{
  "schema_version": "ocme-derivation-reader-v1",
  "guide_id": "derivation-mko-finite-quotient-group-01",
  "mko_id": "mko-finite-quotient-group",
  "title": {
    "zh-Hant": "全部代表pairs的商群表",
    "en": "A quotient table checked on every representative pair"
  },
  "contract": {
    "schema_version": "ocme-derivation-guide-v0.1",
    "guide_id": "derivation-mko-finite-quotient-group-01",
    "source": {
      "mko_id": "mko-finite-quotient-group",
      "commit": "b1b2f1dcf8c50d67a93fe52d95f8eef7b3abba9f",
      "sha256": "f3e42a5b51685f805af8727447418b3b158bf4c512da0e78fd2804f7a04bba24",
      "english_sha256": "036ebd3e0c66d661a3e151b97f378a2d60aae4fc019e98ba7d711292436a7f89"
    },
    "mode": "worked_example",
    "audience_prerequisites": [
      "calc"
    ],
    "goal": {
      "zh-Hant": "G/N的兩cosets A=N、B={s,rs,r²s}具良定乘法AA=A,AB=BA=B,BB=A。",
      "en": "G/N has cosets A=N,B={s,rs,r2s} and well-defined products AA=A,AB=BA=B,BB=A."
    },
    "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)。",
          "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)."
        }
      }
    ],
    "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",
          "S01",
          "S02"
        ],
        "before_tex": "N=\\{e,r,r^2\\}",
        "after_tex": "\\begin{array}{c|ccc}*&e&r&r^2\\\\\\hline e&e&r&r^2\\\\r&r&r^2&e\\\\r^2&r^2&e&r\\\\\\end{array}\\qquad\\begin{array}{c|ccc}gng^{-1}&e&r&r^2\\\\\\hline e&e&r&r^2\\\\r&e&r&r^2\\\\r^2&e&r&r^2\\\\s&e&r^2&r\\\\rs&e&r^2&r\\\\r^2s&e&r^2&r\\\\\\end{array}",
        "relation": "implies",
        "rule": "calc",
        "conditions": [],
        "explanation": {
          "zh-Hant": "九乘積、逆元e,r²,r及完整18格共軛皆留N，因此N先認證normal subgroup。",
          "en": "All nine products, inverses e,r^2,r and the complete 18-entry conjugation table qualify N as a normal subgroup first."
        },
        "reverse_audit": {
          "kind": "premise_audit",
          "required_ids": [
            "G1",
            "S01",
            "S02"
          ],
          "explanation": {
            "zh-Hant": "此資格不是因為直接寫下G/N便自動存在。",
            "en": "Qualification does not follow merely from writing G/N."
          }
        },
        "leaf_basis": "calc"
      },
      {
        "id": "S04",
        "uses": [
          "G1",
          "S01",
          "S03"
        ],
        "before_tex": "A=N,\\quad B=\\{s,rs,r^2s\\}",
        "after_tex": "eN=rN=r^2N=A;\\quad sN=(rs)N=(r^2s)N=B;\\quad A\\sqcup B=G",
        "relation": "implies",
        "rule": "calc",
        "conditions": [],
        "explanation": {
          "zh-Hant": "全部六left cosets去重成兩個非空互斥coset sets，覆蓋G。",
          "en": "All six left cosets deduplicate to two nonempty disjoint coset sets covering G."
        },
        "reverse_audit": {
          "kind": "premise_audit",
          "required_ids": [
            "G1",
            "S01",
            "S03"
          ],
          "explanation": {
            "zh-Hant": "商元素是完整A、B，不是e與s兩個代表元。",
            "en": "Quotient elements are the full sets A,B, not representatives e,s."
