{
  "schema_version": "1.0.0",
  "source": "vibemath",
  "problem_id": "vibemathed:sop-2-sop-3",
  "title": "$SOP_2 = SOP_3$",
  "canonical_statement": "The classes of SOP_2 and SOP_3 first-order theories coincide. This answers a question of Džamonja and Shelah from 2004.",
  "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:sop-2-sop-3:event",
      "problem_id": "vibemathed:sop-2-sop-3",
      "type": "proved",
      "status": "resolved",
      "occurred_at": "2026-08-13T00:00:00.000Z",
      "title": "SOP_2 = SOP_3",
      "summary": "The new content is $SOP_2 \\Rightarrow SOP_3$; the converse implication was known from the start. Dzamonja and Shelah asked whether either implication in $SOP_3 \\Rightarrow SOP_2 \\Rightarrow SOP_1$ reverses: Mutchnik answered the second ($SOP_1 = SOP_2$), and this answers the first, collapsing the bottom of the hierarchy to $SOP_1 = SOP_2 = SOP_3$. The $SOP_n$ hierarchy for $n \\ge 3$ remains, as does everything above it.",
      "ai_contribution": "ai_co_developed",
      "attempt_ids": [
        "vibemathed:sop-2-sop-3:attempt"
      ],
      "method_family_ids": [
        "vibemathed:sop-2-sop-3:method"
      ],
      "source_assertion_ids": [
        "vibemathed:sop-2-sop-3:assertion"
      ]
    }
  ],
  "attempts": [
    {
      "id": "vibemathed:sop-2-sop-3:attempt",
      "problem_id": "vibemathed:sop-2-sop-3",
      "model": "ChatGPT 5.6",
      "provider": "OpenAI",
      "attempted_at": "2026-08-13T00:00:00.000Z",
      "human_collaborators": [
        "Artem Chernikov"
      ],
      "exposure": "unknown",
      "prompt_public": null,
      "outcome": "From the paper: \"The proof was found using ChatGPT 5.6 and simplified and streamlined\nby the author.\"",
      "failed_routes": [],
      "artifacts": [],
      "sources": [
        {
          "label": "SOP_2 = SOP_3",
          "url": "https://arxiv.org/abs/2608.13291",
          "kind": "primary-mathematical-source",
          "license": null
        },
        {
          "label": "VibeMathed: $SOP_2 = SOP_3$",
          "url": "https://vibemathed.com/problem/sop-2-sop-3",
          "kind": "source-assertion",
          "license": "https://vibemathed.com/data-license"
        }
      ],
      "cost_usd": null,
      "independence": "unknown"
    }
  ],
  "verifications": [
    {
      "id": "vibemathed:sop-2-sop-3:verification",
      "problem_id": "vibemathed:sop-2-sop-3",
      "solution_event_id": "vibemathed:sop-2-sop-3:event",
      "level": "unreviewed",
      "mathematical_correctness": "unknown",
      "statement_fidelity": "unaudited",
      "peer_review": "none",
      "verifier": "VibeMathed (source-reported label)",
      "verified_at": null,
      "note": "Checked by this site on 14 August 2026 against the paper's LaTeX source. The paper is real - five pages, math.LO, posted 13 August - and its author, Artem Chernikov, is a leading model theorist in exactly this area. The AI disclosure is verbatim as the entry quotes it. The history checks out in the paper's own introduction: Dzamonja and Shelah introduced the tree configurations $SOP_1$ and $SOP_2$ and asked whether either implication $SOP_3 \\Rightarrow SOP_2 \\Rightarrow SOP_1$ reverses; Mutchnik's breakthrough proved $SOP_2 = SOP_1$, and the question $SOP_2 = SOP_3$ was repeatedly highlighted afterwards, with partial results by Conant, Kaplan-Ramsey-Simon and Mutchnik. This result completes the collapse $SOP_1 = SOP_2 = SOP_3$. The acknowledgements record comments by Itay Kaplan and Scott Mutchnik on a preliminary version - expert eyes, but comments on a draft are not independent verification, and the manuscript is one day old and unrefereed, so the tier is Unreviewed. The proof itself - five pages of tree-indiscernible manipulation - was not checked here; it needs a model theorist.",
      "sources": [
        {
          "label": "VibeMathed: $SOP_2 = SOP_3$",
          "url": "https://vibemathed.com/problem/sop-2-sop-3",
          "kind": "source-assertion",
          "license": "https://vibemathed.com/data-license"
        },
        {
          "label": "SOP_2 = SOP_3",
          "url": "https://arxiv.org/abs/2608.13291",
          "kind": "primary-mathematical-source",
          "license": null
        }
      ]
    }
  ],
  "sources": [
    {
      "label": "SOP_2 = SOP_3",
      "url": "https://arxiv.org/abs/2608.13291",
      "kind": "primary-mathematical-source",
      "license": null
    },
    {
      "label": "VibeMathed: $SOP_2 = SOP_3$",
      "url": "https://vibemathed.com/problem/sop-2-sop-3",
      "kind": "source-assertion",
      "license": "https://vibemathed.com/data-license"
    }
  ],
  "recommended_mode": "verify",
  "recommended_exposure": "result_only",
  "opportunity_signals": [
    {
      "id": "vibemathed:sop-2-sop-3:signal:watch_only",
      "problem_id": "vibemathed:sop-2-sop-3",
      "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"
}