{
  "schema_version": "ocme-reviewed-question-reader-v1",
  "question_id": "q-20261014-advance-20261110-r03-continuous-image-of-compact-0058",
  "pilot_number": "R0043-22033",
  "number": 22033,
  "family_id": "qf-mod41-compactimg-cover-pullback",
  "title": {
    "zh-Hant": "第 22033 題：緊緻集的連續像",
    "en": "Question 22033: Pull an Image Cover Back to K and Preserve Its Subcover Indices"
  },
  "original_question": {
    "schema_version": "ocme-question-v0.1",
    "id": "q-20261014-advance-20261110-r03-continuous-image-of-compact-0058",
    "batch_id": "batch-20261014-advance-20261110-r03-continuous-image-of-compact-foundations",
    "language": "zh-Hant",
    "source_mko_ids": [
      "mko-continuous-image-of-compact"
    ],
    "title_zh": "緊緻集的連續像",
    "stem_zh": "compact K、完整來源與目標拓撲以及total continuous map。操作=compactimg-cover-pullback；資料={\"source_n\":3,\"target_n\":3,\"source_topology\":[[],[0],[0,1],[0,1,2],[0,2],[1],[1,2],[2]],\"target_topology\":[[],[0],[0,1],[0,1,2],[0,2],[1],[1,2],[2]],\"map\":[1,2,0],\"subset\":[0,1,2],\"cover\":[[0],[1],[2]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。",
    "answer": {
      "kind": "expression",
      "value": "([0, 1, 2], [[0], [1], [2]], [[2], [0], [1]], [0, 1, 2], [[0], [1], [2]], 是, finite_subcover_pulled_back_to_K_then_pushed_forward_to_image)",
      "explanation_zh": "Wave6第四層核對=([0, 1, 2], [[0], [1], [2]], [[2], [0], [1]], [0, 1, 2], [[0], [1], [2]], 是, finite_subcover_pulled_back_to_K_then_pushed_forward_to_image)；依完整明示資料獨立重算；對image open cover先拉回K取得finite subcover再推回image；不需injective或surjective；compact image不反推domain compact，preimage compact與image closed皆需額外假設。"
    },
    "difficulty": {
      "band": "undergraduate_core",
      "level": 1,
      "skills": [
        "緊緻集的連續像"
      ]
    },
    "variant": {
      "family_id": "qf-mod41-compactimg-cover-pullback",
      "seed": 202611103058,
      "parameters": {
        "source_n": 3,
        "target_n": 3,
        "source_topology": [
          [],
          [
            0
          ],
          [
            0,
            1
          ],
          [
            0,
            1,
            2
          ],
          [
            0,
            2
          ],
          [
            1
          ],
          [
            1,
            2
          ],
          [
            2
          ]
        ],
        "target_topology": [
          [],
          [
            0
          ],
          [
            0,
            1
          ],
          [
            0,
            1,
            2
          ],
          [
            0,
            2
          ],
          [
            1
          ],
          [
            1,
            2
          ],
          [
            2
          ]
        ],
        "map": [
          1,
          2,
          0
        ],
        "subset": [
          0,
          1,
          2
        ],
        "cover": [
          [
            0
          ],
          [
            1
          ],
          [
            2
          ]
        ],
        "domain": "Finite_topological_compact_image",
        "operation": "compactimg-cover-pullback",
        "semantic_template": "qf-mod41-compactimg-cover-pullback/{\"cover\":[[0],[1],[2]],\"map\":[1,2,0],\"source_n\":3,\"source_topology\":[[],[0,1,2],[0,1],[0,2],[0],[1,2],[1],[2]],\"subset\":[0,1,2],\"target_n\":3,\"target_topology\":[[],[0,1,2],[0,1],[0,2],[0],[1,2],[1],[2]]}"
      }
    },
    "verification": {
      "status": "candidate",
      "checks": [
        "producer_exact_finite_or_symbolic_model",
        "producer_source_snapshot_check",
        "producer_substantive_template_deduplication",
        "producer_json_schema"
      ],
      "warning_zh": "候選須主端重算；不構成人工審定、sampled orbit-stabilizer、sampled MVT、reverse compactness、zero-mass division、underspecified recurrence、incomplete divisor inversion、bounded-search PDA rejection、Lean或Evidence。"
    },
    "provenance": {
      "generator_role": "question_producer",
      "generator_model": "model_unknown",
      "generated_at": "2026-10-14",
      "contract_version": "ocme-question-factory-v0.1"
    }
  },
  "original_english_edition": {
    "schema_version": "ocme-question-localization-v0.1",
    "question_id": "q-20261014-advance-20261110-r03-continuous-image-of-compact-0058",
    "batch_id": "batch-20261014-advance-20261110-r03-continuous-image-of-compact-foundations",
    "locale": "en",
    "translation_state": "ai_candidate",
    "source": {
      "question_sha256": "f5cf4d17f092c2191bd68fc7a6decbb26b5e61b451adff43495c40f0137523d3",
      "family_id": "qf-mod41-compactimg-cover-pullback",
      "seed": 202611103058,
