{
  "schema_version": "ocme-reviewed-question-reader-v1",
  "question_id": "q-20261014-advance-20261107-r03-topological-basis-0047",
  "pilot_number": "R0040-20672",
  "number": 20672,
  "family_id": "qf-mod38-topbasis-generated-topology",
  "title": {
    "zh-Hant": "第 20672 題：拓撲基底",
    "en": "Question 20672: Generate All Unions Only after the Basis Axioms Pass"
  },
  "original_question": {
    "schema_version": "ocme-question-v0.1",
    "id": "q-20261014-advance-20261107-r03-topological-basis-0047",
    "batch_id": "batch-20261014-advance-20261107-r03-topological-basis-foundations",
    "language": "zh-Hant",
    "source_mko_ids": [
      "mko-topological-basis"
    ],
    "title_zh": "拓撲基底",
    "stem_zh": "完整有限carrier與完整basis candidate。操作=topbasis-generated-topology；資料={\"n\":5,\"basis\":[[0],[0,1],[1],[1,2],[2],[3],[3,4],[4]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。",
    "answer": {
      "kind": "expression",
      "value": "([[], [0], [0, 1], [0, 1, 2], [0, 1, 2, 3], [0, 1, 2, 3, 4], [0, 1, 2, 4], [0, 1, 3], [0, 1, 3, 4], [0, 1, 4], [0, 2], [0, 2, 3], [0, 2, 3, 4], [0, 2, 4], [0, 3], [0, 3, 4], [0, 4], [1], [1, 2], [1, 2, 3], [1, 2, 3, 4], [1, 2, 4], [1, 3], [1, 3, 4], [1, 4], [2], [2, 3], [2, 3, 4], [2, 4], [3], [3, 4], [4]], 32, 是, all_unions_of_basis_elements)",
      "explanation_zh": "Wave6根層核對=([[], [0], [0, 1], [0, 1, 2], [0, 1, 2, 3], [0, 1, 2, 3, 4], [0, 1, 2, 4], [0, 1, 3], [0, 1, 3, 4], [0, 1, 4], [0, 2], [0, 2, 3], [0, 2, 3, 4], [0, 2, 4], [0, 3], [0, 3, 4], [0, 4], [1], [1, 2], [1, 2, 3], [1, 2, 3, 4], [1, 2, 4], [1, 3], [1, 3, 4], [1, 4], [2], [2, 3], [2, 3, 4], [2, 4], [3], [3, 4], [4]], 32, 是, all_unions_of_basis_elements)；依完整明示資料獨立重算；basis須覆蓋X並逐點滿足局部交細化；basis elements皆為X子集；相對既有拓撲還須basis elements開且每個開集為basis聯集；basis不可混同subbasis。"
    },
    "difficulty": {
      "band": "undergraduate_core",
      "level": 3,
      "skills": [
        "拓撲基底"
      ]
    },
    "variant": {
      "family_id": "qf-mod38-topbasis-generated-topology",
      "seed": 202611073047,
      "parameters": {
        "n": 5,
        "basis": [
          [
            0
          ],
          [
            0,
            1
          ],
          [
            1
          ],
          [
            1,
            2
          ],
          [
            2
          ],
          [
            3
          ],
          [
            3,
            4
          ],
          [
            4
          ]
        ],
        "domain": "Complete_finite_topological_basis",
        "operation": "topbasis-generated-topology",
        "semantic_template": "qf-mod38-topbasis-generated-topology/{\"basis\":[[0,1],[0],[1,2],[1],[2],[3,4],[3],[4]],\"n\":5}"
      }
    },
    "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、有限sample群作用、sup等於max、basis等於subbasis、signed Markov、pigeonhole converse、nonunit Euler、finite-prefix subsequence、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-20261107-r03-topological-basis-0047",
