tool disclosure states that GPT-5.6 Pro, used in exploratory brainstorming directed by the authors, suggested splitting complete flags into their odd- and even-rank subflags. That suggestion arose in connection with the edge problem but became the halved-flag construction carrying Theorem 1.1 itself, the graph being named for it. The authors developed it, formulated and verified every argument, and take full responsibility. Generative AI also assisted the Lean 4 formalization.", "failed_routes": [], "artifacts": [ { "label": "Asymptotically attaining the Moore bound — Lean 4 formalization", "url": "https://github.com/woutercvb/wewantmoore", "kind": "code", "license": null }, { "label": "Degree diameter problem", "url": "https://en.wikipedia.org/wiki/Degree_diameter_problem", "kind": "wikipedia", "license": null } ], "sources": [ { "label": "Asymptotically attaining the Moore bound", "url": "https://arxiv.org/abs/2608.03965", "kind": "primary-mathematical-source", "license": null }, { "label": "VibeMathed: Asymptotically attaining the Moore bound", "url": "https://vibemathed.com/problem/asymptotically-attaining-the-moore-bound", "kind": "source-assertion", "license": "https://vibemathed.com/data-license" } ], "cost_usd": null, "independence": "unknown" } ], "verifications": [ { "id": "vibemathed:asymptotically-attaining-the-moore-bound:verification", "problem_id": "vibemathed:asymptotically-attaining-the-moore-bound", "solution_event_id": "vibemathed:asymptotically-attaining-the-moore-bound:event", "level": "lean_verified_statement_audited", "mathematical_correctness": "supported", "statement_fidelity": "audited", "peer_review": "none", "verifier": "VibeMathed (source-reported label)", "verified_at": null, "note": "Lean 4 formalization at github.com/woutercvb/wewantmoore, checked by the site on 2026-08-06 at commit 32beb227. `DegreeDiameter.theorem_1_1` states Theorem 1.1 itself, as a limit of nKD k d / d^k, and `corollary_1_2` states Corollary 1.2; neither is a weakened lemma, and a second independent route is proved alongside each under `_via_big_cell`. No sorry or admit appears in the sources, and the committed axiom audit shows both final theorems resting only on propext, Classical.choice and Quot.sound. That audit is not taken on trust: the repository's CI builds the project from the pinned toolchain and manifest, regenerates the axiom and dependency reports, and fails if they differ from the committed ones. It passes on this commit. The formalization was itself AI-assisted, per the paper's disclosure, and the authors note it is not a line-by-line transcription: k = 1 is handled by the same construction rather than by complete graphs, and the order and cap asymptotics go through leading terms rather than the displayed O_k(q^-1) estimates.", "sources": [ { "label": "VibeMathed: Asymptotically attaining the Moore bound", "url": "https://vibemathed.com/problem/asymptotically-attaining-the-moore-bound", "kind": "source-assertion", "license": "https://vibemathed.com/data-license" }, { "label": "Asymptotically attaining the Moore bound", "url": "https://arxiv.org/abs/2608.03965", "kind": "primary-mathematical-source", "license": null } ] } ], "sources": [ { "label": "Asymptotically attaining the Moore bound", "url": "https://arxiv.org/abs/2608.03965", "kind": "primary-mathematical-source", "license": null }, { "label": "VibeMathed: Asymptotically attaining the Moore bound", "url": "https://vibemathed.com/problem/asymptotically-attaining-the-moore-bound", "kind": "source-assertion", "license": "https://vibemathed.com/data-license" } ], "recommended_mode": "replay", "recommended_exposure": "method_aware", "opportunity_signals": [ { "id": "vibemathed:asymptotically-attaining-the-moore-bound:signal:replay_ready", "problem_id": "vibemathed:asymptotically-attaining-the-moore-bound", "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.", "The source did not supply a subject field. VibeMath has not independently audited the mathematical statement, proof, or novelty claim." ], "generated_at": "2026-09-13T16:28:40.178Z" }