      "semantic_template": "qf-mod41-compactimg-cover-pullback/{\"cover\":[[0],[1],[2]],\"map\":[1,2,0],\"source_n\":3,\"source_topology\":[[],[0,1,2],[0,1],[0,2],[0],[1,2],[1],[2]],\"subset\":[0,1,2],\"target_n\":3,\"target_topology\":[[],[0,1,2],[0,1],[0,2],[0],[1,2],[1],[2]]}"
    },
    "fields": {
      "title": "Pull an Image Cover Back to K and Preserve Its Subcover Indices",
      "stem": "Use complete source carrier Fin(3)={0,…,2} with topology=[[], [0], [0, 1], [0, 1, 2], [0, 2], [1], [1, 2], [2]], target carrier Fin(3)={0,…,2} with topology=[[], [0], [0, 1], [0, 1, 2], [0, 2], [1], [1, 2], [2]], total map=[1, 2, 0] and source subset K=[0, 1, 2]. The source/target topologies must qualify; continuity is tested by every target-open preimage. K is compact by finiteness, including empty K. Original indexed target-open cover=[[0], [1], [2]]. Give (image, cover, relative_pullbacks, selected_indices, selected_target_opens, finite_subcover_certificate, certificate).",
      "answer": {
        "kind": "expression",
        "value": "([0, 1, 2], [[0], [1], [2]], [[2], [0], [1]], [0, 1, 2], [[0], [1], [2]], Yes, finite_subcover_pulled_back_to_K_then_pushed_forward_to_image)",
        "explanation": "Require continuity and actual coverage of f(K). Intersect each preimage with K, select a finite subcover of K and use those same indices in the original target cover. The source selects minimum cardinality, then first increasing bitmask among ties, not a unique subcover. No injectivity or surjectivity is required. The exact result, retaining the requested order, is ([0, 1, 2], [[0], [1], [2]], [[2], [0], [1]], [0, 1, 2], [[0], [1], [2]], Yes, finite_subcover_pulled_back_to_K_then_pushed_forward_to_image)."
      },
      "skills": [
        "Pull an Image Cover Back to K and Preserve Its Subcover Indices"
      ],
      "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": "compact K、完整來源與目標拓撲以及total continuous map。操作=compactimg-cover-pullback；資料={\"source_n\":3,\"target_n\":3,\"source_topology\":[[],[0],[0,1],[0,1,2],[0,2],[1],[1,2],[2]],\"target_topology\":[[],[0],[0,1],[0,1,2],[0,2],[1],[1,2],[2]],\"map\":[1,2,0],\"subset\":[0,1,2],\"cover\":[[0],[1],[2]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。",
        "en": "Use complete source carrier Fin(3)={0,…,2} with topology=[[], [0], [0, 1], [0, 1, 2], [0, 2], [1], [1, 2], [2]], target carrier Fin(3)={0,…,2} with topology=[[], [0], [0, 1], [0, 1, 2], [0, 2], [1], [1, 2], [2]], total map=[1, 2, 0] and source subset K=[0, 1, 2]. The source/target topologies must qualify; continuity is tested by every target-open preimage. K is compact by finiteness, including empty K. Original indexed target-open cover=[[0], [1], [2]]. Give (image, cover, relative_pullbacks, selected_indices, selected_target_opens, finite_subcover_certificate, certificate)."
      }
    }
  ],
  "steps": [
    {
      "id": "S001",
      "uses": [
        "G001"
      ],
      "before": "source operation=compactimg-cover-pullback;original snapshot and mathematics retained",
      "after": "explicit ordered source output fields=[image, cover, relative_pullbacks, selected_indices, selected_target_opens, finite_subcover_certificate, certificate]",
      "relation": "implies",
      "rule_id": "source_ordered_interface_explicit_without_javascript",
      "conditions": [],
      "explanation": {
        "zh-Hant": "原題歷史介面標記保留，直接列來源有序欄位，不需JavaScript；MVT由原mode選欄序，不補空欄或看答案形狀猜。tag與固定No保原聲稱範圍。",
        "en": "Retain the historical interface marker and list ordered source fields without JavaScript.MVT fields come from the original mode,without padding or answer-shape inference.Tags and fixed No markers retain their claim scope."
      },
      "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": "SOURCE:ORIGINAL carrier Fin(3);complete canonical opens=[[], [0], [0, 1], [0, 1, 2], [0, 2], [1], [1, 2], [2]];open index=0,U=[]",
      "after": "ALL [other_open_index,union,intersection] rows=[[0, [], []], [1, [0], []], [2, [0, 1], []], [3, [0, 1, 2], []], [4, [0, 2], []], [5, [1], []], [6, [1, 2], []], [7, [2], []]];empty/full opens present;every result in SAME SOURCE topology",
      "relation": "implies",
      "rule_id": "compactimg_qualified_complete_source_or_target_topology",
      "conditions": [],
      "explanation": {
        "zh-Hant": "先分清來源與目標carrier，逐全部open pair核union/intersection並含空／全集。有限family任意union可去重成有限union，不能因有限像就略過拓撲或連續性責任。",
        "en": "Keep source and target carriers distinct.Check all open pairs and empty/full membership.Any union in a finite family reduces to finitely many distinct opens;a finite image does not bypass topology or continuity obligations."
