{
  "schema_version": "ocme-reviewed-question-reader-v1",
  "question_id": "q-20261013-advance-20261026-r07-equivalence-relation-0026",
  "pilot_number": "R0023-12351",
  "number": 12351,
  "family_id": "qf-mod26-equiv-symmetry-audit",
  "title": {
    "zh-Hant": "第 12351 題：等價關係",
    "en": "Question 12351: Record Original Pairs Whose Reverse Is Missing"
  },
  "original_question": {
    "schema_version": "ocme-question-v0.1",
    "id": "q-20261013-advance-20261026-r07-equivalence-relation-0026",
    "batch_id": "batch-20261013-advance-20261026-r07-equivalence-relation-foundations",
    "language": "zh-Hant",
    "source_mko_ids": [
      "mko-equivalence-relation"
    ],
    "title_zh": "等價關係",
    "stem_zh": "固定完整有限載體與關係對集合。操作=equiv-symmetry-audit；資料={\"n\":6,\"pairs\":[[0,0],[1,1],[2,2],[3,3],[4,4],[5,5],[0,1],[1,0]]}。請依 [n,missing_reverses,symmetric] 欄序回答。",
    "answer": {
      "kind": "expression",
      "value": "(6, [], 是)",
      "explanation_zh": "結構深度核對=(6, [], 是)；依完整明示資料重算；完整有限／精確切片不自動升格為任意無限定理；反身對稱傳遞三責任分開；類別只在完整等價資格後輸出。"
    },
    "difficulty": {
      "band": "undergraduate_core",
      "level": 1,
      "skills": [
        "等價關係"
      ]
    },
    "variant": {
      "family_id": "qf-mod26-equiv-symmetry-audit",
      "seed": 202610267026,
      "parameters": {
        "n": 6,
        "pairs": [
          [
            0,
            0
          ],
          [
            1,
            1
          ],
          [
            2,
            2
          ],
          [
            3,
            3
          ],
          [
            4,
            4
          ],
          [
            5,
            5
          ],
          [
            0,
            1
          ],
          [
            1,
            0
          ]
        ],
        "domain": "Complete_finite_equivalence_relation",
        "operation": "equiv-symmetry-audit",
        "semantic_template": "qf-mod26-equiv-symmetry-audit/{\"n\":6,\"pairs\":[[0,0],[0,1],[1,0],[1,1],[2,2],[3,3],[4,4],[5,5]]}"
      }
    },
    "verification": {
      "status": "candidate",
      "checks": [
        "producer_exact_finite_model",
        "producer_source_snapshot_check",
        "producer_substantive_template_deduplication",
        "producer_json_schema"
      ],
      "warning_zh": "產生端候選須由主端重算；不構成人工審定、一般無限定理、獨立性、一般圖同構、不限維分析、Lean執行或形式Evidence。"
    },
    "provenance": {
      "generator_role": "question_producer",
      "generator_model": "model_unknown",
      "generated_at": "2026-10-13",
      "contract_version": "ocme-question-factory-v0.1"
    }
  },
  "original_english_edition": {
    "schema_version": "ocme-question-localization-v0.1",
    "question_id": "q-20261013-advance-20261026-r07-equivalence-relation-0026",
    "batch_id": "batch-20261013-advance-20261026-r07-equivalence-relation-foundations",
    "locale": "en",
    "translation_state": "ai_candidate",
    "source": {
      "question_sha256": "4cb41fbf7e6ab04b71c237758791791fa0386ac347b842f9cc56cb00b3ab6a4c",
      "family_id": "qf-mod26-equiv-symmetry-audit",
      "seed": 202610267026,
      "semantic_template": "qf-mod26-equiv-symmetry-audit/{\"n\":6,\"pairs\":[[0,0],[0,1],[1,0],[1,1],[2,2],[3,3],[4,4],[5,5]]}"
    },
    "fields": {
      "title": "Record Original Pairs Whose Reverse Is Missing",
      "stem": "Use the full fixed carrier Fin(6)={0,…,5} and complete distinct ordered relation pairs=[[0, 0], [1, 1], [2, 2], [3, 3], [4, 4], [5, 5], [0, 1], [1, 0]]. Check reflexivity, symmetry and transitivity separately on the whole carrier. Equivalence classes may be emitted only after all three qualifications pass; one counterexample refutes its corresponding axiom. Give (n,lexicographically ordered original [a,b] pairs for which [b,a] is missing,symmetric?). The records are the original pairs, not their absent reverses.",
      "answer": {
        "kind": "expression",
        "value": "(6, [], Yes)",
        "explanation": "For every supplied ordered pair check its reverse. An empty relation passes symmetry vacuously but need not pass reflexivity. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (6, [], Yes)."