    "batch_id": "batch-20261014-advance-20261107-r03-topological-basis-foundations",
    "locale": "en",
    "translation_state": "ai_candidate",
    "source": {
      "question_sha256": "538c459c383aea485a572051040c7e0c854134eb9b7a31b552e2f9e592dfe87a",
      "family_id": "qf-mod38-topbasis-generated-topology",
      "seed": 202611073047,
      "semantic_template": "qf-mod38-topbasis-generated-topology/{\"basis\":[[0,1],[0],[1,2],[1],[2],[3,4],[3],[4]],\"n\":5}"
    },
    "fields": {
      "title": "Generate All Unions Only after the Basis Axioms Pass",
      "stem": "Use finite carrier Fin(5)={0,…,4} and the complete canonical candidate family basis=[[0], [0, 1], [1], [1, 2], [2], [3], [3, 4], [4]]. Its union must cover the carrier; for every point in each pairwise intersection a candidate element containing that point must lie inside the intersection. Complete finite data, not sampled intersections, supply this audit. Give (topology, open_count, topology_certificate, certificate).",
      "answer": {
        "kind": "expression",
        "value": "([[], [0], [0, 1], [0, 1, 2], [0, 1, 2, 3], [0, 1, 2, 3, 4], [0, 1, 2, 4], [0, 1, 3], [0, 1, 3, 4], [0, 1, 4], [0, 2], [0, 2, 3], [0, 2, 3, 4], [0, 2, 4], [0, 3], [0, 3, 4], [0, 4], [1], [1, 2], [1, 2, 3], [1, 2, 3, 4], [1, 2, 4], [1, 3], [1, 3, 4], [1, 4], [2], [2, 3], [2, 3, 4], [2, 4], [3], [3, 4], [4]], 32, Yes, all_unions_of_basis_elements)",
        "explanation": "Require a valid basis and enumerate every distinct union, including the empty union. Retain the source canonical set-family order and exact open-set count. This is an exact certificate for the stated finite carrier, not an arbitrary-space sampling proof. The exact result, retaining the requested order, is ([[], [0], [0, 1], [0, 1, 2], [0, 1, 2, 3], [0, 1, 2, 3, 4], [0, 1, 2, 4], [0, 1, 3], [0, 1, 3, 4], [0, 1, 4], [0, 2], [0, 2, 3], [0, 2, 3, 4], [0, 2, 4], [0, 3], [0, 3, 4], [0, 4], [1], [1, 2], [1, 2, 3], [1, 2, 3, 4], [1, 2, 4], [1, 3], [1, 3, 4], [1, 4], [2], [2, 3], [2, 3, 4], [2, 4], [3], [3, 4], [4]], 32, Yes, all_unions_of_basis_elements)."
      },
      "skills": [
        "Generate All Unions Only after the Basis Axioms Pass"
      ],
      "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": "完整有限carrier與完整basis candidate。操作=topbasis-generated-topology；資料={\"n\":5,\"basis\":[[0],[0,1],[1],[1,2],[2],[3],[3,4],[4]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。",
        "en": "Use finite carrier Fin(5)={0,…,4} and the complete canonical candidate family basis=[[0], [0, 1], [1], [1, 2], [2], [3], [3, 4], [4]]. Its union must cover the carrier; for every point in each pairwise intersection a candidate element containing that point must lie inside the intersection. Complete finite data, not sampled intersections, supply this audit. Give (topology, open_count, topology_certificate, certificate)."