      },
      "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": "SOURCE:ORIGINAL carrier Fin(3);complete canonical opens=[[], [0], [0, 1], [0, 1, 2], [0, 2], [1], [1, 2], [2]];open index=1,U=[0]",
      "after": "ALL [other_open_index,union,intersection] rows=[[0, [0], []], [1, [0], [0]], [2, [0, 1], [0]], [3, [0, 1, 2], [0]], [4, [0, 2], [0]], [5, [0, 1], []], [6, [0, 1, 2], []], [7, [0, 2], []]];empty/full opens present;every result in SAME SOURCE topology",
      "relation": "implies",
      "rule_id": "compactimg_qualified_complete_source_or_target_topology",
      "conditions": [],
      "explanation": {
        "zh-Hant": "先分清來源與目標carrier，逐全部open pair核union/intersection並含空／全集。有限family任意union可去重成有限union，不能因有限像就略過拓撲或連續性責任。",
        "en": "Keep source and target carriers distinct.Check all open pairs and empty/full membership.Any union in a finite family reduces to finitely many distinct opens;a finite image does not bypass topology or continuity obligations."
      },
      "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": "SOURCE:ORIGINAL carrier Fin(3);complete canonical opens=[[], [0], [0, 1], [0, 1, 2], [0, 2], [1], [1, 2], [2]];open index=2,U=[0, 1]",
      "after": "ALL [other_open_index,union,intersection] rows=[[0, [0, 1], []], [1, [0, 1], [0]], [2, [0, 1], [0, 1]], [3, [0, 1, 2], [0, 1]], [4, [0, 1, 2], [0]], [5, [0, 1], [1]], [6, [0, 1, 2], [1]], [7, [0, 1, 2], []]];empty/full opens present;every result in SAME SOURCE topology",
      "relation": "implies",
      "rule_id": "compactimg_qualified_complete_source_or_target_topology",
      "conditions": [],
      "explanation": {
        "zh-Hant": "先分清來源與目標carrier，逐全部open pair核union/intersection並含空／全集。有限family任意union可去重成有限union，不能因有限像就略過拓撲或連續性責任。",
        "en": "Keep source and target carriers distinct.Check all open pairs and empty/full membership.Any union in a finite family reduces to finitely many distinct opens;a finite image does not bypass topology or continuity obligations."
      },
      "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": "SOURCE:ORIGINAL carrier Fin(3);complete canonical opens=[[], [0], [0, 1], [0, 1, 2], [0, 2], [1], [1, 2], [2]];open index=3,U=[0, 1, 2]",
      "after": "ALL [other_open_index,union,intersection] rows=[[0, [0, 1, 2], []], [1, [0, 1, 2], [0]], [2, [0, 1, 2], [0, 1]], [3, [0, 1, 2], [0, 1, 2]], [4, [0, 1, 2], [0, 2]], [5, [0, 1, 2], [1]], [6, [0, 1, 2], [1, 2]], [7, [0, 1, 2], [2]]];empty/full opens present;every result in SAME SOURCE topology",
      "relation": "implies",
      "rule_id": "compactimg_qualified_complete_source_or_target_topology",
      "conditions": [],
      "explanation": {
        "zh-Hant": "先分清來源與目標carrier，逐全部open pair核union/intersection並含空／全集。有限family任意union可去重成有限union，不能因有限像就略過拓撲或連續性責任。",
        "en": "Keep source and target carriers distinct.Check all open pairs and empty/full membership.Any union in a finite family reduces to finitely many distinct opens;a finite image does not bypass topology or continuity obligations."
      },
      "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": "SOURCE:ORIGINAL carrier Fin(3);complete canonical opens=[[], [0], [0, 1], [0, 1, 2], [0, 2], [1], [1, 2], [2]];open index=4,U=[0, 2]",
      "after": "ALL [other_open_index,union,intersection] rows=[[0, [0, 2], []], [1, [0, 2], [0]], [2, [0, 1, 2], [0]], [3, [0, 1, 2], [0, 2]], [4, [0, 2], [0, 2]], [5, [0, 1, 2], []], [6, [0, 1, 2], [2]], [7, [0, 2], [2]]];empty/full opens present;every result in SAME SOURCE topology",
      "relation": "implies",
      "rule_id": "compactimg_qualified_complete_source_or_target_topology",
      "conditions": [],
      "explanation": {
        "zh-Hant": "先分清來源與目標carrier，逐全部open pair核union/intersection並含空／全集。有限family任意union可去重成有限union，不能因有限像就略過拓撲或連續性責任。",
        "en": "Keep source and target carriers distinct.Check all open pairs and empty/full membership.Any union in a finite family reduces to finitely many distinct opens;a finite image does not bypass topology or continuity obligations."