      },
      "skills": [
        "Record Original Pairs Whose Reverse Is Missing"
      ],
      "verification_warning": "This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."
    }
  },
  "givens": [
    {
      "id": "G001",
      "text": {
        "zh-Hant": "固定完整有限載體與關係對集合。操作=equiv-symmetry-audit；資料={\"n\":6,\"pairs\":[[0,0],[1,1],[2,2],[3,3],[4,4],[5,5],[0,1],[1,0]]}。請依 [n,missing_reverses,symmetric] 欄序回答。",
        "en": "Use the full fixed carrier Fin(6)={0,…,5} and complete distinct ordered relation pairs=[[0, 0], [1, 1], [2, 2], [3, 3], [4, 4], [5, 5], [0, 1], [1, 0]]. Check reflexivity, symmetry and transitivity separately on the whole carrier. Equivalence classes may be emitted only after all three qualifications pass; one counterexample refutes its corresponding axiom. Give (n,lexicographically ordered original [a,b] pairs for which [b,a] is missing,symmetric?). The records are the original pairs, not their absent reverses."
      }
    }
  ],
  "steps": [
    {
      "id": "S001",
      "uses": [
        "G001"
      ],
      "before": "raw relation pairs=[[0, 0], [1, 1], [2, 2], [3, 3], [4, 4], [5, 5], [0, 1], [1, 0]];carrier Fin(6)",
      "after": "lexicographically sorted distinct ordered pairs=[[0, 0], [0, 1], [1, 0], [1, 1], [2, 2], [3, 3], [4, 4], [5, 5]]",
      "relation": "implies",
      "rule_id": "complete_relation_no_duplicate_or_implicit_repair",
      "conditions": [],
      "explanation": {
        "zh-Hant": "方向有別的關係對按詞典序列出；重複不是合法輸入，不自行補反向、對角或傳遞閉包。",
        "en": "List distinct directed pairs lexicographically.Duplicates are invalid;do not silently add reverses,diagonals or a transitive closure."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S002",
      "uses": [
        "G001",
        "S001"
      ],
      "before": "original pair=[0, 0];required reverse=[0, 0]",
      "after": "reverse present=是",
      "relation": "implies",
      "rule_id": "relation_original_pair_missing_reverse_record",
      "conditions": [],
      "explanation": {
        "zh-Hant": "檢查逆向對是否存在，但失敗清單按協定記原來的[a,b]，不是缺少的[b,a]。空關係對稱是空泛真。",
        "en": "Check for the reverse,but record the original[a,b] on failure,not its missing[b,a] reverse.An empty relation is vacuously symmetric."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S003",
      "uses": [
        "G001",
        "S002"
      ],
      "before": "original pair=[0, 1];required reverse=[1, 0]",
      "after": "reverse present=是",
      "relation": "implies",
      "rule_id": "relation_original_pair_missing_reverse_record",
      "conditions": [],
      "explanation": {
        "zh-Hant": "檢查逆向對是否存在，但失敗清單按協定記原來的[a,b]，不是缺少的[b,a]。空關係對稱是空泛真。",
        "en": "Check for the reverse,but record the original[a,b] on failure,not its missing[b,a] reverse.An empty relation is vacuously symmetric."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S004",
      "uses": [
        "G001",
        "S003"
      ],
      "before": "original pair=[1, 0];required reverse=[0, 1]",
      "after": "reverse present=是",
      "relation": "implies",
      "rule_id": "relation_original_pair_missing_reverse_record",
      "conditions": [],
      "explanation": {
        "zh-Hant": "檢查逆向對是否存在，但失敗清單按協定記原來的[a,b]，不是缺少的[b,a]。空關係對稱是空泛真。",
        "en": "Check for the reverse,but record the original[a,b] on failure,not its missing[b,a] reverse.An empty relation is vacuously symmetric."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S005",
      "uses": [
        "G001",
        "S004"
      ],
      "before": "original pair=[1, 1];required reverse=[1, 1]",