      }
    }
  ],
  "steps": [
    {
      "id": "S001",
      "uses": [
        "G001"
      ],
      "before": "source operation=topbasis-generated-topology;original snapshot and mathematics retained",
      "after": "explicit ordered source output fields=[topology, open_count, topology_certificate, certificate]",
      "relation": "implies",
      "rule_id": "source_ordered_interface_explicit_without_javascript",
      "conditions": [],
      "explanation": {
        "zh-Hant": "原題快照保留歷史介面標記；本頁直接列來源欄位順序，不需JavaScript或外部表。標籤不自動是數學證據，No保持來源的聲稱範圍。",
        "en": "Keep the historical interface marker in the original snapshot while listing source fields here without JavaScript or external lookup.Labels are not mathematical evidence,and No retains its source 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": "original basis candidate:ORIGINAL complete candidate family=[[0], [0, 1], [1], [1, 2], [2], [3], [3, 4], [4]];carrier Fin(5)",
      "after": "within-set numeric normalization=[[0], [0, 1], [1], [1, 2], [2], [3], [3, 4], [4]];no duplicate points or duplicate FAMILY elements;WHOLE original family already in required key order;empty family allowed as candidate",
      "relation": "implies",
      "rule_id": "topbasis_canonical_candidate_family_not_semantic_reordering",
      "conditions": [],
      "explanation": {
        "zh-Hant": "basis是subset族，不是向量basis；個數可大於n。只規範每個集合內的點順序，不重排原族索引，不刪重複來修資料；空candidate族仍可受資格審查。",
        "en": "A topological basis is a family of subsets,not a vector-space basis;its size may exceed n.Normalize point order within each set without changing original family indices or deleting duplicates to repair data.An empty candidate family can still be audited."
      },
      "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": "COMPLETE original candidate basis=[[0], [0, 1], [1], [1, 2], [2], [3], [3, 4], [4]];ALL carrier points0..4",
      "after": "ALL [point,containing_basis_indices] rows=[[0, [0, 1]], [1, [1, 2, 3]], [2, [3, 4]], [3, [5, 6]], [4, [6, 7]]];covers WHOLE carrier=是",
      "relation": "implies",
      "rule_id": "topbasis_every_point_cover_separate_from_refinement",
      "conditions": [],
      "explanation": {
        "zh-Hant": "每個點至少落一個basis候選才cover；這項不能代替局部交細化。族為空時n≥1，所以cover失敗而不是拋棄該candidate。",
        "en": "Every carrier point needs a containing candidate element for coverage.This does not replace local intersection refinement.With n≥1 an empty family fails coverage rather than being discarded as an unauditable candidate."
      },
      "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 ordered basis positions(left,right)=[0, 0];actual basis elements=[[0], [0]];FULL intersection=[0]",
      "after": "ALL [intersection_point,ALL refining original basis indices] rows=[[0, [0]]];empty intersection gives VACUOUS local obligation;first failure retained in source order",
      "relation": "implies",
      "rule_id": "topbasis_full_local_point_refinement_not_intersection_membership",
      "conditions": [],
      "explanation": {
        "zh-Hant": "對交集內每個x要存在某basis元素含x且包含在交集中，不要求整個交集本身在basis。逐全部候選refiner才可證不存在，首失敗保left/right/point的LEX順序。",
        "en": "For each intersection point x require some candidate basis element containing x and contained in the intersection.The entire intersection need not itself be a basis element.Check all refiners before declaring absence,and keep the first left/right/point failure in source order."
      },
      "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 ordered basis positions(left,right)=[0, 1];actual basis elements=[[0], [0, 1]];FULL intersection=[0]",
      "after": "ALL [intersection_point,ALL refining original basis indices] rows=[[0, [0]]];empty intersection gives VACUOUS local obligation;first failure retained in source order",
      "relation": "implies",
      "rule_id": "topbasis_full_local_point_refinement_not_intersection_membership",
      "conditions": [],
      "explanation": {
        "zh-Hant": "對交集內每個x要存在某basis元素含x且包含在交集中，不要求整個交集本身在basis。逐全部候選refiner才可證不存在，首失敗保left/right/point的LEX順序。",
        "en": "For each intersection point x require some candidate basis element containing x and contained in the intersection.The entire intersection need not itself be a basis element.Check all refiners before declaring absence,and keep the first left/right/point failure in source order."
