{
  "$schema": "../../../schemas/mko-v0.4.schema.json",
  "schema_version": "mko-v0.4",
  "id": "mko-finite-group-orbit",
  "semantic_family_id": "sf-finite-group-orbit",
  "type": "definition",
  "version": "0.10.0",
  "language": "zh-Hant",
  "titles": {
    "zh-Hant": "有限群作用的軌道"
  },
  "summary": {
    "zh-Hant": "有限群作用下，一點被全部群元素送到的像集合形成軌道，全部軌道分割carrier。"
  },
  "statement": {
    "zh-Hant": "在完整已認證有限群作用上，G·x={g·x:g∈G}須遍歷全部群元素。x約y iff存在g送x到y形成等價關係，不同orbits相等或不交並覆蓋carrier。單次作用結果、單generator有限步或stabilizer都不是orbit。"
  },
  "formula": {
    "tex": "O(x)",
    "mathml": "<math xmlns=\"http://www.w3.org/1998/Math/MathML\"><mrow><mi>O</mi><mo>(</mo><mi>x</mi><mo>)</mo></mrow></math>",
    "semantic_ast": {
      "type": "function_call",
      "callee": {
        "type": "symbol",
        "name": "O",
        "glyph": "O"
      },
      "arguments": [
        {
          "type": "symbol",
          "name": "x",
          "glyph": "x"
        }
      ]
    },
    "compiler": {
      "id": "ocme-formula-core",
      "version": "0.4.0",
      "source_sha256": "5df48944e5db48d9b78e2c8bd0b0eab27aa6fd3b340ec3473b4ecfa9e0f49be5"
    }
  },
  "assumptions": [
    {
      "id": "a1",
      "zh-Hant": "完整有限群Cayley table與total action table已全稱通過action axioms。"
    },
    {
      "id": "a2",
      "zh-Hant": "orbit與equivalence witness遍歷全部群元素。"
    }
  ],
  "symbols": [
    {
      "token": "O",
      "role": "group_action_orbit",
      "role_zh": "群作用軌道",
      "scope": "representative_formula"
    },
    {
      "token": "x",
      "role": "acted_point",
      "role_zh": "被作用點",
      "scope": "representative_formula"
    }
  ],
  "dependencies": [
    {
      "id": "mko-finite-group-action",
      "title_zh": "有限群作用",
      "reason_zh": "完整合法作用。"
    },
    {
      "id": "mko-equivalence-class",
      "title_zh": "等價類",
      "reason_zh": "orbit partition。"
    }
  ],
  "explanation": {
    "paragraphs_zh": [
      "orbit使用全部group elements。",
      "同orbit關係具反身對稱傳遞。",
      "orbits partition整個carrier。"
    ],
    "common_misconception_zh": "把g·x單一像、時間軌道或stabilizer稱為orbit。"
  },
  "proofs": [],
  "computational_companions": [],
  "verification": {
    "computational_status": "not_evaluated_no_computational_claim",
    "warning_zh": "只處理完整有限作用表；不推連續或拓撲群作用。",
    "evidence_refs": [],
    "producers": [
      {
        "id": "lean-mathlib",
        "adapter": "formal_system",
        "config_path": "formal/lean/lakefile.toml",
        "status": "configured"
      }
    ]
  },
  "formalization": {
    "status": "not_formalized",
    "target_systems": [
      "Lean 4",
      "Mathlib"
    ],
    "next_obligation_zh": "加入stabilizer與orbit-stabilizer theorem。"
  },
  "provenance": {
    "origin": "ocme_local_ai_materialization_candidate",
    "note_zh": "依 research/atlas-finite-group-orbit.md 整理；2026-10-14 脈衝 R01（使用者手動接續 2026-11-08 工作序列；實際日期仍為 2026-10-14），英文及 Lean 依使用者指示延後。",
    "content_license": "CC-BY-SA-4.0",
    "code_license": "MIT",
    "created_at": "2026-10-14"
  }
}