{
  "schema_version": "1.0.0",
  "source": "vibemath",
  "problem_id": "vibemathed:four-color-rado-number-of-x-y-c-z-40c-41",
  "title": "The 4-Color Rado Number of $x+y+c=z$: $R(c)=40c+41$ Whenever $c+1$ Is Divisible by 3, 4, 5 or 7",
  "canonical_statement": "$R(c) = 40c+41$ for every $c \\geq 2$ such that $c+1$ is divisible by 3, 4, 5, or 7 (covering $\\approx 66\\%$ of all $c$); the full conjecture (Myers 2015 Conj. 4.9, ABEMRS16 §5.5) reduces to prime cases $p \\geq 89$, all smaller primes settled by SAT. Twenty-eight exact values, nineteen new primes $p = 11, \\ldots, 83$, zero deviations from the conjectured line.",
  "plain_summary": "VibeMathed reports this item as partial. VibeMath preserves that report as a source assertion and has not independently authored a plain-language mathematical explanation.",
  "current_status": "partial",
  "solution_events": [
    {
      "id": "vibemathed:four-color-rado-number-of-x-y-c-z-40c-41:event",
      "problem_id": "vibemathed:four-color-rado-number-of-x-y-c-z-40c-41",
      "type": "proved",
      "status": "partial",
      "occurred_at": "2026-08-14T00:00:00.000Z",
      "title": "GitHub repo (synthesis theorem + SAT certificates + dual-encoder verification + independent checker)",
      "summary": "Twenty-eight individual exact values, each proved by SAT certificate (coloring at n-1, UNSAT at n). The synthesis theorem covers every c >= 2 whose c+1 is divisible by 3, 4, 5, or 7 (~66% of integers). The prime-reduction corollary shows the full conjecture (R(c)=40c+41 for all c >= 2) is equivalent to checking primes p >= 89; all primes through 83 are settled. What stays open: the conjecture at c=88 (p=89) and every larger c whose c+1 has all prime factors >= 89. The scaling lemma's attribution is hedged relative to Malo 2000 (full text not accessed). No Lean formalization; the SAT certificates and dual-encoder architecture are the verification tier.",
      "ai_contribution": "ai_discovered",
      "attempt_ids": [
        "vibemathed:four-color-rado-number-of-x-y-c-z-40c-41:attempt"
      ],
      "method_family_ids": [
        "vibemathed:four-color-rado-number-of-x-y-c-z-40c-41:method"
      ],
      "source_assertion_ids": [
        "vibemathed:four-color-rado-number-of-x-y-c-z-40c-41:assertion"
      ]
    }
  ],
  "attempts": [
    {
      "id": "vibemathed:four-color-rado-number-of-x-y-c-z-40c-41:attempt",
      "problem_id": "vibemathed:four-color-rado-number-of-x-y-c-z-40c-41",
      "model": "Claude Fable",
      "provider": "Anthropic",
      "attempted_at": "2026-08-14T00:00:00.000Z",
      "human_collaborators": [],
      "exposure": "unknown",
      "prompt_public": null,
      "outcome": "An autonomous clean-room Claude session chose the target problem (4-color Rado numbers), surveyed three mutually-unaware literatures (Malo 2000, Myers 2015, ABEMRS16 2016), re-derived the scaling lemma, proved the synthesis theorem and prime-reduction corollary, built the SAT pipeline and independent verifier, solved all nineteen prime cases, and ran the full two-tier certification (DRAT + independent second encoder). Two independent AI referee agents verified the proof (both CONFIRMED). Human direction limited to run design, operational supervision, and posting.",
      "failed_routes": [],
      "artifacts": [
        {
          "label": "ABEMRS16, On the n-color Rado number for x_1+...+x_k+c = x_{k+1} (Math. Comp. 85, section 5.5 poses the conjecture)",
          "url": "https://doi.org/10.1090/mcom3034",
          "kind": "problem-record",
          "license": null
        },
        {
          "label": "Myers, Computational Advances in Rado Numbers (Rutgers Ph.D. thesis, 2015) - Conjecture 4.9",
          "url": "https://sites.math.rutgers.edu/~zeilberg/Theses/KellenMyersThesis.pdf",
          "kind": "problem-record",
          "license": null
        },
        {
          "label": "Malo, Four Color Rado Numbers for x_1+x_2+c=x_3 (South Dakota State M.S. thesis, 2000)",
          "url": "https://openprairie.sdstate.edu/etd2/760/",
          "kind": "problem-record",
          "license": null
        }
      ],
      "sources": [
        {
          "label": "GitHub repo (synthesis theorem + SAT certificates + dual-encoder verification + independent checker)",
