{
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  "source": "vibemath",
  "problem_id": "vibemathed:an-improved-lower-bound-for-the-shannon-capacity-of-c-11",
  "title": "An improved lower bound for the Shannon capacity of $C_{11}$",
  "canonical_statement": "Determine the Shannon capacity of the eleven-cycle $C_{11}$, or improve its best explicit lower bound. The preceding BPZ construction, updated on 10 August 2026, gives an independent set of cardinality $N_0$ in dimension 207 and the lower bound $\\Theta(C_{11}) \\ge N_0^{1/207} = 5.29549231578462014255\\ldots$. The exact capacity remains open.",
  "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:an-improved-lower-bound-for-the-shannon-capacity-of-c-11:event",
      "problem_id": "vibemathed:an-improved-lower-bound-for-the-shannon-capacity-of-c-11",
      "type": "proved",
      "status": "partial",
      "occurred_at": "2026-09-08T00:00:00.000Z",
      "title": "R10 C11 capacity bound: pinned Lean construction from v0.5.0",
      "summary": "A sequence of explicitly constructed independent sets in strong powers of $C_{11}$, with successive root improvements certified by exact integer comparisons, using cross-powers when dimensions differ. The current bound is $\\Theta(C_{11}) \\ge 5.295526013632343$, from an independent set in $C_{11}^{\\boxtimes 213}$ (R10, 9 September 2026), improving the R3 result of $5.295492477500681$ in dimension 207 that this entry was first listed for, and the Buys-Polak-Zuiddam baseline of $5.295492315784620$. The intermediate steps R5 and R6 and the superseded R9 are recorded on the frontier. The exact capacity remains open, and no upper bound is claimed. The R5, R6, R9 and R10 builds carry disclosed native-evaluation dependencies rather than being kernel-only.",
      "ai_contribution": "ai_co_developed",
      "attempt_ids": [
        "vibemathed:an-improved-lower-bound-for-the-shannon-capacity-of-c-11:attempt"
      ],
      "method_family_ids": [
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      ],
      "source_assertion_ids": [
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    }
  ],
  "attempts": [
    {
      "id": "vibemathed:an-improved-lower-bound-for-the-shannon-capacity-of-c-11:attempt",
      "problem_id": "vibemathed:an-improved-lower-bound-for-the-shannon-capacity-of-c-11",
      "model": "Astra 6 Pro; Codex GPT-6 Astra Extra-High",
      "provider": "OpenAI",
      "attempted_at": "2026-09-08T00:00:00.000Z",
      "human_collaborators": [
        "Matthew Protti"
      ],
      "exposure": "unknown",
      "prompt_public": null,
      "outcome": "AI co-developed with OpenAI's Astra 6 Pro and Codex GPT-6 Astra Extra-High. Matthew Protti directed the research, evaluated proposals, required exact checks, set the scope and approved disclosure. The ChatGPT research collaboration supplied the R5 reassembly, R6 typed refinements and R9/R10 terminal certificates and written arguments. Codex replayed the finite certificates, wrote and compiled the Lean formalizations of actual independent sets, exact cardinalities and capacity bounds, inspected statement types and axiom dependencies, ran mathematical negative controls and prepared the releases. R9 and R10 import the checked R6 typed base and every child construction unchanged; only their terminal codes change. The underlying framework, base constructions and generic product machinery are due to Pjotr Buys, Sven Polak and Jeroen Zuiddam. These are author-side checks, not independent expert review.",
      "failed_routes": [],
      "artifacts": [
        {
          "label": "R3 history: pinned construction and capacity theorems (v0.1.0)",
          "url": "https://github.com/matthewprotti/c11-shannon-capacity-lower-bound/blob/0f3baddf9223e7a62eb67df2d7248da07b6d047d/checked_release/derived_source/lean/ShannonBounds/C11AuxiliaryTrade.lean",
          "kind": "lean-proof",
          "license": null
        },
        {
          "label": "R3 history: exact N0, N1 and frozen replacement table",
          "url": "https://github.com/matthewprotti/c11-shannon-capacity-lower-bound/blob/0f3baddf9223e7a62eb67df2d7248da07b6d047d/checked_release/frozen_inputs/CANDIDATE.json",
          "kind": "code",
          "license": null
        },
        {
          "label": "R3 history: checked build, native trust and negative controls",
          "url": "https://github.com/matthewprotti/c11-shannon-capacity-lower-bound/blob/0f3baddf9223e7a62eb67df2d7248da07b6d047d/checked_release/BUILD_SUMMARY.md",
          "kind": "other",
          "license": null
        },
        {
          "label": "v0.1.0 release with unchanged checked R3 archive",