      },
      "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": "SOURCE:ORIGINAL carrier Fin(3);complete canonical opens=[[], [0], [0, 1], [0, 1, 2], [0, 2], [1], [1, 2], [2]];open index=5,U=[1]",
      "after": "ALL [other_open_index,union,intersection] rows=[[0, [1], []], [1, [0, 1], []], [2, [0, 1], [1]], [3, [0, 1, 2], [1]], [4, [0, 1, 2], []], [5, [1], [1]], [6, [1, 2], [1]], [7, [1, 2], []]];empty/full opens present;every result in SAME SOURCE topology",
      "relation": "implies",
      "rule_id": "compactimg_qualified_complete_source_or_target_topology",
      "conditions": [],
      "explanation": {
        "zh-Hant": "先分清來源與目標carrier，逐全部open pair核union/intersection並含空／全集。有限family任意union可去重成有限union，不能因有限像就略過拓撲或連續性責任。",
        "en": "Keep source and target carriers distinct.Check all open pairs and empty/full membership.Any union in a finite family reduces to finitely many distinct opens;a finite image does not bypass topology or continuity obligations."
      },
      "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": "SOURCE:ORIGINAL carrier Fin(3);complete canonical opens=[[], [0], [0, 1], [0, 1, 2], [0, 2], [1], [1, 2], [2]];open index=6,U=[1, 2]",
      "after": "ALL [other_open_index,union,intersection] rows=[[0, [1, 2], []], [1, [0, 1, 2], []], [2, [0, 1, 2], [1]], [3, [0, 1, 2], [1, 2]], [4, [0, 1, 2], [2]], [5, [1, 2], [1]], [6, [1, 2], [1, 2]], [7, [1, 2], [2]]];empty/full opens present;every result in SAME SOURCE topology",
      "relation": "implies",
      "rule_id": "compactimg_qualified_complete_source_or_target_topology",
      "conditions": [],
      "explanation": {
        "zh-Hant": "先分清來源與目標carrier，逐全部open pair核union/intersection並含空／全集。有限family任意union可去重成有限union，不能因有限像就略過拓撲或連續性責任。",
        "en": "Keep source and target carriers distinct.Check all open pairs and empty/full membership.Any union in a finite family reduces to finitely many distinct opens;a finite image does not bypass topology or continuity obligations."
      },
      "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": "SOURCE:ORIGINAL carrier Fin(3);complete canonical opens=[[], [0], [0, 1], [0, 1, 2], [0, 2], [1], [1, 2], [2]];open index=7,U=[2]",
      "after": "ALL [other_open_index,union,intersection] rows=[[0, [2], []], [1, [0, 2], []], [2, [0, 1, 2], []], [3, [0, 1, 2], [2]], [4, [0, 2], [2]], [5, [1, 2], []], [6, [1, 2], [2]], [7, [2], [2]]];empty/full opens present;every result in SAME SOURCE topology",
      "relation": "implies",
      "rule_id": "compactimg_qualified_complete_source_or_target_topology",
      "conditions": [],
      "explanation": {
        "zh-Hant": "先分清來源與目標carrier，逐全部open pair核union/intersection並含空／全集。有限family任意union可去重成有限union，不能因有限像就略過拓撲或連續性責任。",
        "en": "Keep source and target carriers distinct.Check all open pairs and empty/full membership.Any union in a finite family reduces to finitely many distinct opens;a finite image does not bypass topology or continuity obligations."
      },
      "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": "S010",
      "uses": [
        "G001",
        "S009"
      ],
      "before": "TARGET:ORIGINAL carrier Fin(3);complete canonical opens=[[], [0], [0, 1], [0, 1, 2], [0, 2], [1], [1, 2], [2]];open index=0,U=[]",
      "after": "ALL [other_open_index,union,intersection] rows=[[0, [], []], [1, [0], []], [2, [0, 1], []], [3, [0, 1, 2], []], [4, [0, 2], []], [5, [1], []], [6, [1, 2], []], [7, [2], []]];empty/full opens present;every result in SAME TARGET topology",
      "relation": "implies",
      "rule_id": "compactimg_qualified_complete_source_or_target_topology",
      "conditions": [],
      "explanation": {
        "zh-Hant": "先分清來源與目標carrier，逐全部open pair核union/intersection並含空／全集。有限family任意union可去重成有限union，不能因有限像就略過拓撲或連續性責任。",
        "en": "Keep source and target carriers distinct.Check all open pairs and empty/full membership.Any union in a finite family reduces to finitely many distinct opens;a finite image does not bypass topology or continuity obligations."
      },
      "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": "S011",
      "uses": [
        "G001",
        "S010"
      ],
      "before": "TARGET:ORIGINAL carrier Fin(3);complete canonical opens=[[], [0], [0, 1], [0, 1, 2], [0, 2], [1], [1, 2], [2]];open index=1,U=[0]",
      "after": "ALL [other_open_index,union,intersection] rows=[[0, [0], []], [1, [0], [0]], [2, [0, 1], [0]], [3, [0, 1, 2], [0]], [4, [0, 2], [0]], [5, [0, 1], []], [6, [0, 1, 2], []], [7, [0, 2], []]];empty/full opens present;every result in SAME TARGET topology",
      "relation": "implies",
      "rule_id": "compactimg_qualified_complete_source_or_target_topology",
      "conditions": [],
      "explanation": {
        "zh-Hant": "先分清來源與目標carrier，逐全部open pair核union/intersection並含空／全集。有限family任意union可去重成有限union，不能因有限像就略過拓撲或連續性責任。",
        "en": "Keep source and target carriers distinct.Check all open pairs and empty/full membership.Any union in a finite family reduces to finitely many distinct opens;a finite image does not bypass topology or continuity obligations."