          "url": "https://github.com/ZestyWombat854/rado-number-4color",
          "kind": "primary-mathematical-source",
          "license": null
        },
        {
          "label": "VibeMathed: The 4-Color Rado Number of $x+y+c=z$: $R(c)=40c+41$ Whenever $c+1$ Is Divisible by 3, 4, 5 or 7",
          "url": "https://vibemathed.com/problem/four-color-rado-number-of-x-y-c-z-40c-41",
          "kind": "source-assertion",
          "license": "https://vibemathed.com/data-license"
        }
      ],
      "cost_usd": null,
      "independence": "unknown"
    }
  ],
  "verifications": [
    {
      "id": "vibemathed:four-color-rado-number-of-x-y-c-z-40c-41:verification",
      "problem_id": "vibemathed:four-color-rado-number-of-x-y-c-z-40c-41",
      "solution_event_id": "vibemathed:four-color-rado-number-of-x-y-c-z-40c-41:event",
      "level": "unreviewed",
      "mathematical_correctness": "unknown",
      "statement_fidelity": "unaudited",
      "peer_review": "none",
      "verifier": "VibeMathed (source-reported label)",
      "verified_at": null,
      "note": "Reproduced in substance by this site on 14 August 2026, independently of the repo's code. All 28 coloring certificates were re-checked by an own-code scanner over every monochromatic triple: 28/28 valid, so every lower bound holds outright. Five base cells were fully re-solved with an independently written encoder (own variable layout, own symmetry breaking): satisfiable at $n-1$ and unsatisfiable at $n$ for $c = 0, 2, 3, 4, 5$, matching $R(0)=45$ and the $40c+41$ line exactly. The scaling lemma, its sharpness against the universal lower bound, the synthesis theorem and the prime-reduction corollary were verified by hand; the algebra is elementary and correct. The literature was verified independently: ABEMRS16 is Math. Comp. 85 (2016) 2047-2064 with exactly the claimed authors; Myers' Conjecture 4.9 appears verbatim in the Rutgers thesis; Malo's 2000 thesis is real (Open Prairie, South Dakota State) with $R(1..3)$ in its public abstract, and its full text is bot-gated - so the submitter's hedge about the scaling lemma possibly being Malo's is accurate and could not be resolved from here either. The 2026 papers on this equation were spot-checked and are two-color, as claimed. Not reproduced: the nineteen prime-case UNSAT certificates ($n$ up to 3321), which rest on the bundle's kissat DRAT proofs, drat-trim VERIFIED, with a second independent encoder agreeing on every instance both ran; and no human peer review exists - produced and refereed by AI agents in one pipeline.",
      "sources": [
        {
          "label": "VibeMathed: The 4-Color Rado Number of $x+y+c=z$: $R(c)=40c+41$ Whenever $c+1$ Is Divisible by 3, 4, 5 or 7",
          "url": "https://vibemathed.com/problem/four-color-rado-number-of-x-y-c-z-40c-41",
          "kind": "source-assertion",
          "license": "https://vibemathed.com/data-license"
        },
        {
          "label": "GitHub repo (synthesis theorem + SAT certificates + dual-encoder verification + independent checker)",
          "url": "https://github.com/ZestyWombat854/rado-number-4color",
          "kind": "primary-mathematical-source",
          "license": null
        }
      ]
    }
  ],
  "sources": [
    {
      "label": "GitHub repo (synthesis theorem + SAT certificates + dual-encoder verification + independent checker)",
      "url": "https://github.com/ZestyWombat854/rado-number-4color",
      "kind": "primary-mathematical-source",
      "license": null
    },
    {
      "label": "VibeMathed: The 4-Color Rado Number of $x+y+c=z$: $R(c)=40c+41$ Whenever $c+1$ Is Divisible by 3, 4, 5 or 7",
      "url": "https://vibemathed.com/problem/four-color-rado-number-of-x-y-c-z-40c-41",
      "kind": "source-assertion",
      "license": "https://vibemathed.com/data-license"
    }
  ],
  "recommended_mode": "expand",
  "recommended_exposure": "result_only",
  "opportunity_signals": [
    {
      "id": "vibemathed:four-color-rado-number-of-x-y-c-z-40c-41:signal:recent_partial_result",
      "problem_id": "vibemathed:four-color-rado-number-of-x-y-c-z-40c-41",
      "kind": "recent_partial_result",
      "reason": "A source-reported partial result may support bounded expansion.",
      "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"
}