      },
      "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 ordered basis positions(left,right)=[0, 2];actual basis elements=[[0], [1]];FULL intersection=[]",
      "after": "ALL [intersection_point,ALL refining original basis indices] rows=[];empty intersection gives VACUOUS local obligation;first failure retained in source order",
      "relation": "implies",
      "rule_id": "topbasis_full_local_point_refinement_not_intersection_membership",
      "conditions": [],
      "explanation": {
        "zh-Hant": "對交集內每個x要存在某basis元素含x且包含在交集中，不要求整個交集本身在basis。逐全部候選refiner才可證不存在，首失敗保left/right/point的LEX順序。",
        "en": "For each intersection point x require some candidate basis element containing x and contained in the intersection.The entire intersection need not itself be a basis element.Check all refiners before declaring absence,and keep the first left/right/point failure in source order."
      },
      "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 ordered basis positions(left,right)=[0, 3];actual basis elements=[[0], [1, 2]];FULL intersection=[]",
      "after": "ALL [intersection_point,ALL refining original basis indices] rows=[];empty intersection gives VACUOUS local obligation;first failure retained in source order",
      "relation": "implies",
      "rule_id": "topbasis_full_local_point_refinement_not_intersection_membership",
      "conditions": [],
      "explanation": {
        "zh-Hant": "對交集內每個x要存在某basis元素含x且包含在交集中，不要求整個交集本身在basis。逐全部候選refiner才可證不存在，首失敗保left/right/point的LEX順序。",
        "en": "For each intersection point x require some candidate basis element containing x and contained in the intersection.The entire intersection need not itself be a basis element.Check all refiners before declaring absence,and keep the first left/right/point failure in source order."
      },
      "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 ordered basis positions(left,right)=[0, 4];actual basis elements=[[0], [2]];FULL intersection=[]",
      "after": "ALL [intersection_point,ALL refining original basis indices] rows=[];empty intersection gives VACUOUS local obligation;first failure retained in source order",
      "relation": "implies",
      "rule_id": "topbasis_full_local_point_refinement_not_intersection_membership",
      "conditions": [],
      "explanation": {
        "zh-Hant": "對交集內每個x要存在某basis元素含x且包含在交集中，不要求整個交集本身在basis。逐全部候選refiner才可證不存在，首失敗保left/right/point的LEX順序。",
        "en": "For each intersection point x require some candidate basis element containing x and contained in the intersection.The entire intersection need not itself be a basis element.Check all refiners before declaring absence,and keep the first left/right/point failure in source order."
      },
      "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 ordered basis positions(left,right)=[0, 5];actual basis elements=[[0], [3]];FULL intersection=[]",
      "after": "ALL [intersection_point,ALL refining original basis indices] rows=[];empty intersection gives VACUOUS local obligation;first failure retained in source order",
      "relation": "implies",
      "rule_id": "topbasis_full_local_point_refinement_not_intersection_membership",
      "conditions": [],
      "explanation": {
        "zh-Hant": "對交集內每個x要存在某basis元素含x且包含在交集中，不要求整個交集本身在basis。逐全部候選refiner才可證不存在，首失敗保left/right/point的LEX順序。",
        "en": "For each intersection point x require some candidate basis element containing x and contained in the intersection.The entire intersection need not itself be a basis element.Check all refiners before declaring absence,and keep the first left/right/point failure in source order."
      },
      "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": "ORIGINAL ordered basis positions(left,right)=[0, 6];actual basis elements=[[0], [3, 4]];FULL intersection=[]",
      "after": "ALL [intersection_point,ALL refining original basis indices] rows=[];empty intersection gives VACUOUS local obligation;first failure retained in source order",
      "relation": "implies",
      "rule_id": "topbasis_full_local_point_refinement_not_intersection_membership",
      "conditions": [],
      "explanation": {
        "zh-Hant": "對交集內每個x要存在某basis元素含x且包含在交集中，不要求整個交集本身在basis。逐全部候選refiner才可證不存在，首失敗保left/right/point的LEX順序。",
        "en": "For each intersection point x require some candidate basis element containing x and contained in the intersection.The entire intersection need not itself be a basis element.Check all refiners before declaring absence,and keep the first left/right/point failure in source order."
