{
  "schema_version": "ocme-derivation-reader-v1",
  "guide_id": "derivation-mko-finite-group-stabilizer-01",
  "mko_id": "mko-finite-group-stabilizer",
  "title": {
    "zh-Hant": "point0的完整固定群元素",
    "en": "All group elements fixing point zero"
  },
  "contract": {
    "schema_version": "ocme-derivation-guide-v0.1",
    "guide_id": "derivation-mko-finite-group-stabilizer-01",
    "source": {
      "mko_id": "mko-finite-group-stabilizer",
      "commit": "b1b2f1dcf8c50d67a93fe52d95f8eef7b3abba9f",
      "sha256": "49304c0f16f563ddb9f6ef0acf3ad8e4e8d4361d20320142e367f75e49497c65",
      "english_sha256": "6a97398d871602e0d4587369bd44c2e589da9db81d48beb29fdcd7597b303436"
    },
    "mode": "worked_example",
    "audience_prerequisites": [
      "calc"
    ],
    "goal": {
      "zh-Hant": "Stab_G(0)={e,s}為子群；Stab_G(3)=G，與orbit及action kernel不同。",
      "en": "Stab_G(0)={e,s} is a subgroup; Stab_G(3)=G, distinct from orbit and action-kernel constructions."
    },
    "givens": [
      {
        "id": "G1",
        "statement": {
          "zh-Hant": "G={e,r,r²,s,rs,r²s}，依序編碼(0,0),(1,0),(2,0),(0,1),(1,1),(2,1)。乘法(i,j)(k,l)=(i+(−1)^j k mod3,j+l mod2)。點載體X={0,1,2,3}；(i,j)作用x<3為i+(−1)^j x mod3，並固定3。",
          "en": "G={e,r,r2,s,rs,r2s}, encoded respectively (0,0),(1,0),(2,0),(0,1),(1,1),(2,1), with (i,j)(k,l)=(i+(-1)^j*k mod3,j+l mod2). Points X={0,1,2,3}; (i,j) acts on x<3 by i+(-1)^j*x mod3 and fixes 3."
        }
      }
    ],
    "steps": [
      {
        "id": "S01",
        "uses": [
          "G1"
        ],
        "before_tex": "G=\\{e,r,r^2,s,rs,r^2s\\}",
        "after_tex": "\\begin{array}{c|cccccc}*&e&r&r^2&s&rs&r^2s\\\\\\hline e&e&r&r^2&s&rs&r^2s\\\\r&r&r^2&e&rs&r^2s&s\\\\r^2&r^2&e&r&r^2s&s&rs\\\\s&s&r^2s&rs&e&r^2&r\\\\rs&rs&s&r^2s&r&e&r^2\\\\r^2s&r^2s&rs&s&r^2&r&e\\\\\\end{array}",
        "relation": "implies",
        "rule": "calc",
        "conditions": [],
        "explanation": {
          "zh-Hant": "將給定配對公式逐36對展開，全部輸出在同一六元素載體，沒有缺格。",
          "en": "Expand the pair formula for all 36 products; every output belongs to the same six-element carrier and no cell is missing."
        },
        "reverse_audit": {
          "kind": "premise_audit",
          "required_ids": [
            "G1"
          ],
          "explanation": {
            "zh-Hant": "labels依序綁定六個pair，故同名r²不代表額外第七元素。",
            "en": "The six labels bind the six pairs; r^2 is not an additional seventh element."
          }
        },
        "leaf_basis": "calc"
      },
      {
        "id": "S02",
        "uses": [
          "G1",
          "S01"
        ],
        "before_tex": "(i,j)(k,l)=(i+(-1)^jk,j+l)\\quad(\\bmod(3,2))",
        "after_tex": "((i,j)(k,l))(m,n)=(i+(-1)^jk+(-1)^{j+l}m,j+l+n)=(i,j)((k,l)(m,n));\\quad e=(0,0);\\quad(i,j)^{-1}=(-(-1)^ji,j)",
        "relation": "implies",
        "rule": "calc",
        "conditions": [],
        "explanation": {
          "zh-Hant": "兩種括號的座標公式相同，證全部216 triples結合律；(0,0)兩側中性，列出的逆元兩側乘回(0,0)。",
          "en": "Both bracketings have identical coordinates, proving associativity for all 216 triples; (0,0) is a two-sided identity and the inverse formula multiplies back on both sides."