          }
        },
        "leaf_basis": "calc"
      },
      {
        "id": "S05",
        "uses": [
          "G1",
          "S01",
          "S04"
        ],
        "before_tex": "g,h=e,r,r^2,s,rs,r^2s",
        "after_tex": "\\begin{array}{c|cccccc}(gh)N&e&r&r^2&s&rs&r^2s\\\\\\hline e&A&A&A&B&B&B\\\\r&A&A&A&B&B&B\\\\r^2&A&A&A&B&B&B\\\\s&B&B&B&A&A&A\\\\rs&B&B&B&A&A&A\\\\r^2s&B&B&B&A&A&A\\\\\\end{array}",
        "relation": "implies",
        "rule": "calc",
        "conditions": [],
        "explanation": {
          "zh-Hant": "完整36格記錄每個gh落哪個coset，四個3×3區塊各是單一類。",
          "en": "All 36 entries record the coset of gh; each of the four 3-by-3 blocks has a single class."
        },
        "reverse_audit": {
          "kind": "premise_audit",
          "required_ids": [
            "G1",
            "S01",
            "S04"
          ],
          "explanation": {
            "zh-Hant": "因此任意換同coset的兩個代表都不改product coset，已完整核而非只看e、s。",
            "en": "Thus any replacement of either representative within its coset preserves the product class, checking every pair rather than just e,s."
          }
        },
        "leaf_basis": "calc"
      },
      {
        "id": "S06",
        "uses": [
          "G1",
          "S04",
          "S05"
        ],
        "before_tex": "Q=\\{A,B\\}",
        "after_tex": "\\begin{array}{c|cc}*&A&B\\\\\\hline A&A&B\\\\B&B&A\\\\\\end{array}",
        "relation": "implies",
        "rule": "calc",
        "conditions": [],
        "explanation": {
          "zh-Hant": "把四個常值區塊縮成商乘法表，所有四乘法均良定。",
          "en": "Compress the four constant blocks into the quotient multiplication table, with all four products well-defined."
        },
        "reverse_audit": {
          "kind": "premise_audit",
          "required_ids": [
            "G1",
            "S04",
            "S05"
          ],
          "explanation": {
            "zh-Hant": "表的兩個labels綁coset sets，沒有另加母群元素。",
            "en": "The labels bind coset sets without adding parent-group elements."
          }
        },
        "leaf_basis": "calc"
      },
      {
        "id": "S07",
        "uses": [
          "G1",
          "S02",
          "S06"
        ],
        "before_tex": "A=N,\\quad B^{-1}=B",
        "after_tex": "A\\text{ identity},\\quad A^{-1}=A,\\quad B^{-1}=B",
        "relation": "implies",
        "rule": "calc",
        "conditions": [],
        "explanation": {
          "zh-Hant": "商表直接核identity與全部兩逆元；g⁻¹N與gN的此逆類相符。",
          "en": "The table directly checks identity and both inverses; these agree with the classes g inverse N."
        },
        "reverse_audit": {
          "kind": "premise_audit",
          "required_ids": [
            "G1",
            "S02",
            "S06"
          ],
          "explanation": {
            "zh-Hant": "B²=A而非B，不能把reflection類當identity。",
            "en": "B^2=A, not B, so the reflection class is not the identity."
          }
        },
        "leaf_basis": "calc"
      },
      {
        "id": "S08",
        "uses": [
          "G1",
          "S02",
          "S05",
          "S06",
          "S07"
        ],
        "before_tex": "((gN)(hN))(kN)=(gh)kN",
        "after_tex": "(gh)kN=g(hk)N=(gN)((hN)(kN))\\Rightarrow G/N\\text{ group}",
        "relation": "implies",
        "rule": "mko_definition",
        "conditions": [],
        "explanation": {
          "zh-Hant": "先有良定性，再由父群結合律下降；連同封閉、identity、inverse完成商群。",
          "en": "After well-definedness, associativity descends from G; closure, identity and inverses complete the quotient group."
        },
        "reverse_audit": {
          "kind": "premise_audit",
          "required_ids": [
            "G1",
            "S02",
            "S05",
            "S06",
            "S07"
          ],
          "explanation": {
            "zh-Hant": "本例商交換來自實際兩元素表，不宣稱任意商皆交換。",
            "en": "Commutativity here comes from the actual two-element table, not a claim about every quotient."