      },
      "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": "S012",
      "uses": [
        "G001",
        "S011"
      ],
      "before": "TARGET:ORIGINAL carrier Fin(3);complete canonical opens=[[], [0], [0, 1], [0, 1, 2], [0, 2], [1], [1, 2], [2]];open index=2,U=[0, 1]",
      "after": "ALL [other_open_index,union,intersection] rows=[[0, [0, 1], []], [1, [0, 1], [0]], [2, [0, 1], [0, 1]], [3, [0, 1, 2], [0, 1]], [4, [0, 1, 2], [0]], [5, [0, 1], [1]], [6, [0, 1, 2], [1]], [7, [0, 1, 2], []]];empty/full opens present;every result in SAME TARGET topology",
      "relation": "implies",
      "rule_id": "compactimg_qualified_complete_source_or_target_topology",
      "conditions": [],
      "explanation": {
        "zh-Hant": "先分清來源與目標carrier，逐全部open pair核union/intersection並含空／全集。有限family任意union可去重成有限union，不能因有限像就略過拓撲或連續性責任。",
        "en": "Keep source and target carriers distinct.Check all open pairs and empty/full membership.Any union in a finite family reduces to finitely many distinct opens;a finite image does not bypass topology or continuity obligations."
      },
      "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": "S013",
      "uses": [
        "G001",
        "S012"
      ],
      "before": "TARGET:ORIGINAL carrier Fin(3);complete canonical opens=[[], [0], [0, 1], [0, 1, 2], [0, 2], [1], [1, 2], [2]];open index=3,U=[0, 1, 2]",
      "after": "ALL [other_open_index,union,intersection] rows=[[0, [0, 1, 2], []], [1, [0, 1, 2], [0]], [2, [0, 1, 2], [0, 1]], [3, [0, 1, 2], [0, 1, 2]], [4, [0, 1, 2], [0, 2]], [5, [0, 1, 2], [1]], [6, [0, 1, 2], [1, 2]], [7, [0, 1, 2], [2]]];empty/full opens present;every result in SAME TARGET topology",
      "relation": "implies",
      "rule_id": "compactimg_qualified_complete_source_or_target_topology",
      "conditions": [],
      "explanation": {
        "zh-Hant": "先分清來源與目標carrier，逐全部open pair核union/intersection並含空／全集。有限family任意union可去重成有限union，不能因有限像就略過拓撲或連續性責任。",
        "en": "Keep source and target carriers distinct.Check all open pairs and empty/full membership.Any union in a finite family reduces to finitely many distinct opens;a finite image does not bypass topology or continuity obligations."
      },
      "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": "S014",
      "uses": [
        "G001",
        "S013"
      ],
      "before": "TARGET:ORIGINAL carrier Fin(3);complete canonical opens=[[], [0], [0, 1], [0, 1, 2], [0, 2], [1], [1, 2], [2]];open index=4,U=[0, 2]",
      "after": "ALL [other_open_index,union,intersection] rows=[[0, [0, 2], []], [1, [0, 2], [0]], [2, [0, 1, 2], [0]], [3, [0, 1, 2], [0, 2]], [4, [0, 2], [0, 2]], [5, [0, 1, 2], []], [6, [0, 1, 2], [2]], [7, [0, 2], [2]]];empty/full opens present;every result in SAME TARGET topology",
      "relation": "implies",
      "rule_id": "compactimg_qualified_complete_source_or_target_topology",
      "conditions": [],
      "explanation": {
        "zh-Hant": "先分清來源與目標carrier，逐全部open pair核union/intersection並含空／全集。有限family任意union可去重成有限union，不能因有限像就略過拓撲或連續性責任。",
        "en": "Keep source and target carriers distinct.Check all open pairs and empty/full membership.Any union in a finite family reduces to finitely many distinct opens;a finite image does not bypass topology or continuity obligations."
      },
      "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": "S015",
      "uses": [
        "G001",
        "S014"
      ],
      "before": "TARGET:ORIGINAL carrier Fin(3);complete canonical opens=[[], [0], [0, 1], [0, 1, 2], [0, 2], [1], [1, 2], [2]];open index=5,U=[1]",
      "after": "ALL [other_open_index,union,intersection] rows=[[0, [1], []], [1, [0, 1], []], [2, [0, 1], [1]], [3, [0, 1, 2], [1]], [4, [0, 1, 2], []], [5, [1], [1]], [6, [1, 2], [1]], [7, [1, 2], []]];empty/full opens present;every result in SAME TARGET topology",
      "relation": "implies",
      "rule_id": "compactimg_qualified_complete_source_or_target_topology",
      "conditions": [],
      "explanation": {
        "zh-Hant": "先分清來源與目標carrier，逐全部open pair核union/intersection並含空／全集。有限family任意union可去重成有限union，不能因有限像就略過拓撲或連續性責任。",
        "en": "Keep source and target carriers distinct.Check all open pairs and empty/full membership.Any union in a finite family reduces to finitely many distinct opens;a finite image does not bypass topology or continuity obligations."