        },
        "reverse_audit": {
          "kind": "premise_audit",
          "required_ids": [
            "G1",
            "S01"
          ],
          "explanation": {
            "zh-Hant": "指數j+l只依mod2，因(−1)²=1，公式對所有代表一致；逆元清單e,r²,r,s,rs,r²s完整。",
            "en": "The sign depends only on j+l modulo 2 since (-1)^2=1; the full inverse list is e,r^2,r,s,rs,r^2s."
          }
        },
        "leaf_basis": "calc"
      },
      {
        "id": "S03",
        "uses": [
          "G1",
          "S02"
        ],
        "before_tex": "X=\\{0,1,2,3\\}",
        "after_tex": "\\begin{array}{c|cccc}\\cdot&0&1&2&3\\\\\\hline e&0&1&2&3\\\\r&1&2&0&3\\\\r^2&2&0&1&3\\\\s&0&2&1&3\\\\rs&1&0&2&3\\\\r^2s&2&1&0&3\\\\\\end{array}",
        "relation": "implies",
        "rule": "calc",
        "conditions": [],
        "explanation": {
          "zh-Hant": "全部24個作用值由同一公式展開。對x<3，g·(h·x)=i+(−1)^j k+(−1)^{j+l}x=(gh)·x；3兩邊皆固定。e逐點固定。",
          "en": "Expand all 24 action values. For x<3, g*(h*x)=i+(-1)^j*k+(-1)^(j+l)*x=(gh)*x; both sides fix 3, and e fixes every point."
        },
        "reverse_audit": {
          "kind": "premise_audit",
          "required_ids": [
            "G1",
            "S02"
          ],
          "explanation": {
            "zh-Hant": "表格對每個g與每個point恰有一值；相容式涵蓋全部6²·4=144 triples，不是只核每列像排列。",
            "en": "Every group-point pair has one value; the formula covers all 144 triples, not merely permutation rows."
          }
        },
        "leaf_basis": "calc"
      },
      {
        "id": "S04",
        "uses": [
          "G1",
          "S03"
        ],
        "before_tex": "g=e,r,r^2,s,rs,r^2s",
        "after_tex": "g\\cdot0=0,1,2,0,1,2",
        "relation": "implies",
        "rule": "calc",
        "conditions": [],
        "explanation": {
          "zh-Hant": "逐群元素核固定0條件，只有e與s給0。",
          "en": "Check the fixed-zero condition for each group element; only e and s give zero."
        },
        "reverse_audit": {
          "kind": "premise_audit",
          "required_ids": [
            "G1",
            "S03"
          ],
          "explanation": {
            "zh-Hant": "找到e不能停止，還有非單位s也固定0。",
            "en": "Finding e is not enough; nonidentity s fixes zero too."
          }
        },
        "leaf_basis": "calc"
      },
      {
        "id": "S05",
        "uses": [
          "G1",
          "S04"
        ],
        "before_tex": "H=\\{g:g\\cdot0=0\\}",
        "after_tex": "H=\\{e,s\\},\\quad e\\in H",
        "relation": "implies",
        "rule": "mko_definition",
        "conditions": [],
        "explanation": {
          "zh-Hant": "完整column給整個stabilizer，identity membership明列。",
          "en": "The complete column gives the whole stabilizer, with explicit identity membership."
        },
        "reverse_audit": {
          "kind": "premise_audit",
          "required_ids": [
            "G1",
            "S04"
          ],
          "explanation": {
            "zh-Hant": "H屬G的子集；orbit0則屬point載體X。",
            "en": "H is a subset of G, while orbit 0 is a subset of point carrier X."
          }
        },
        "leaf_basis": "mko_definition"
      },
      {
        "id": "S06",
        "uses": [
          "G1",
          "S01",
          "S05"
        ],
        "before_tex": "H=\\{e,s\\}",
        "after_tex": "\\begin{array}{c|cc}*&e&s\\\\\\hline e&e&s\\\\s&s&e\\end{array}",
        "relation": "implies",
        "rule": "calc",
        "conditions": [],
        "explanation": {
          "zh-Hant": "H全部四乘積落H，直接驗product closure而非僅抽樣s²。",
          "en": "All four products in H stay in H, directly checking closure rather than sampling only s^2."
