{
  "schema_version": "ocme-derivation-reader-v1",
  "guide_id": "derivation-mko-finite-group-orbit-01",
  "mko_id": "mko-finite-group-orbit",
  "title": {
    "zh-Hant": "完整orbits與partition",
    "en": "Complete orbits and their partition"
  },
  "contract": {
    "schema_version": "ocme-derivation-guide-v0.1",
    "guide_id": "derivation-mko-finite-group-orbit-01",
    "source": {
      "mko_id": "mko-finite-group-orbit",
      "commit": "b1b2f1dcf8c50d67a93fe52d95f8eef7b3abba9f",
      "sha256": "090d257b39b4d7071dedd320490e40cf67e01ea92431b0b6dc042f6127849022",
      "english_sha256": "6f0d82e7b440502269691a0b3252ad89327583a2459e8dd927850ba70396e4d2"
    },
    "mode": "worked_example",
    "audience_prerequisites": [
      "calc"
    ],
    "goal": {
      "zh-Hant": "X的全部互異orbits為{0,1,2}與{3}，同orbit關係是等價關係。",
      "en": "The distinct orbits are {0,1,2} and {3}; same-orbit is an equivalence relation."
    },
    "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\\Rightarrow G\\cdot0=\\{0,1,2\\}",
        "relation": "implies",
        "rule": "calc",
        "conditions": [],
        "explanation": {
          "zh-Hant": "完整0-column遍歷全部六群元素，再去重得完整orbit。",
          "en": "Enumerate all six group elements in the 0-column and deduplicate to obtain the whole orbit."
        },
        "reverse_audit": {
          "kind": "premise_audit",
          "required_ids": [
            "G1",
            "S03"
          ],
          "explanation": {
            "zh-Hant": "只取e或單一次r·0=1都不是完整orbit。",
            "en": "Taking only e or the single image r*0=1 is not the whole orbit."
          }
        },
        "leaf_basis": "calc"
      },
      {
        "id": "S05",
        "uses": [
          "G1",
          "S03"
        ],
        "before_tex": "g\\in G",
        "after_tex": "g\\cdot3=3\\Rightarrow G\\cdot3=\\{3\\}",
        "relation": "implies",
        "rule": "calc",
        "conditions": [],
        "explanation": {
          "zh-Hant": "完整3-column全部固定3，故orbit只有自身。",
          "en": "Every entry of the complete 3-column fixes 3, so this orbit is a singleton."
        },
        "reverse_audit": {
          "kind": "premise_audit",
          "required_ids": [
            "G1",
            "S03"
          ],
          "explanation": {
            "zh-Hant": "orbit非空因e·3=3；固定point並不表示沒有orbit。",
            "en": "The orbit is nonempty since e*3=3; a fixed point still has an orbit."
          }
        },
        "leaf_basis": "calc"
      },
      {
        "id": "S06",
        "uses": [
          "G1",
          "S02",
          "S03"
        ],
        "before_tex": "x\\sim y\\iff\\exists g:\\ g\\cdot x=y",
        "after_tex": "e\\cdot x=x;\\quad g\\cdot x=y\\Rightarrow g^{-1}\\cdot y=x;\\quad h\\cdot y=z\\Rightarrow(hg)\\cdot x=z",
        "relation": "implies",
        "rule": "calc",
        "conditions": [],
        "explanation": {
          "zh-Hant": "identity、inverse與composition分別證反身、對稱、傳遞，故同orbit為等價關係。",
          "en": "Identity, inverse and composition establish reflexivity, symmetry and transitivity, proving same-orbit equivalence."
        },
        "reverse_audit": {
          "kind": "premise_audit",
          "required_ids": [
            "G1",
            "S02",
            "S03"
          ],
          "explanation": {
            "zh-Hant": "兩個witness的組合順序是hg，因先g再h。",
            "en": "The combined witness is hg because g acts first and then h."
          }
        },
        "leaf_basis": "calc"
      },
      {
        "id": "S07",
        "uses": [
          "G1",
          "S03",
          "S04",
          "S06"
        ],
        "before_tex": "g\\cdot1=1,2,0,2,0,1;\\quad g\\cdot2=2,0,1,1,2,0",
        "after_tex": "G\\cdot1=G\\cdot2=\\{0,1,2\\}",
        "relation": "implies",
        "rule": "calc",
        "conditions": [],
        "explanation": {
          "zh-Hant": "另兩columns也完整去重，確定全部四點的orbit已列完。",
          "en": "Deduplicating the other two complete columns accounts for the orbits of all four points."
