{
  "$schema": "../../../schemas/mko-v0.4.schema.json",
  "schema_version": "mko-v0.4",
  "id": "mko-orbit-stabilizer-theorem",
  "semantic_family_id": "sf-orbit-stabilizer-theorem",
  "type": "theorem",
  "version": "0.10.0",
  "language": "zh-Hant",
  "titles": {
    "zh-Hant": "軌道－穩定子定理"
  },
  "summary": {
    "zh-Hant": "完整有限群作用中，同一點的orbit大小乘以stabilizer大小等於群階。"
  },
  "statement": {
    "zh-Hant": "固定同一完整有限作用G作用X與x。action map phi_x(g)=g·x對orbit滿射；若g0·x=y，fiber精確等於左陪集g0 Stab_G(x)，故|G|=|G·x||Stab_G(x)|。不要求action free、faithful或transitive；proper sample、不同point綁定或只碰巧數值相等皆不足。"
  },
  "formula": {
    "tex": "c(G)=c(O(x))*c(S(x))",
    "mathml": "<math xmlns=\"http://www.w3.org/1998/Math/MathML\"><mrow><mrow><mi>c</mi><mo>(</mo><mi>G</mi><mo>)</mo></mrow><mo>=</mo><mrow><mrow><mi>c</mi><mo>(</mo><mrow><mi>O</mi><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo>)</mo></mrow><mo>×</mo><mrow><mi>c</mi><mo>(</mo><mrow><mi>S</mi><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo>)</mo></mrow></mrow></mrow></math>",
    "semantic_ast": {
      "type": "equation",
      "lhs": {
        "type": "function_call",
        "callee": {
          "type": "symbol",
          "name": "c",
          "glyph": "c"
        },
        "arguments": [
          {
            "type": "symbol",
            "name": "G",
            "glyph": "G"
          }
        ]
      },
      "rhs": {
        "type": "multiplication",
        "operands": [
          {
            "type": "function_call",
            "callee": {
              "type": "symbol",
              "name": "c",
              "glyph": "c"
            },
            "arguments": [
              {
                "type": "function_call",
                "callee": {
                  "type": "symbol",
                  "name": "O",
                  "glyph": "O"
                },
                "arguments": [
                  {
                    "type": "symbol",
                    "name": "x",
                    "glyph": "x"
                  }
                ]
              }
            ]
          },
          {
            "type": "function_call",
            "callee": {
              "type": "symbol",
              "name": "c",
              "glyph": "c"
            },
            "arguments": [
              {
                "type": "function_call",
                "callee": {
                  "type": "symbol",
                  "name": "S",
                  "glyph": "S"
                },
                "arguments": [
                  {
                    "type": "symbol",
                    "name": "x",
                    "glyph": "x"
                  }
                ]
              }
            ]
          }
        ]
      }
    },
    "compiler": {
      "id": "ocme-formula-core",
      "version": "0.4.0",
      "source_sha256": "e13e6985c58b72ee887b8179273166010b5768d98c4e77c41c26d08cb6fae4ca"
    }
  },
  "assumptions": [
    {
      "id": "a1",
      "zh-Hant": "完整有限群表與total action table已通過全部作用公理。"
    },
    {
      "id": "a2",
      "zh-Hant": "orbit、stabilizer與fibers均相對同一指定point完整枚舉。"
    }
  ],
  "symbols": [
    {
      "token": "c",
      "role": "finite_cardinality",
      "role_zh": "有限基數",
      "scope": "representative_formula"
    },
    {
      "token": "G",
      "role": "acting_finite_group",
      "role_zh": "作用有限群",
      "scope": "representative_formula"
    },
    {
      "token": "O",
      "role": "orbit_operator",
      "role_zh": "軌道算子",
      "scope": "representative_formula"
    },
    {
      "token": "x",
      "role": "acted_point",
      "role_zh": "指定點",
      "scope": "representative_formula"
    },
    {
      "token": "S",
      "role": "stabilizer_operator",
      "role_zh": "穩定子算子",
      "scope": "representative_formula"
    }
  ],
  "dependencies": [
    {
      "id": "mko-finite-group-orbit",
      "title_zh": "有限群作用的軌道",
      "reason_zh": "完整orbit。"
    },
    {
      "id": "mko-finite-group-stabilizer",
      "title_zh": "有限群作用的穩定子群",
      "reason_zh": "同一point stabilizer。"
    },
    {
      "id": "mko-cardinality-comparison",
      "title_zh": "基數比較",
      "reason_zh": "finite coset/fiber counting。"
    }
  ],
  "explanation": {
    "paragraphs_zh": [
      "action map的fibers是stabilizer左陪集。",
      "每個fiber大小相同且orbit是像。",
      "計數identity不加入free或transitive假設。"
    ],
    "common_misconception_zh": "只驗|G|數值乘積，卻未綁定同一作用、同一點或fiber-coset bijection。"
  },
  "proofs": [],
  "computational_companions": [],
  "verification": {
    "computational_status": "not_evaluated_no_computational_claim",
    "warning_zh": "本輪限完整有限作用；不由sampled group elements推一般定理。",
    "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Cauchy定理與同態kernel。"
  },
  "provenance": {
    "origin": "ocme_local_ai_materialization_candidate",
    "note_zh": "依 research/atlas-orbit-stabilizer-theorem.md 整理；2026-10-14 脈衝 R01（使用者手動接續 2026-11-10 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"
  }
}