{
  "schema_version": "1.0.0",
  "source": "vibemath",
  "problem_id": "vibemathed:erdos-problem-1-sum-distinct-sets",
  "title": "Erdős Problem #1: sum-distinct sets",
  "canonical_statement": "A finite set $A\\subseteq\\{1,\\dots,N\\}$ is sum-distinct if all subset sums\n$$\n\\sum_{a\\in S} a,\\qquad S\\subseteq A,\n$$\nare distinct. Erdős asked whether there is an absolute constant $C>0$ such that every sum-distinct set $A\\subseteq\\{1,\\dots,N\\}$ satisfies\n$$\nN>C\\,2^{|A|}.\n$$\nThe conjecture is false: for every $\\varepsilon>0$ there are arbitrarily large $n$ and sum-distinct sets $A\\subseteq\\{1,\\dots,N\\}$ with\n$$\n|A|=n,\\qquad N\\le \\varepsilon 2^n.\n$$",
  "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:erdos-problem-1-sum-distinct-sets:event",
      "problem_id": "vibemathed:erdos-problem-1-sum-distinct-sets",
      "type": "disproved",
      "status": "resolved",
      "occurred_at": "2026-08-28T00:00:00.000Z",
      "title": "Github",
      "summary": "The formal theorem proves that no universal constant $C>0$ can satisfy\n$$\nN>C\\,2^{|A|}\n$$\nfor every nonempty interval bound $N$ and every sum-distinct $A\\subseteq\\{1,\\dots,N\\}$.\n\nEquivalently, for every $\\varepsilon>0$ there are arbitrarily large $n$ and sum-distinct $n$-element sets contained in $\\{1,\\dots,N\\}$ with\n$$\nN\\le\\varepsilon 2^n.\n$$\n\nThe proof is ineffective: it establishes the existence of arbitrarily large such $n$ but gives no explicit bound for how large $n$ must be in terms of $\\varepsilon$.",
      "ai_contribution": "ai_discovered",
      "attempt_ids": [
        "vibemathed:erdos-problem-1-sum-distinct-sets:attempt"
      ],
      "method_family_ids": [
        "vibemathed:erdos-problem-1-sum-distinct-sets:method"
      ],
      "source_assertion_ids": [
        "vibemathed:erdos-problem-1-sum-distinct-sets:assertion"
      ]
    }
  ],
  "attempts": [
    {
      "id": "vibemathed:erdos-problem-1-sum-distinct-sets:attempt",
      "problem_id": "vibemathed:erdos-problem-1-sum-distinct-sets",
      "model": "GPT-6 Astra (pre-release)",
      "provider": "OpenAI",
      "attempted_at": "2026-08-28T00:00:00.000Z",
      "human_collaborators": [
        "Tom Adamczewski"
      ],
      "exposure": "unknown",
      "prompt_public": null,
      "outcome": "A pre-release GPT-6 Astra autonomously solved the Formal Conjectures benchmark statement with no human steering during proof search. The primary run constructs, for large $n$, rational $n\\times n$ matrices with small determinant and additional admissibility properties, then uses integral changes of basis, saturated bidiagonal perturbations, and a binary-block construction to obtain sum-distinct sets with $N\\le\\varepsilon 2^n$. An independent larger-budget Astra run found an essentially equivalent lattice-based argument. Astra also wrote the Lean proofs; Claude was later used to prepare repository documentation from the completed runs.",
      "failed_routes": [],
      "artifacts": [
        {
          "label": "erdosproblems.com/1: status and Thomas Bloom's proof exposition",
          "url": "https://www.erdosproblems.com/1",
          "kind": "problem-record",
          "license": null
        },
        {
          "label": "Challenge.lean: the compared statement",
          "url": "https://github.com/tadamcz/erdos1/blob/main/Challenge.lean",
          "kind": "lean-statement",
          "license": null
        },
        {
          "label": "Solution.lean and the Lean development",
          "url": "https://github.com/tadamcz/erdos1/blob/main/Solution.lean",
          "kind": "lean-proof",
          "license": null
        },
        {
          "label": "Epoch AI, Announcing FrontierMath Erdős (1 September 2026)",
          "url": "https://epoch.ai/latest/announcing-frontiermath-erdos",
          "kind": "announcement",
          "license": null
        }
      ],
      "sources": [
        {
          "label": "Github",
          "url": "https://github.com/tadamcz/erdos1",
          "kind": "primary-mathematical-source",
          "license": null
        },
        {
          "label": "VibeMathed: Erdős Problem #1: sum-distinct sets",
          "url": "https://vibemathed.com/problem/erdos-problem-1-sum-distinct-sets",
          "kind": "source-assertion",
          "license": "https://vibemathed.com/data-license"
        }
      ],
      "cost_usd": 405,
      "independence": "unknown"
    }
  ],
  "verifications": [
    {
      "id": "vibemathed:erdos-problem-1-sum-distinct-sets:verification",
      "problem_id": "vibemathed:erdos-problem-1-sum-distinct-sets",
      "solution_event_id": "vibemathed:erdos-problem-1-sum-distinct-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": "Lean-verified. Checked here on 6 September 2026 from a clone of tadamcz/erdos1 at db6f909: 4,608 lines of Lean, zero sorry outside the statement stubs, zero axiom declarations, no native_decide, unsafe or implemented_by; Comparator configuration present and CI runs it with only propext, Quot.sound and Classical.choice. The statement is copied verbatim from Formal Conjectures' ErdosProblems/1.lean at commit 488aade2, the human-curated formalization the model was given, and the proved theorem is its negation. Two independent runs found essentially equivalent lattice-based arguments. The proof is ineffective: it gives no bound on how large n must be. erdosproblems.com, the field's own record, marks the problem DISPROVED (LEAN) with a proof exposition by Thomas Bloom, which is why this is Resolved rather than Candidate: the canonical tracker has accepted it.",
      "sources": [
        {
          "label": "VibeMathed: Erdős Problem #1: sum-distinct sets",
          "url": "https://vibemathed.com/problem/erdos-problem-1-sum-distinct-sets",
          "kind": "source-assertion",
          "license": "https://vibemathed.com/data-license"
        },
        {
          "label": "Github",
          "url": "https://github.com/tadamcz/erdos1",
          "kind": "primary-mathematical-source",
          "license": null
        }
      ]
    }
  ],
  "sources": [
    {
      "label": "Github",
      "url": "https://github.com/tadamcz/erdos1",
      "kind": "primary-mathematical-source",
      "license": null
    },
    {
      "label": "VibeMathed: Erdős Problem #1: sum-distinct sets",
      "url": "https://vibemathed.com/problem/erdos-problem-1-sum-distinct-sets",
      "kind": "source-assertion",
      "license": "https://vibemathed.com/data-license"
    }
  ],
  "recommended_mode": "replay",
  "recommended_exposure": "method_aware",
  "opportunity_signals": [
    {
      "id": "vibemathed:erdos-problem-1-sum-distinct-sets:signal:replay_ready",
      "problem_id": "vibemathed:erdos-problem-1-sum-distinct-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"
}