      },
      "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": "S016",
      "uses": [
        "G001",
        "S015"
      ],
      "before": "TARGET:ORIGINAL carrier Fin(3);complete canonical opens=[[], [0], [0, 1], [0, 1, 2], [0, 2], [1], [1, 2], [2]];open index=6,U=[1, 2]",
      "after": "ALL [other_open_index,union,intersection] rows=[[0, [1, 2], []], [1, [0, 1, 2], []], [2, [0, 1, 2], [1]], [3, [0, 1, 2], [1, 2]], [4, [0, 1, 2], [2]], [5, [1, 2], [1]], [6, [1, 2], [1, 2]], [7, [1, 2], [2]]];empty/full opens present;every result in SAME TARGET topology",
      "relation": "implies",
      "rule_id": "compactimg_qualified_complete_source_or_target_topology",
      "conditions": [],
      "explanation": {
        "zh-Hant": "先分清來源與目標carrier，逐全部open pair核union/intersection並含空／全集。有限family任意union可去重成有限union，不能因有限像就略過拓撲或連續性責任。",
        "en": "Keep source and target carriers distinct.Check all open pairs and empty/full membership.Any union in a finite family reduces to finitely many distinct opens;a finite image does not bypass topology or continuity obligations."
      },
      "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": "S017",
      "uses": [
        "G001",
        "S016"
      ],
      "before": "TARGET:ORIGINAL carrier Fin(3);complete canonical opens=[[], [0], [0, 1], [0, 1, 2], [0, 2], [1], [1, 2], [2]];open index=7,U=[2]",
      "after": "ALL [other_open_index,union,intersection] rows=[[0, [2], []], [1, [0, 2], []], [2, [0, 1, 2], []], [3, [0, 1, 2], [2]], [4, [0, 2], [2]], [5, [1, 2], []], [6, [1, 2], [2]], [7, [2], [2]]];empty/full opens present;every result in SAME TARGET topology",
      "relation": "implies",
      "rule_id": "compactimg_qualified_complete_source_or_target_topology",
      "conditions": [],
      "explanation": {
        "zh-Hant": "先分清來源與目標carrier，逐全部open pair核union/intersection並含空／全集。有限family任意union可去重成有限union，不能因有限像就略過拓撲或連續性責任。",
        "en": "Keep source and target carriers distinct.Check all open pairs and empty/full membership.Any union in a finite family reduces to finitely many distinct opens;a finite image does not bypass topology or continuity obligations."
      },
      "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": "S018",
      "uses": [
        "G001",
        "S017"
      ],
      "before": "ORIGINAL target-open index=0,O=[];total original map=[1, 2, 0];ENTIRE source carrier Fin(3)",
      "after": "full source preimage=[];source-open=是;NOT restricted to K",
      "relation": "implies",
      "rule_id": "compactimg_EVERY_target_open_FULL_source_preimage",
      "conditions": [],
      "explanation": {
        "zh-Hant": "連續性查每個目標open在整個source的逆像，不先交K，也不只抽幾個open。這個逆像可不是source-open，失敗旗標保False。",
        "en": "Continuity checks every target-open preimage in the entire source,not just K or sampled opens.A failed source-open test remains false."
      },
      "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": "S019",
      "uses": [
        "G001",
        "S018"
      ],
      "before": "ORIGINAL target-open index=1,O=[0];total original map=[1, 2, 0];ENTIRE source carrier Fin(3)",
      "after": "full source preimage=[2];source-open=是;NOT restricted to K",
      "relation": "implies",
      "rule_id": "compactimg_EVERY_target_open_FULL_source_preimage",
      "conditions": [],
      "explanation": {
        "zh-Hant": "連續性查每個目標open在整個source的逆像，不先交K，也不只抽幾個open。這個逆像可不是source-open，失敗旗標保False。",
        "en": "Continuity checks every target-open preimage in the entire source,not just K or sampled opens.A failed source-open test remains false."
      },
      "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": "S020",
      "uses": [
        "G001",
        "S019"
      ],
      "before": "ORIGINAL target-open index=2,O=[0, 1];total original map=[1, 2, 0];ENTIRE source carrier Fin(3)",
      "after": "full source preimage=[0, 2];source-open=是;NOT restricted to K",
      "relation": "implies",
      "rule_id": "compactimg_EVERY_target_open_FULL_source_preimage",
      "conditions": [],
      "explanation": {
        "zh-Hant": "連續性查每個目標open在整個source的逆像，不先交K，也不只抽幾個open。這個逆像可不是source-open，失敗旗標保False。",
        "en": "Continuity checks every target-open preimage in the entire source,not just K or sampled opens.A failed source-open test remains false."
      },
      "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": "S021",
      "uses": [
        "G001",
        "S020"
      ],
      "before": "ORIGINAL target-open index=3,O=[0, 1, 2];total original map=[1, 2, 0];ENTIRE source carrier Fin(3)",
      "after": "full source preimage=[0, 1, 2];source-open=是;NOT restricted to K",
      "relation": "implies",
      "rule_id": "compactimg_EVERY_target_open_FULL_source_preimage",
      "conditions": [],
      "explanation": {
        "zh-Hant": "連續性查每個目標open在整個source的逆像，不先交K，也不只抽幾個open。這個逆像可不是source-open，失敗旗標保False。",
        "en": "Continuity checks every target-open preimage in the entire source,not just K or sampled opens.A failed source-open test remains false."