      "after": "reverse present=是",
      "relation": "implies",
      "rule_id": "relation_original_pair_missing_reverse_record",
      "conditions": [],
      "explanation": {
        "zh-Hant": "檢查逆向對是否存在，但失敗清單按協定記原來的[a,b]，不是缺少的[b,a]。空關係對稱是空泛真。",
        "en": "Check for the reverse,but record the original[a,b] on failure,not its missing[b,a] reverse.An empty relation is vacuously symmetric."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S006",
      "uses": [
        "G001",
        "S005"
      ],
      "before": "original pair=[2, 2];required reverse=[2, 2]",
      "after": "reverse present=是",
      "relation": "implies",
      "rule_id": "relation_original_pair_missing_reverse_record",
      "conditions": [],
      "explanation": {
        "zh-Hant": "檢查逆向對是否存在，但失敗清單按協定記原來的[a,b]，不是缺少的[b,a]。空關係對稱是空泛真。",
        "en": "Check for the reverse,but record the original[a,b] on failure,not its missing[b,a] reverse.An empty relation is vacuously symmetric."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S007",
      "uses": [
        "G001",
        "S006"
      ],
      "before": "original pair=[3, 3];required reverse=[3, 3]",
      "after": "reverse present=是",
      "relation": "implies",
      "rule_id": "relation_original_pair_missing_reverse_record",
      "conditions": [],
      "explanation": {
        "zh-Hant": "檢查逆向對是否存在，但失敗清單按協定記原來的[a,b]，不是缺少的[b,a]。空關係對稱是空泛真。",
        "en": "Check for the reverse,but record the original[a,b] on failure,not its missing[b,a] reverse.An empty relation is vacuously symmetric."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S008",
      "uses": [
        "G001",
        "S007"
      ],
      "before": "original pair=[4, 4];required reverse=[4, 4]",
      "after": "reverse present=是",
      "relation": "implies",
      "rule_id": "relation_original_pair_missing_reverse_record",
      "conditions": [],
      "explanation": {
        "zh-Hant": "檢查逆向對是否存在，但失敗清單按協定記原來的[a,b]，不是缺少的[b,a]。空關係對稱是空泛真。",
        "en": "Check for the reverse,but record the original[a,b] on failure,not its missing[b,a] reverse.An empty relation is vacuously symmetric."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S009",
      "uses": [
        "G001",
        "S008"
      ],
      "before": "original pair=[5, 5];required reverse=[5, 5]",
      "after": "reverse present=是",
      "relation": "implies",
      "rule_id": "relation_original_pair_missing_reverse_record",
      "conditions": [],
      "explanation": {
        "zh-Hant": "檢查逆向對是否存在，但失敗清單按協定記原來的[a,b]，不是缺少的[b,a]。空關係對稱是空泛真。",
        "en": "Check for the reverse,but record the original[a,b] on failure,not its missing[b,a] reverse.An empty relation is vacuously symmetric."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    }
  ],
  "math_model": {
    "result": [
      6,
      [],
      true
    ]
  },
  "conclusion": {
    "canonical_value": "(6, [], 是)",
    "explanation": {
      "zh-Hant": "依上列完整代值及條件核對得到原題答案；原題編號、格式與狀態保持不變。",
      "en": "The full substitutions and premise checks give the original answer; source identity,format and status are unchanged."
    }
  },
  "definition_note": {
    "zh-Hant": "沿用原題明示的論域、假設與來源 MKO 定義。本題已完成第二對話數理與中英語義覆核，並由原始參數逐題獨立重算模型、答案與每個算術步驟；原題仍為 candidate。",
    "en": "Use the original question domain, assumptions and cited MKO definitions. This item has second-context mathematical and bilingual review, with the model, answer and every arithmetic step independently recomputed from the original parameters; the source question remains a candidate."