      },
      "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": "ORIGINAL ordered basis positions(left,right)=[0, 7];actual basis elements=[[0], [4]];FULL intersection=[]",
      "after": "ALL [intersection_point,ALL refining original basis indices] rows=[];empty intersection gives VACUOUS local obligation;first failure retained in source order",
      "relation": "implies",
      "rule_id": "topbasis_full_local_point_refinement_not_intersection_membership",
      "conditions": [],
      "explanation": {
        "zh-Hant": "對交集內每個x要存在某basis元素含x且包含在交集中，不要求整個交集本身在basis。逐全部候選refiner才可證不存在，首失敗保left/right/point的LEX順序。",
        "en": "For each intersection point x require some candidate basis element containing x and contained in the intersection.The entire intersection need not itself be a basis element.Check all refiners before declaring absence,and keep the first left/right/point failure in source order."
      },
      "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": "ORIGINAL ordered basis positions(left,right)=[1, 0];actual basis elements=[[0, 1], [0]];FULL intersection=[0]",
      "after": "ALL [intersection_point,ALL refining original basis indices] rows=[[0, [0]]];empty intersection gives VACUOUS local obligation;first failure retained in source order",
      "relation": "implies",
      "rule_id": "topbasis_full_local_point_refinement_not_intersection_membership",
      "conditions": [],
      "explanation": {
        "zh-Hant": "對交集內每個x要存在某basis元素含x且包含在交集中，不要求整個交集本身在basis。逐全部候選refiner才可證不存在，首失敗保left/right/point的LEX順序。",
        "en": "For each intersection point x require some candidate basis element containing x and contained in the intersection.The entire intersection need not itself be a basis element.Check all refiners before declaring absence,and keep the first left/right/point failure in source order."
      },
      "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": "ORIGINAL ordered basis positions(left,right)=[1, 1];actual basis elements=[[0, 1], [0, 1]];FULL intersection=[0, 1]",
      "after": "ALL [intersection_point,ALL refining original basis indices] rows=[[0, [0, 1]], [1, [1, 2]]];empty intersection gives VACUOUS local obligation;first failure retained in source order",
      "relation": "implies",
      "rule_id": "topbasis_full_local_point_refinement_not_intersection_membership",
      "conditions": [],
      "explanation": {
        "zh-Hant": "對交集內每個x要存在某basis元素含x且包含在交集中，不要求整個交集本身在basis。逐全部候選refiner才可證不存在，首失敗保left/right/point的LEX順序。",
        "en": "For each intersection point x require some candidate basis element containing x and contained in the intersection.The entire intersection need not itself be a basis element.Check all refiners before declaring absence,and keep the first left/right/point failure in source order."
      },
      "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": "ORIGINAL ordered basis positions(left,right)=[1, 2];actual basis elements=[[0, 1], [1]];FULL intersection=[1]",
      "after": "ALL [intersection_point,ALL refining original basis indices] rows=[[1, [2]]];empty intersection gives VACUOUS local obligation;first failure retained in source order",
      "relation": "implies",
      "rule_id": "topbasis_full_local_point_refinement_not_intersection_membership",
      "conditions": [],
      "explanation": {
        "zh-Hant": "對交集內每個x要存在某basis元素含x且包含在交集中，不要求整個交集本身在basis。逐全部候選refiner才可證不存在，首失敗保left/right/point的LEX順序。",
        "en": "For each intersection point x require some candidate basis element containing x and contained in the intersection.The entire intersection need not itself be a basis element.Check all refiners before declaring absence,and keep the first left/right/point failure in source order."
