{
  "schema_version": "1.0.0",
  "source": "vibemath",
  "problem_id": "vibemathed:two-adic-absolute-galois-presentation",
  "title": "Explicit Presentation of the 2-adic Absolute Galois Group",
  "canonical_statement": "Give an explicit profinite presentation of $\\operatorname{Gal}(\\overline{\\mathbb{Q}}_2 / \\mathbb{Q}_2)$. The tame local cases were settled by the early 1980s; the dyadic case was the last one missing. The new presentation has four generators, two word relations and a pro-$2$ condition on the wild generators.",
  "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:two-adic-absolute-galois-presentation:event",
      "problem_id": "vibemathed:two-adic-absolute-galois-presentation",
      "type": "proved",
      "status": "resolved",
      "occurred_at": "2026-07-26T00:00:00.000Z",
      "title": "A presentation of the absolute Galois group of Q2 (project site)",
      "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.",
      "ai_contribution": "ai_discovered",
      "attempt_ids": [
        "vibemathed:two-adic-absolute-galois-presentation:attempt"
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      "source_assertion_ids": [
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  "attempts": [
    {
      "id": "vibemathed:two-adic-absolute-galois-presentation:attempt",
      "problem_id": "vibemathed:two-adic-absolute-galois-presentation",
      "model": "ChatGPT-5.5 Pro (GPT-5.6 A/B), Claude Fable 5, Claude Opus 4.8",
      "provider": "OpenAI / Anthropic",
      "attempted_at": "2026-07-26T00:00:00.000Z",
      "human_collaborators": [
        "David Roe",
        "David Turturean"
      ],
      "exposure": "unknown",
      "prompt_public": null,
      "outcome": "A ChatGPT Pro conversation produced the candidate presentation with an informal proof; it passed Roe's finite-quotient verifier on all 5,402 test groups, and the proof was then formalized twice in Lean 4 with coding agents.",
      "failed_routes": [],
      "artifacts": [
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          "label": "Finite-quotient verifier (5,402 test groups)",
          "url": "https://roed314.github.io/gq2/verifier/",
          "kind": "code",
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        }
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          "url": "https://roed314.github.io/gq2/",
          "kind": "primary-mathematical-source",
          "license": null
        },
        {
          "label": "VibeMathed: Explicit Presentation of the 2-adic Absolute Galois Group",
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          "kind": "source-assertion",
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      "problem_id": "vibemathed:two-adic-absolute-galois-presentation",
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      "peer_review": "none",
      "verifier": "VibeMathed (source-reported label)",
      "verified_at": null,
      "note": "Two separately initiated Lean 4 formalizations, checked modulo 7 and 9 named interfaces to the classical literature respectively. Manuscript public with an interactive web edition; not yet externally peer-reviewed.",
      "sources": [
        {
          "label": "VibeMathed: Explicit Presentation of the 2-adic Absolute Galois Group",
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          "kind": "source-assertion",
          "license": "https://vibemathed.com/data-license"
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          "label": "A presentation of the absolute Galois group of Q2 (project site)",
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          "kind": "primary-mathematical-source",
          "license": null
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  "sources": [
    {
      "label": "A presentation of the absolute Galois group of Q2 (project site)",
      "url": "https://roed314.github.io/gq2/",
      "kind": "primary-mathematical-source",
      "license": null
    },
    {
      "label": "VibeMathed: Explicit Presentation of the 2-adic Absolute Galois Group",
      "url": "https://vibemathed.com/problem/two-adic-absolute-galois-presentation",
      "kind": "source-assertion",
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      "id": "vibemathed:two-adic-absolute-galois-presentation:signal:replay_ready",
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      "generated_at": "2026-09-13T16:28:40.178Z"
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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."
  ],
  "generated_at": "2026-09-13T16:28:40.178Z"
}