      },
      "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": "S022",
      "uses": [
        "G001",
        "S021"
      ],
      "before": "ORIGINAL target-open index=4,O=[0, 2];total original map=[1, 2, 0];ENTIRE source carrier Fin(3)",
      "after": "full source preimage=[1, 2];source-open=是;NOT restricted to K",
      "relation": "implies",
      "rule_id": "compactimg_EVERY_target_open_FULL_source_preimage",
      "conditions": [],
      "explanation": {
        "zh-Hant": "連續性查每個目標open在整個source的逆像，不先交K，也不只抽幾個open。這個逆像可不是source-open，失敗旗標保False。",
        "en": "Continuity checks every target-open preimage in the entire source,not just K or sampled opens.A failed source-open test remains false."
      },
      "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": "S023",
      "uses": [
        "G001",
        "S022"
      ],
      "before": "ORIGINAL target-open index=5,O=[1];total original map=[1, 2, 0];ENTIRE source carrier Fin(3)",
      "after": "full source preimage=[0];source-open=是;NOT restricted to K",
      "relation": "implies",
      "rule_id": "compactimg_EVERY_target_open_FULL_source_preimage",
      "conditions": [],
      "explanation": {
        "zh-Hant": "連續性查每個目標open在整個source的逆像，不先交K，也不只抽幾個open。這個逆像可不是source-open，失敗旗標保False。",
        "en": "Continuity checks every target-open preimage in the entire source,not just K or sampled opens.A failed source-open test remains false."
      },
      "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": "S024",
      "uses": [
        "G001",
        "S023"
      ],
      "before": "ORIGINAL target-open index=6,O=[1, 2];total original map=[1, 2, 0];ENTIRE source carrier Fin(3)",
      "after": "full source preimage=[0, 1];source-open=是;NOT restricted to K",
      "relation": "implies",
      "rule_id": "compactimg_EVERY_target_open_FULL_source_preimage",
      "conditions": [],
      "explanation": {
        "zh-Hant": "連續性查每個目標open在整個source的逆像，不先交K，也不只抽幾個open。這個逆像可不是source-open，失敗旗標保False。",
        "en": "Continuity checks every target-open preimage in the entire source,not just K or sampled opens.A failed source-open test remains false."
      },
      "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": "S025",
      "uses": [
        "G001",
        "S024"
      ],
      "before": "ORIGINAL target-open index=7,O=[2];total original map=[1, 2, 0];ENTIRE source carrier Fin(3)",
      "after": "full source preimage=[1];source-open=是;NOT restricted to K",
      "relation": "implies",
      "rule_id": "compactimg_EVERY_target_open_FULL_source_preimage",
      "conditions": [],
      "explanation": {
        "zh-Hant": "連續性查每個目標open在整個source的逆像，不先交K，也不只抽幾個open。這個逆像可不是source-open，失敗旗標保False。",
        "en": "Continuity checks every target-open preimage in the entire source,not just K or sampled opens.A failed source-open test remains false."
      },
      "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": "S026",
      "uses": [
        "G001",
        "S025"
      ],
      "before": "ORIGINAL source subset K=[0, 1, 2];ALL [x,map(x)] pairs=[[0, 1], [1, 2], [2, 0]];target carrier Fin(3)",
      "after": "exact sorted distinct image=[0, 1, 2];K and image finite hence compact;continuous=是 is the ALL-target-open conjunction",
      "relation": "implies",
      "rule_id": "compactimg_finite_image_and_continuity_separate",
      "conditions": [],
      "explanation": {
        "zh-Hant": "每個像由原K中的點見證，反向每個K像均已列，所以集合恰等f(K)。有限集合任意open cover可逐點選一個open給有限子cover；不需先證連續，但一般連續像定理另有拉回證明。",
        "en": "Every image value has an original K witness and every K image is included,so the set is exactly f(K).A finite set has a finite subcover by choosing one cover member per point.This finite argument is separate from continuity and from the general pullback theorem."
      },
      "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": "S027",
      "uses": [
        "G001",
        "S026"
      ],
      "before": "ORIGINAL indexed TARGET-open cover=[[0], [1], [2]] ACTUALLY covers f(K)=[0, 1, 2];continuous=是",
      "after": "ALL ORIGINAL-index relative pullbacks K intersect f^-1(O_i)=[[2], [0], [1]];they cover K and are relative-open;compact K gives finite ORIGINAL indices,which cover image again",
      "relation": "implies",
      "rule_id": "compactimg_original_image_cover_relative_pullbacks_general_proof",
      "conditions": [],
      "explanation": {
        "zh-Hant": "連續性給整個source的open逆像，再交K；cover像才保拉回覆K。任意image點取原K的preimage，有限拉回子cover中的同一索引便覆蓋它；不需injective或surjective到全部target。",
        "en": "Continuity gives ambient-open preimages,then intersect with K.Image coverage makes them cover K.For any image point,choose its K preimage;the same finite subcover index covers the image point.No injectivity or surjectivity onto the entire target is needed."