  },
  "source": {
    "input_commit": "0f7be61e2c22317a4075561dc822c87d881a50db",
    "question_binding": {
      "batch_id": "batch-20261013-advance-20261026-r07-equivalence-relation-foundations",
      "batch_path": "public/data/questions/batches/batch-20261013-advance-20261026-r07-equivalence-relation-foundations/questions.jsonl",
      "batch_sha256": "cb362754e8943f3ceca46c9953885155b9af4b21a087a4ee9548cd615af55137",
      "original_line": 26,
      "original_line_sha256": "976c3dc185e26c45cb963cd8253b21a724ab31ca1c5fdf377214d7496723fd54",
      "question_compact_sha256": "4cb41fbf7e6ab04b71c237758791791fa0386ac347b842f9cc56cb00b3ab6a4c",
      "english_row_compact_sha256": "3f929d712f75aa5ec698ab398f28c897337d9139eef3dff78e4b3d2d1aeb0c6d",
      "english_manifest_path": "public/data/i18n/editions/batch-20261013-advance-20261026-r07-equivalence-relation-foundations.en.json",
      "english_rows_path": "public/data/i18n/questions/batch-20261013-advance-20261026-r07-equivalence-relation-foundations.en.jsonl"
    },
    "mko_bindings": {
      "mko-equivalence-relation": {
        "sha256": "ceccd69c4881e4b2eba57eb1a7c6ab6c6616ab09fa42eb8591999522a5948fd8",
        "english_sha256": "f2bdfa53ace4565feb3a764d528c99663030234943ed2139aa6de2f8e878eb57",
        "original_path": "public/data/mko/equivalence-relation.json",
        "english_path": "public/data/i18n/editions/mko-equivalence-relation.en.json"
      }
    },
    "selected_manifest_sha256": "5233edba9654a6cffd8b6b2d83c932f82ded15f9c0b7c733bd049943e672acff",
    "selected_part_sha256": "c6ae588be9bc8b3dd8461be32c073abfa25a0f827b4602cf28bf291ce532d507",
    "original_candidate_sha256": "c2ddc573cad764cf5c45215ceb11f19425e9fba373119acb19911ed23bc87fc2",
    "original_candidate_line_sha256": "c2ddc573cad764cf5c45215ceb11f19425e9fba373119acb19911ed23bc87fc2"
  },
  "historical_source_state": {
    "review_state": {
      "derivation": "candidate_authored_and_locally_checked",
      "source_question_state": "candidate",
      "independent_AI_or_human_review": "NOT_PERFORMED",
      "formalization": "not_formalized",
      "Lean": "NOT_RUN"
    },
    "scope_note": {
      "zh-Hant": "來源綁定教學推導候選；精確核對不等於獨立AI／人工審查或一般形式證明。原題、答案、MKO、公式、AST與歷史不變。",
      "en": "Source-bound pedagogical derivation candidate.Exact checking is not independent AI/human review or general formal proof.Original questions,answers,MKOs,formulas,ASTs and history remain unchanged."
    },
    "label": "Frozen producer metadata; current per-question review is separate below."
  },
  "revision": {
    "id": "review-20261026-006-p24",
    "review_date": "2026-10-26",
    "owner_authorized_reader_publication": true,
    "scope": "One hundred sixty family prototypes and their mathematical/bilingual reasoning variants, stems, boundary statements and every distinct explanation shape were reviewed in a context independent of the producer; parameter substitutions, source bindings, full math models, both answer strings and every arithmetic step were independently reconstructed per question from the original parameters only, with negative controls. This grouped method does not claim a separate literal AI reading of every repeated sentence or an extra AI voter.",
    "review_receipts": {
      "semantic_review_sha256": "f783e81f44ff0faa13dbf948657e9187244a59a809ebdf96f22cca24966649ff",
      "question_first_witness_sha256": "49a50ed0550be55124d6562260c2ccf4fa6d3f316d7ad9d0d32174400e362e18"
    },
    "status": "second_context_reviewed",
    "mathematical_semantics": true,
    "bilingual_semantics": true,
    "steps_reviewed": 9,
    "method": "family_template_semantic_review_plus_independent_per_instance_reconstruction",
    "exact_question_first_check": {
      "model_sha256": "8667279becfed437e379c98d117fb030bfa71c8b91db0ec05e33cf17335f61e3",
      "arithmetic_steps": 0
    },
    "uses_answer_as_premise": false,
    "human_review": "NOT_PERFORMED",
    "formalization": "not_formalized",
    "Lean": "NOT_RUN",
    "Canonical_truth_promotion": false,
    "authority": "Owner-authorized reviewed reader publication; original question/candidate states remain historical and unchanged."
  }
}