{
  "schema_version": "1.0.0",
  "source": "vibemath",
  "problem_id": "vibemathed:shannon-capacity-odd-cycles-records",
  "title": "Record Lower Bounds for the Shannon Capacity of Odd Cycles",
  "canonical_statement": "Determine the Shannon capacities of odd cycles beyond $C_5$, or improve the best explicit bounds. Lovasz's theta function settled $C_5$ in 1979 and every longer odd cycle has stayed open since. The current records, all obtained with model assistance and formally verified, are $\\Theta(C_7) \\ge 3.258805369885$, $\\Theta(C_{11}) \\ge 5.294502522149$, $\\Theta(C_{13}) \\ge 6.302455083464$, $\\Theta(C_{15}) \\ge 7.301600534487$, $\\Theta(C_{19}) \\ge 9.357192705918$, $\\Theta(C_{21}) \\ge 10.342455853338$ and $\\Theta(C_{23}) \\ge 11.328224257774$.",
  "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:shannon-capacity-odd-cycles-records:event",
      "problem_id": "vibemathed:shannon-capacity-odd-cycles-records",
      "type": "proved",
      "status": "partial",
      "occurred_at": "2026-07-31T00:00:00.000Z",
      "title": "arXiv:2607.21517 - Improved lower bounds for the Shannon capacity of odd cycles",
      "summary": "record lower bounds for seven odd cycles; the exact capacities remain open for every odd cycle beyond C5",
      "ai_contribution": "ai_discovered",
      "attempt_ids": [
        "vibemathed:shannon-capacity-odd-cycles-records:attempt"
      ],
      "method_family_ids": [
        "vibemathed:shannon-capacity-odd-cycles-records:method"
      ],
      "source_assertion_ids": [
        "vibemathed:shannon-capacity-odd-cycles-records:assertion"
      ]
    }
  ],
  "attempts": [
    {
      "id": "vibemathed:shannon-capacity-odd-cycles-records:attempt",
      "problem_id": "vibemathed:shannon-capacity-odd-cycles-records",
      "model": "ChatGPT 5.6 Sol Pro, ChatGPT 5.6 Sol, Claude Opus 5",
      "provider": "OpenAI / Anthropic",
      "attempted_at": "2026-07-31T00:00:00.000Z",
      "human_collaborators": [
        "Pjotr Buys",
        "Sven Polak",
        "Jeroen Zuiddam",
        "Yu Gao",
        "Nathaniel Itty",
        "Christopher D. Rosin",
        "Chase Carstensen",
        "Daniel Reichman"
      ],
      "exposure": "unknown",
      "prompt_public": null,
      "outcome": "Three model-assisted papers in eleven days, each beating the last. Itty, Rosin, Carstensen and Reichman had ChatGPT-5.6 Sol Pro generate and run search programs across repeated prompts, returning explicit independent sets in strong graph powers that the authors checked. Gao then improved $C_7$ with a recursive construction and states that ChatGPT 5.6 Sol implemented all the code and expanded the proofs. Buys, Polak and Zuiddam followed both methods using ChatGPT 5.6 Sol Pro and Claude Opus 5, beat every previous bound, added three more cycles, and formalised the lot in Lean.",
      "failed_routes": [],
      "artifacts": [
        {
          "label": "Current record, Lean-verified (Buys, Polak, Zuiddam), arXiv:2607.29681",
          "url": "https://arxiv.org/abs/2607.29681",
          "kind": "paper",
          "license": null
        },
        {
          "label": "Lean 4 formalization of all seven bounds",
          "url": "https://github.com/spectra-research/shannon-capacity-lean",
          "kind": "lean-proof",
          "license": null
        },
        {
          "label": "Intermediate C7 record by recursive construction (Gao), arXiv:2607.27869",
          "url": "https://arxiv.org/abs/2607.27869",
          "kind": "paper",
          "license": null
        }
      ],
      "sources": [
        {
          "label": "arXiv:2607.21517 - Improved lower bounds for the Shannon capacity of odd cycles",
          "url": "https://arxiv.org/abs/2607.21517",
          "kind": "primary-mathematical-source",
          "license": null
        },
        {
          "label": "VibeMathed: Record Lower Bounds for the Shannon Capacity of Odd Cycles",
          "url": "https://vibemathed.com/problem/shannon-capacity-odd-cycles-records",
          "kind": "source-assertion",
          "license": "https://vibemathed.com/data-license"
        }
      ],
      "cost_usd": null,
      "independence": "unknown"
    }
  ],
  "verifications": [
    {
      "id": "vibemathed:shannon-capacity-odd-cycles-records:verification",
      "problem_id": "vibemathed:shannon-capacity-odd-cycles-records",
      "solution_event_id": "vibemathed:shannon-capacity-odd-cycles-records:event",
      "level": "lean_verified_statement_audited",
      "mathematical_correctness": "supported",
      "statement_fidelity": "audited",
      "peer_review": "none",
      "verifier": "VibeMathed (source-reported label)",
      "verified_at": null,
      "note": "The current records are formalised in Lean 4 at the linked repository, one base tuple per bound, so the seven stated inequalities are machine-checked rather than author-checked. We have not compiled it. Gao's intermediate record ships exact-integer verification code pinned to a fixed commit, and the earlier Itty-Rosin-Carstensen-Reichman constructions came with public data, prompts and checking code. All three are arXiv preprints; none is peer-reviewed.",
      "sources": [
        {
          "label": "VibeMathed: Record Lower Bounds for the Shannon Capacity of Odd Cycles",
          "url": "https://vibemathed.com/problem/shannon-capacity-odd-cycles-records",
          "kind": "source-assertion",
          "license": "https://vibemathed.com/data-license"
        },
        {
          "label": "arXiv:2607.21517 - Improved lower bounds for the Shannon capacity of odd cycles",
          "url": "https://arxiv.org/abs/2607.21517",
          "kind": "primary-mathematical-source",
          "license": null
        }
      ]
    }
  ],
  "sources": [
    {
      "label": "arXiv:2607.21517 - Improved lower bounds for the Shannon capacity of odd cycles",
      "url": "https://arxiv.org/abs/2607.21517",
      "kind": "primary-mathematical-source",
      "license": null
    },
    {
      "label": "VibeMathed: Record Lower Bounds for the Shannon Capacity of Odd Cycles",
      "url": "https://vibemathed.com/problem/shannon-capacity-odd-cycles-records",
      "kind": "source-assertion",
      "license": "https://vibemathed.com/data-license"
    }
  ],
  "recommended_mode": "replay",
  "recommended_exposure": "method_aware",
  "opportunity_signals": [
    {
      "id": "vibemathed:shannon-capacity-odd-cycles-records:signal:recent_partial_result",
      "problem_id": "vibemathed:shannon-capacity-odd-cycles-records",
      "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"
}