        },
        "reverse_audit": {
          "kind": "premise_audit",
          "required_ids": [
            "G1",
            "S01",
            "S05"
          ],
          "explanation": {
            "zh-Hant": "使用同一父群表，不能另外替H設乘法。",
            "en": "Use the same parent multiplication, not a new operation on H."
          }
        },
        "leaf_basis": "calc"
      },
      {
        "id": "S07",
        "uses": [
          "G1",
          "S02",
          "S05",
          "S06"
        ],
        "before_tex": "e^{-1}=e,\\quad s^{-1}=s",
        "after_tex": "H^{-1}=H\\Rightarrow H\\le G",
        "relation": "implies",
        "rule": "calc",
        "conditions": [],
        "explanation": {
          "zh-Hant": "全部兩逆元在H，配合identity與product closure完成子群資格。",
          "en": "Both inverses lie in H; with identity and product closure this qualifies H as a subgroup."
        },
        "reverse_audit": {
          "kind": "premise_audit",
          "required_ids": [
            "G1",
            "S02",
            "S05",
            "S06"
          ],
          "explanation": {
            "zh-Hant": "stabilizer的子群結論已直接核，不因大小2自動成立。",
            "en": "The subgroup conclusion is directly checked, not inferred from size two."
          }
        },
        "leaf_basis": "calc"
      },
      {
        "id": "S08",
        "uses": [
          "G1",
          "S03",
          "S05"
        ],
        "before_tex": "g\\cdot3=3\\quad\\forall g\\in G",
        "after_tex": "\\operatorname{Stab}_G(3)=G\\ne H",
        "relation": "implies",
        "rule": "calc",
        "conditions": [],
        "explanation": {
          "zh-Hant": "完整3-column給不同point的stabilizer，顯示其依point而變。",
          "en": "The complete 3-column gives a different point stabilizer, showing point dependence."
        },
        "reverse_audit": {
          "kind": "premise_audit",
          "required_ids": [
            "G1",
            "S03",
            "S05"
          ],
          "explanation": {
            "zh-Hant": "不能把point3的群大小代入point0的公式。",
            "en": "The stabilizer size at 3 cannot be substituted for the one at 0."
          }
        },
        "leaf_basis": "calc"
      },
      {
        "id": "S09",
        "uses": [
          "G1",
          "S03",
          "S05",
          "S08"
        ],
        "before_tex": "s\\cdot1=2,\\quad rs\\cdot0=1,\\quad r^2s\\cdot0=2,\\quad r\\cdot0=1,\\quad r^2\\cdot0=2",
        "after_tex": "\\{g:g\\cdot x=x\\ \\forall x\\in X\\}=\\{e\\}\\ne\\operatorname{Stab}_G(0)",
        "relation": "implies",
        "rule": "calc",
        "conditions": [],
        "explanation": {
          "zh-Hant": "完整排除其餘五元素的全點固定性，action kernel只有e，而point stabilizer含s。",
          "en": "Explicitly exclude all five other elements from fixing every point; the action kernel is {e}, whereas the point stabilizer contains s."
        },
        "reverse_audit": {
          "kind": "premise_audit",
          "required_ids": [
            "G1",
            "S03",
            "S05",
            "S08"
          ],
          "explanation": {
            "zh-Hant": "action kernel要求所有point，stabilizer只要求指定0，量詞不同。",
            "en": "The action kernel fixes every point, while the stabilizer fixes the specified 0; the quantifiers differ."
          }
        },
        "leaf_basis": "calc"
      }
    ],
    "result": {
      "zh-Hant": "Stab_G(0)={e,s}為子群；Stab_G(3)=G，與orbit及action kernel不同。",
      "en": "Stab_G(0)={e,s} is a subgroup; Stab_G(3)=G, distinct from orbit and action-kernel constructions."
    },
    "negative_controls": [
      {
        "claim": {
          "zh-Hant": "Stab(0)={e}因只核identity。",
          "en": "Stab(0)={e} after checking only identity."
        },
        "expected": "reject",
        "reason": {
          "zh-Hant": "s·0=0亦成立。",
          "en": "s*0=0 also holds."
        }
      },
      {
        "claim": {
          "zh-Hant": "Stab(0)=orbit0。",
          "en": "Stab(0)=orbit 0."