        },
        "reverse_audit": {
          "kind": "premise_audit",
          "required_ids": [
            "G1",
            "S03",
            "S04",
            "S06"
          ],
          "explanation": {
            "zh-Hant": "同orbit的不同起點可有相同集合，不應重算成三個互異orbits。",
            "en": "Different base points may have equal orbit sets, which must not be counted as distinct orbits."
          }
        },
        "leaf_basis": "calc"
      },
      {
        "id": "S08",
        "uses": [
          "G1",
          "S04",
          "S05",
          "S07"
        ],
        "before_tex": "O=\\{0,1,2\\},\\quad P=\\{3\\}",
        "after_tex": "O\\cap P=\\varnothing,\\quad O\\cup P=X,\\quad X/G=\\{O,P\\}",
        "relation": "implies",
        "rule": "mko_definition",
        "conditions": [],
        "explanation": {
          "zh-Hant": "全部互異orbits非空、互斥且覆蓋X，直接認證partition。",
          "en": "All distinct orbits are nonempty, disjoint and cover X, directly certifying a partition."
        },
        "reverse_audit": {
          "kind": "premise_audit",
          "required_ids": [
            "G1",
            "S04",
            "S05",
            "S07"
          ],
          "explanation": {
            "zh-Hant": "這是point的子集partition，不是群元素的stabilizer或cosets。",
            "en": "This partitions points and is not a collection of stabilizers or group cosets."
          }
        },
        "leaf_basis": "mko_definition"
      }
    ],
    "result": {
      "zh-Hant": "X的全部互異orbits為{0,1,2}與{3}，同orbit關係是等價關係。",
      "en": "The distinct orbits are {0,1,2} and {3}; same-orbit is an equivalence relation."
    },
    "negative_controls": [
      {
        "claim": {
          "zh-Hant": "r·0=1就是orbit。",
          "en": "r*0=1 is the orbit."
        },
        "expected": "reject",
        "reason": {
          "zh-Hant": "orbit是全體像的集合{0,1,2}，不是單一point1。",
          "en": "The orbit is the full image set {0,1,2}, not point 1."
        }
      },
      {
        "claim": {
          "zh-Hant": "point3沒有orbit因不動。",
          "en": "Point 3 has no orbit because it is fixed."
        },
        "expected": "reject",
        "reason": {
          "zh-Hant": "orbit恰為{3}。",
          "en": "Its orbit is exactly {3}."
        }
      },
      {
        "claim": {
          "zh-Hant": "orbit0與orbit1互斥。",
          "en": "The orbits of 0 and 1 are disjoint."
        },
        "expected": "reject",
        "reason": {
          "zh-Hant": "兩者皆為{0,1,2}，完全相等。",
          "en": "Both are {0,1,2} and are equal."
        }
      }
    ],
    "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。",
          "en": "Start with a fully qualified finite group action and enumerate G*x={g*x:g∈G} using every group element. The relation x~y exactly when some g sends x to y is an equivalence relation. Orbits are equal or disjoint and cover the carrier. One action image, finitely sampled generator steps and a stabilizer are not the whole orbit."
        },
        "source": {
          "mko_id": "mko-finite-group-orbit",
          "sha256": "090d257b39b4d7071dedd320490e40cf67e01ea92431b0b6dc042f6127849022",
          "commit": "b1b2f1dcf8c50d67a93fe52d95f8eef7b3abba9f"
        }
      }
    ],
    "revision_log": [],
    "numeric_checks": []
  },
  "presentation": {
    "md_scope": {
      "zh-Hant": "同一完整六元素四點作用，遍歷全部g，不用單generator有界步。",
      "en": "The same complete six-element/four-point action, using every g rather than bounded generator steps."
    },
    "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"
      ],
      [
        "G\\cdot x",
        "指定point的全部群像",
        "all group images of a specified point"
      ]
    ],
    "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": {
    "id": "ocme-new142-reader-adoption-20261021",
    "collection": "coverage-20261021",
    "source_commit": "b1b2f1dcf8c50d67a93fe52d95f8eef7b3abba9f",
    "adopted_source_sha256": "15c74347343414557ec306e3bda093c9fb53df12da58f4a045f551f1a6c45b11",
    "original_source_sha256": "15c74347343414557ec306e3bda093c9fb53df12da58f4a045f551f1a6c45b11",
    "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."
  }
}