      },
      "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": "ORIGINAL ordered basis positions(left,right)=[1, 3];actual basis elements=[[0, 1], [1, 2]];FULL intersection=[1]",
      "after": "ALL [intersection_point,ALL refining original basis indices] rows=[[1, [2]]];empty intersection gives VACUOUS local obligation;first failure retained in source order",
      "relation": "implies",
      "rule_id": "topbasis_full_local_point_refinement_not_intersection_membership",
      "conditions": [],
      "explanation": {
        "zh-Hant": "對交集內每個x要存在某basis元素含x且包含在交集中，不要求整個交集本身在basis。逐全部候選refiner才可證不存在，首失敗保left/right/point的LEX順序。",
        "en": "For each intersection point x require some candidate basis element containing x and contained in the intersection.The entire intersection need not itself be a basis element.Check all refiners before declaring absence,and keep the first left/right/point failure in source order."
      },
      "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": "ORIGINAL ordered basis positions(left,right)=[1, 4];actual basis elements=[[0, 1], [2]];FULL intersection=[]",
      "after": "ALL [intersection_point,ALL refining original basis indices] rows=[];empty intersection gives VACUOUS local obligation;first failure retained in source order",
      "relation": "implies",
      "rule_id": "topbasis_full_local_point_refinement_not_intersection_membership",
      "conditions": [],
      "explanation": {
        "zh-Hant": "對交集內每個x要存在某basis元素含x且包含在交集中，不要求整個交集本身在basis。逐全部候選refiner才可證不存在，首失敗保left/right/point的LEX順序。",
        "en": "For each intersection point x require some candidate basis element containing x and contained in the intersection.The entire intersection need not itself be a basis element.Check all refiners before declaring absence,and keep the first left/right/point failure in source order."
      },
      "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": "ORIGINAL ordered basis positions(left,right)=[1, 5];actual basis elements=[[0, 1], [3]];FULL intersection=[]",
      "after": "ALL [intersection_point,ALL refining original basis indices] rows=[];empty intersection gives VACUOUS local obligation;first failure retained in source order",
      "relation": "implies",
      "rule_id": "topbasis_full_local_point_refinement_not_intersection_membership",
      "conditions": [],
      "explanation": {
        "zh-Hant": "對交集內每個x要存在某basis元素含x且包含在交集中，不要求整個交集本身在basis。逐全部候選refiner才可證不存在，首失敗保left/right/point的LEX順序。",
        "en": "For each intersection point x require some candidate basis element containing x and contained in the intersection.The entire intersection need not itself be a basis element.Check all refiners before declaring absence,and keep the first left/right/point failure in source order."
      },
      "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 ordered basis positions(left,right)=[1, 6];actual basis elements=[[0, 1], [3, 4]];FULL intersection=[]",
      "after": "ALL [intersection_point,ALL refining original basis indices] rows=[];empty intersection gives VACUOUS local obligation;first failure retained in source order",
      "relation": "implies",
      "rule_id": "topbasis_full_local_point_refinement_not_intersection_membership",
      "conditions": [],
      "explanation": {
        "zh-Hant": "對交集內每個x要存在某basis元素含x且包含在交集中，不要求整個交集本身在basis。逐全部候選refiner才可證不存在，首失敗保left/right/point的LEX順序。",
        "en": "For each intersection point x require some candidate basis element containing x and contained in the intersection.The entire intersection need not itself be a basis element.Check all refiners before declaring absence,and keep the first left/right/point failure in source order."
      },
      "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 ordered basis positions(left,right)=[1, 7];actual basis elements=[[0, 1], [4]];FULL intersection=[]",
      "after": "ALL [intersection_point,ALL refining original basis indices] rows=[];empty intersection gives VACUOUS local obligation;first failure retained in source order",
      "relation": "implies",
      "rule_id": "topbasis_full_local_point_refinement_not_intersection_membership",
      "conditions": [],
      "explanation": {
        "zh-Hant": "對交集內每個x要存在某basis元素含x且包含在交集中，不要求整個交集本身在basis。逐全部候選refiner才可證不存在，首失敗保left/right/point的LEX順序。",
        "en": "For each intersection point x require some candidate basis element containing x and contained in the intersection.The entire intersection need not itself be a basis element.Check all refiners before declaring absence,and keep the first left/right/point failure in source order."
