{
  "schema_version": "1.0.0",
  "source": "vibemath",
  "problem_id": "vibemathed:modified-lyons-sidorova-conjecture",
  "title": "The Modified Lyons–Sidorova Conjecture for Bounded-Variation Paths",
  "canonical_statement": "For a continuous bounded-variation path with signature $g$, logarithmic signature $l$ and increment $v$, the modified Lyons–Sidorova conjecture predicts the structure of $g$ when $R(l)=\\infty$. The paper proves it: $g=1$ when $v=0$, and otherwise a prefix $\\alpha$ of the centred path gives $S(\\gamma) = S(\\alpha)e^{v}S(\\alpha)^{-1}$.",
  "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:modified-lyons-sidorova-conjecture:event",
      "problem_id": "vibemathed:modified-lyons-sidorova-conjecture",
      "type": "proved",
      "status": "resolved",
      "occurred_at": "2026-07-29T00:00:00.000Z",
      "title": "arXiv",
      "summary": "This is the MODIFIED conjecture, not the original Lyons-Sidorova one, and it is proved for continuous bounded-variation paths. Prior work had a line-image result under the stronger assumption of infinite radius on every subinterval; this removes that assumption.",
      "ai_contribution": "ai_assisted",
      "attempt_ids": [
        "vibemathed:modified-lyons-sidorova-conjecture:attempt"
      ],
      "method_family_ids": [
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      "source_assertion_ids": [
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      ]
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  ],
  "attempts": [
    {
      "id": "vibemathed:modified-lyons-sidorova-conjecture:attempt",
      "problem_id": "vibemathed:modified-lyons-sidorova-conjecture",
      "model": "ChatGPT",
      "provider": "OpenAI",
      "attempted_at": "2026-07-29T00:00:00.000Z",
      "human_collaborators": [
        "Elena Boguslavskaya"
      ],
      "exposure": "unknown",
      "prompt_public": null,
      "outcome": "The author reports using ChatGPT for mathematical exploration, critical examination of arguments, testing of intermediate proof strategies, and drafting and editing, and states she reviewed and revised all AI-assisted material and takes full responsibility. The model is not credited with producing the proof.",
      "failed_routes": [],
      "artifacts": [],
      "sources": [
        {
          "label": "arXiv",
          "url": "https://arxiv.org/abs/2607.26377",
          "kind": "primary-mathematical-source",
          "license": null
        },
        {
          "label": "VibeMathed: The Modified Lyons–Sidorova Conjecture for Bounded-Variation Paths",
          "url": "https://vibemathed.com/problem/modified-lyons-sidorova-conjecture",
          "kind": "source-assertion",
          "license": "https://vibemathed.com/data-license"
        }
      ],
      "cost_usd": null,
      "independence": "unknown"
    }
  ],
  "verifications": [
    {
      "id": "vibemathed:modified-lyons-sidorova-conjecture:verification",
      "problem_id": "vibemathed:modified-lyons-sidorova-conjecture",
      "solution_event_id": "vibemathed:modified-lyons-sidorova-conjecture:event",
      "level": "unreviewed",
      "mathematical_correctness": "unknown",
      "statement_fidelity": "unaudited",
      "peer_review": "none",
      "verifier": "VibeMathed (source-reported label)",
      "verified_at": null,
      "note": "No independent check, and the disclosure places the model in a supporting role rather than crediting it with the argument. Preprint, not refereed.",
      "sources": [
        {
          "label": "VibeMathed: The Modified Lyons–Sidorova Conjecture for Bounded-Variation Paths",
          "url": "https://vibemathed.com/problem/modified-lyons-sidorova-conjecture",
          "kind": "source-assertion",
          "license": "https://vibemathed.com/data-license"
        },
        {
          "label": "arXiv",
          "url": "https://arxiv.org/abs/2607.26377",
          "kind": "primary-mathematical-source",
          "license": null
        }
      ]
    }
  ],
  "sources": [
    {
      "label": "arXiv",
      "url": "https://arxiv.org/abs/2607.26377",
      "kind": "primary-mathematical-source",
      "license": null
    },
    {
      "label": "VibeMathed: The Modified Lyons–Sidorova Conjecture for Bounded-Variation Paths",
      "url": "https://vibemathed.com/problem/modified-lyons-sidorova-conjecture",
      "kind": "source-assertion",
      "license": "https://vibemathed.com/data-license"
    }
  ],
  "recommended_mode": "verify",
  "recommended_exposure": "result_only",
  "opportunity_signals": [
    {
      "id": "vibemathed:modified-lyons-sidorova-conjecture:signal:watch_only",
      "problem_id": "vibemathed:modified-lyons-sidorova-conjecture",
      "kind": "watch_only",
      "reason": "Current public evidence does not meet the default replay threshold.",
      "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"
}