          }
        },
        "leaf_basis": "mko_definition"
      },
      {
        "id": "S09",
        "uses": [
          "G1",
          "S01",
          "S02",
          "S05",
          "S08"
        ],
        "before_tex": "H=\\{e,s\\},\\quad eH=sH=H",
        "after_tex": "er=r\\in rH=\\{r,rs\\};\\quad sr=r^2s\\in r^2H=\\{r^2,r^2s\\}",
        "relation": "implies",
        "rule": "calc",
        "conditions": [],
        "explanation": {
          "zh-Hant": "對非正規H，第一coset同代表e或s乘同一r卻落不同cosets，直接證乘法不良定。",
          "en": "For nonnormal H, representatives e and s of the same first coset multiplied by r land in different cosets, directly refuting well-defined multiplication."
        },
        "reverse_audit": {
          "kind": "premise_audit",
          "required_ids": [
            "G1",
            "S01",
            "S02",
            "S05",
            "S08"
          ],
          "explanation": {
            "zh-Hant": "H自身仍是子群，四乘積與逆元由父表核；失敗在代表獨立性。",
            "en": "H remains a subgroup by its four products and inverses; the failure is representative independence."
          }
        },
        "leaf_basis": "calc"
      }
    ],
    "result": {
      "zh-Hant": "G/N的兩cosets A=N、B={s,rs,r²s}具良定乘法AA=A,AB=BA=B,BB=A。",
      "en": "G/N has cosets A=N,B={s,rs,r2s} and well-defined products AA=A,AB=BA=B,BB=A."
    },
    "negative_controls": [
      {
        "claim": {
          "zh-Hant": "A就是單一元素e。",
          "en": "A is just the element e."
        },
        "expected": "reject",
        "reason": {
          "zh-Hant": "A={e,r,r²}是三元素coset。",
          "en": "A={e,r,r^2} is a three-element coset."
        }
      },
      {
        "claim": {
          "zh-Hant": "H不正規也可照選e、r代表乘成商群。",
          "en": "Nonnormal H permits a quotient by choosing e,r representatives."
        },
        "expected": "reject",
        "reason": {
          "zh-Hant": "e與s同coset，但er與sr的product cosets不同。",
          "en": "e and s share a coset while er and sr have different product cosets."
        }
      },
      {
        "claim": {
          "zh-Hant": "任何商群皆交換。",
          "en": "Every quotient group is commutative."
        },
        "expected": "reject",
        "reason": {
          "zh-Hant": "取N={e}則商保留本群的sr=r²s≠rs。",
          "en": "For N={e}, the quotient retains sr=r^2s distinct from rs."
        }
      }
    ],
    "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": "N須先由完整表認證為normal subgroup。全部left cosets形成disjoint complete partition，quotient elements是coset sets而非代表元。乘法表以coset index記錄，且對每一對cosets及所有代表元替換完整驗product落入同一coset。identity為N，inverse由g^{-1}N給出。若N不正規，明示代表元選擇可產生不同product cosets，因此不能建商群。商群不自動commutative，也不提前宣稱第一同構定理。",
          "en": "First certify N as a normal subgroup of the complete group table. All left cosets form a disjoint complete partition; quotient elements are coset sets, not representatives. Record multiplication by coset indices and verify every replacement of either representative lands in the same product coset. N is the identity coset and g^(-1)N is the inverse. A nonnormal subgroup can have representative-dependent products, blocking this quotient multiplication. Quotients are not automatically commutative, and no first-isomorphism claim is added."
        },
        "source": {
          "mko_id": "mko-finite-quotient-group",
          "sha256": "f3e42a5b51685f805af8727447418b3b158bf4c512da0e78fd2804f7a04bba24",
          "commit": "b1b2f1dcf8c50d67a93fe52d95f8eef7b3abba9f"
        }
      }
    ],
    "revision_log": [],
    "numeric_checks": [
      {
        "id": "pair_count",
        "lhs": "6*6",
        "rhs": "36",
        "step": "S05"
      }
    ]
  },
  "presentation": {
    "md_scope": {
      "zh-Hant": "完整正規資格、partition及36個代表pairs類指標；另列nonnormal失敗witness。",
      "en": "Complete normality, partition and 36 representative-pair class indices, with a nonnormal failure witness."
    },
    "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"
      ],
      [
        "A,B",
        "完整商cosets",
        "whole quotient cosets"
      ]
    ],
    "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": []
  },
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    "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": [],
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}