        },
        "expected": "reject",
        "reason": {
          "zh-Hant": "前者是群元素{e,s}，後者points{0,1,2}。",
          "en": "The former is group elements {e,s}; the latter points {0,1,2}."
        }
      },
      {
        "claim": {
          "zh-Hant": "point stabilizer就是action kernel。",
          "en": "A point stabilizer is the action kernel."
        },
        "expected": "reject",
        "reason": {
          "zh-Hant": "s固定0但把1送2，所以不在action kernel。",
          "en": "s fixes 0 but sends 1 to 2 and is not in the action kernel."
        }
      }
    ],
    "status": "candidate",
    "gaps": [],
    "verification": {
      "structure": "checked_local_script",
      "mathematics": "candidate_argument",
      "formalization": "not_formalized"
    },
    "rule_glossary": [
      {
        "id": "calc",
        "kind": "prerequisite",
        "statement": {
          "zh-Hant": "假設精確模3、模2整數算術、有限集合外延相等與群／作用的定義；不預設此運算已為群。",
          "en": "Assume exact arithmetic modulo 3 and 2, extensional finite-set equality, and definitions of groups/actions; this operation is not pre-assumed to be a group."
        }
      },
      {
        "id": "mko_definition",
        "kind": "cited",
        "statement": {
          "zh-Hant": "在同一完整認證作用G作用X上，Stab_G(x)={g:g·x=x}須遍歷全部g。完整驗identity、product closure與inverse closure；單一fixed element、action kernel與orbit均不可混同stabilizer。",
          "en": "For the same fully certified action of G on X, Stab_G(x)={g:g*x=x} requires enumeration of every g. Check identity membership, product closure and inverse closure completely in the same group. One fixing element does not give the full stabilizer. A point stabilizer, an orbit and the action kernel are different constructions, even when particular underlying sets happen to agree."
        },
        "source": {
          "mko_id": "mko-finite-group-stabilizer",
          "sha256": "49304c0f16f563ddb9f6ef0acf3ad8e4e8d4361d20320142e367f75e49497c65",
          "commit": "b1b2f1dcf8c50d67a93fe52d95f8eef7b3abba9f"
        }
      }
    ],
    "revision_log": [],
    "numeric_checks": []
  },
  "presentation": {
    "md_scope": {
      "zh-Hant": "同一完整作用的point stabilizer，直接核全部乘積與逆元。",
      "en": "A point stabilizer in the same complete action, directly checking all products and inverses."
    },
    "md_symbols": [
      [
        "G",
        "六元素群載體",
        "six-element group carrier"
      ],
      [
        "r,s",
        "r³=e,s²=e,sr=r²s的labels",
        "labels with r^3=e,s^2=e,sr=r^2s"
      ],
      [
        "H",
        "point0的stabilizer",
        "stabilizer of point zero"
      ]
    ],
    "md_error_controls": [
      {
        "title": "同一群表 / Same group table",
        "zh": "全部積、逆元與cosets綁定這個完整Cayley表。",
        "en": "Bind every product, inverse and coset to this complete Cayley table."
      },
      {
        "title": "完整枚舉 / Complete enumeration",
        "zh": "任何全稱有限性質都需全載體或全稱公式，不用proper sample。",
        "en": "Every universal finite property needs the full carrier or a universal formula, not a proper sample."
      }
    ],
    "md_open_items": []
  },
  "revision": {
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    "collection": "coverage-20261021",
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    "original_source_sha256": "7a379848dfe73153d408591aeb7867aa3fea39f96d117af769783f52c3e4b936",
    "owner_adopted": true,
    "bilingual_sync": "complete_source_bound",
    "formal_Lean": "NOT_RUN",
    "reader_review": "142_of_142_source_bound",
    "wording_review": "original_Lead_decisions_resolved",
    "glossary_reader_paraphrases": [],
    "independent_AI_concurrence": "SCOPED_LEAD_AND_BILINGUAL_CONTEXT_REVIEW",
    "issue_ids": [],
    "parent_manifest_sha256": "d29f4cae8229ebd24f60d496cb1d3cd4ad8c0a4dca9d44daab0c3245c544283a",
    "scope": "Owner-adopted reader revision. Lead and bilingual review concern the recorded text and premises; no new universal or Lean proof, canonical promotion or historical-label rewrite."
  }
}