          "url": "https://github.com/matthewprotti/c11-shannon-capacity-lower-bound/releases/tag/v0.1.0",
          "kind": "announcement",
          "license": null
        },
        {
          "label": "Prior BPZ C11 certificate: full August 10 baseline",
          "url": "https://github.com/spectra-research/shannon-capacity-lean/blob/aa21eeb12b75b0413d3fa9fb4208b5d0bf2c4d65/ShannonBounds/CertC11.lean",
          "kind": "code",
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        },
        {
          "label": "R6 v0.3.0: checked release archive and checksum",
          "url": "https://github.com/matthewprotti/c11-shannon-capacity-lower-bound/releases/tag/v0.3.0",
          "kind": "code",
          "license": null
        },
        {
          "label": "R5 v0.2.0: checked release archive and checksum",
          "url": "https://github.com/matthewprotti/c11-shannon-capacity-lower-bound/releases/tag/v0.2.0",
          "kind": "code",
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        },
        {
          "label": "v0.5.0: current checked R10 release and archive",
          "url": "https://github.com/matthewprotti/c11-shannon-capacity-lower-bound/releases/tag/v0.5.0",
          "kind": "code",
          "license": null
        },
        {
          "label": "v0.5.0: consolidated verification and trust disclosure",
          "url": "https://github.com/matthewprotti/c11-shannon-capacity-lower-bound/blob/dfaef37e60e55c55b1744d9badd1f26c5364c7d5/shannon_checked_release/README.md",
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        {
          "label": "VibeMathed: An improved lower bound for the Shannon capacity of $C_{11}$",
          "url": "https://vibemathed.com/problem/an-improved-lower-bound-for-the-shannon-capacity-of-c-11",
          "kind": "source-assertion",
          "license": "https://vibemathed.com/data-license"
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      "cost_usd": null,
      "independence": "unknown"
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  ],
  "verifications": [
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      "id": "vibemathed:an-improved-lower-bound-for-the-shannon-capacity-of-c-11:verification",
      "problem_id": "vibemathed:an-improved-lower-bound-for-the-shannon-capacity-of-c-11",
      "solution_event_id": "vibemathed:an-improved-lower-bound-for-the-shannon-capacity-of-c-11:event",
      "level": "lean_checked_statement_unaudited",
      "mathematical_correctness": "supported",
      "statement_fidelity": "unaudited",
      "peer_review": "none",
      "verifier": "VibeMathed (source-reported label)",
      "verified_at": null,
      "note": "The current R10 result is checked in v0.5.0 with Lean 4.32.2, BPZ commit aa21eeb12b75b0413d3fa9fb4208b5d0bf2c4d65 and Mathlib commit 905b95818eb32af7874a58b427f50c1711a5e96c. ShannonBounds.C11R10D213 proves an actual finite independent set in the 213th strong power of Mathlib's cycleGraph 11, its exact cardinality N, the capacity lower bound 5.295526013632343 and strict root comparisons with R6 and R9. A development build and separate fresh replay passed. Across the whole v0.5.0 consolidation, each run audited 184 numerical theorems and 261 symbolic declarations, with 246 distinct native Boolean-check axioms in the numerical audit, and rejected nine mathematical negative controls. Those counts are consolidation-wide, not R10-only. Numerical theorems retain inherited and new native-evaluation dependencies. Symbolic CellRetyping uses only propext, Classical.choice and Quot.sound; that does not remove numerical native trust. Both routes trust the pinned compiler/kernel, runtime and dependency binaries. Independent statement review, expert endorsement and exhaustive novelty clearance remain pending. Earlier R3 evidence is retained as history.",
      "sources": [
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          "label": "VibeMathed: An improved lower bound for the Shannon capacity of $C_{11}$",
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  "recommended_mode": "verify",
  "recommended_exposure": "result_only",
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      "id": "vibemathed:an-improved-lower-bound-for-the-shannon-capacity-of-c-11:signal:recent_partial_result",
      "problem_id": "vibemathed:an-improved-lower-bound-for-the-shannon-capacity-of-c-11",
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  "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."
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  "generated_at": "2026-09-13T16:28:40.178Z"
}