      },
      "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": "S028",
      "uses": [
        "G001",
        "S027"
      ],
      "before": "ORIGINAL cover indices0..2;relative pullbacks=[[2], [0], [1]];test increasing cardinality then NUMERIC increasing bitmask",
      "after": "ALL tested [mask,indices,covered_K_points,covers_K] rows=[[0, [], [], 否], [1, [0], [2], 否], [2, [1], [0], 否], [4, [2], [1], 否], [3, [0, 1], [0, 2], 否], [5, [0, 2], [1, 2], 否], [6, [1, 2], [0, 1], 否], [7, [0, 1, 2], [0, 1, 2], 是]];FIRST passing selected=[0, 1, 2];selected target opens=[[0], [1], [2]]",
      "relation": "implies",
      "rule_id": "compactimg_minimum_cardinality_then_first_NUMERIC_mask",
      "conditions": [],
      "explanation": {
        "zh-Hant": "較少索引的候選全部失敗，同數量中較小numeric mask也都失敗，才取來源首解。這不是索引列表的LEX順序；空K的空選擇合法，不把[]當未找到；最小不代表唯一。",
        "en": "All smaller-cardinality candidates and earlier numeric masks at this size fail before the selected source witness.This is not lexicographic index-list order.Empty K legitimately selects[];minimum does not mean unique."
      },
      "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": [
      [
        0,
        1,
        2
      ],
      [
        [
          0
        ],
        [
          1
        ],
        [
          2
        ]
      ],
      [
        [
          2
        ],
        [
          0
        ],
        [
          1
        ]
      ],
      [
        0,
        1,
        2
      ],
      [
        [
          0
        ],
        [
          1
        ],
        [
          2
        ]
      ],
      true,
      "finite_subcover_pulled_back_to_K_then_pushed_forward_to_image"
    ]
  },
  "conclusion": {
    "canonical_value": "([0, 1, 2], [[0], [1], [2]], [[2], [0], [1]], [0, 1, 2], [[0], [1], [2]], 是, finite_subcover_pulled_back_to_K_then_pushed_forward_to_image)",
    "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-20261014-advance-20261110-r03-continuous-image-of-compact-foundations",
      "batch_path": "public/data/questions/batches/batch-20261014-advance-20261110-r03-continuous-image-of-compact-foundations/questions.jsonl",
      "batch_sha256": "eaacf2d24560b914960985059a27fe82ab74c756c627b78e147e10bddc70846e",
      "original_line": 58,
      "original_line_sha256": "6e44b7643297c18bbdee5419601bdbc8bd2cc2e7294ff636b6089e49732e7445",
      "question_compact_sha256": "f5cf4d17f092c2191bd68fc7a6decbb26b5e61b451adff43495c40f0137523d3",
      "english_row_compact_sha256": "56a8b17c17aeb29e09c716e7a051a67cbe61f60a9d108864f6fab984c5cb75fb",
      "english_manifest_path": "public/data/i18n/editions/batch-20261014-advance-20261110-r03-continuous-image-of-compact-foundations.en.json",
      "english_rows_path": "public/data/i18n/questions/batch-20261014-advance-20261110-r03-continuous-image-of-compact-foundations.en.jsonl"
    },
    "mko_bindings": {
      "mko-continuous-image-of-compact": {
        "sha256": "033a72b2e9f25c030d4348c2fe05cda62d3477182efb9d2fb0e32b8c1178138c",
        "english_sha256": "86415941d2cdb6419d41013c6d3c1611923949c466f9e07156b1aa77671625b5",
        "original_path": "public/data/mko/continuous-image-of-compact.json",
        "english_path": "public/data/i18n/editions/mko-continuous-image-of-compact.en.json"
      }
    },
    "selected_manifest_sha256": "5233edba9654a6cffd8b6b2d83c932f82ded15f9c0b7c733bd049943e672acff",
    "selected_part_sha256": "747b98e75085a6b55251b2d7e04bbf7c60cb5500e5047439b2798fb5f45538fa",
    "original_candidate_sha256": "2ad66a7ba6d324ab4adf1fff64fb27b9d672e23b17da57b0d44e408e374220a5",
    "original_candidate_line_sha256": "2ad66a7ba6d324ab4adf1fff64fb27b9d672e23b17da57b0d44e408e374220a5"
  },
  "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-20261026-009-p44",
    "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": "49cf6ec60211c92d1c0c302663af4d44479635dc8181fb61a20c3374288e21c5",
      "question_first_witness_sha256": "10fd25b4643e5ff12753daceaa61f9f2499bf381f65ef96e9d21185c7c0d0c5e"
    },
    "status": "second_context_reviewed",
    "mathematical_semantics": true,
    "bilingual_semantics": true,
    "steps_reviewed": 28,
    "method": "family_template_semantic_review_plus_independent_per_instance_reconstruction",
    "exact_question_first_check": {
      "model_sha256": "593791f89a703733c58f39492a4af4fb457359a4d15cf4d4216a655bf893a51f",
      "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."
  }
}