      },
      "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 ordered basis positions(left,right)=[2, 0];actual basis elements=[[1], [0]];FULL intersection=[]",
      "after": "ALL [intersection_point,ALL refining original basis indices] rows=[];empty intersection gives VACUOUS local obligation;first failure retained in source order",
      "relation": "implies",
      "rule_id": "topbasis_full_local_point_refinement_not_intersection_membership",
      "conditions": [],
      "explanation": {
        "zh-Hant": "對交集內每個x要存在某basis元素含x且包含在交集中，不要求整個交集本身在basis。逐全部候選refiner才可證不存在，首失敗保left/right/point的LEX順序。",
        "en": "For each intersection point x require some candidate basis element containing x and contained in the intersection.The entire intersection need not itself be a basis element.Check all refiners before declaring absence,and keep the first left/right/point failure in source order."
      },
      "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 ordered basis positions(left,right)=[2, 1];actual basis elements=[[1], [0, 1]];FULL intersection=[1]",
      "after": "ALL [intersection_point,ALL refining original basis indices] rows=[[1, [2]]];empty intersection gives VACUOUS local obligation;first failure retained in source order",
      "relation": "implies",
      "rule_id": "topbasis_full_local_point_refinement_not_intersection_membership",
      "conditions": [],
      "explanation": {
        "zh-Hant": "對交集內每個x要存在某basis元素含x且包含在交集中，不要求整個交集本身在basis。逐全部候選refiner才可證不存在，首失敗保left/right/point的LEX順序。",
        "en": "For each intersection point x require some candidate basis element containing x and contained in the intersection.The entire intersection need not itself be a basis element.Check all refiners before declaring absence,and keep the first left/right/point failure in source order."
      },
      "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 ordered basis positions(left,right)=[2, 2];actual basis elements=[[1], [1]];FULL intersection=[1]",
      "after": "ALL [intersection_point,ALL refining original basis indices] rows=[[1, [2]]];empty intersection gives VACUOUS local obligation;first failure retained in source order",
      "relation": "implies",
      "rule_id": "topbasis_full_local_point_refinement_not_intersection_membership",
      "conditions": [],
      "explanation": {
        "zh-Hant": "對交集內每個x要存在某basis元素含x且包含在交集中，不要求整個交集本身在basis。逐全部候選refiner才可證不存在，首失敗保left/right/point的LEX順序。",
        "en": "For each intersection point x require some candidate basis element containing x and contained in the intersection.The entire intersection need not itself be a basis element.Check all refiners before declaring absence,and keep the first left/right/point failure in source order."
      },
      "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 ordered basis positions(left,right)=[2, 3];actual basis elements=[[1], [1, 2]];FULL intersection=[1]",
      "after": "ALL [intersection_point,ALL refining original basis indices] rows=[[1, [2]]];empty intersection gives VACUOUS local obligation;first failure retained in source order",
      "relation": "implies",
      "rule_id": "topbasis_full_local_point_refinement_not_intersection_membership",
      "conditions": [],
      "explanation": {
        "zh-Hant": "對交集內每個x要存在某basis元素含x且包含在交集中，不要求整個交集本身在basis。逐全部候選refiner才可證不存在，首失敗保left/right/point的LEX順序。",
        "en": "For each intersection point x require some candidate basis element containing x and contained in the intersection.The entire intersection need not itself be a basis element.Check all refiners before declaring absence,and keep the first left/right/point failure in source order."
      },
      "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 ordered basis positions(left,right)=[2, 4];actual basis elements=[[1], [2]];FULL intersection=[]",
      "after": "ALL [intersection_point,ALL refining original basis indices] rows=[];empty intersection gives VACUOUS local obligation;first failure retained in source order",
      "relation": "implies",
      "rule_id": "topbasis_full_local_point_refinement_not_intersection_membership",
      "conditions": [],
      "explanation": {
        "zh-Hant": "對交集內每個x要存在某basis元素含x且包含在交集中，不要求整個交集本身在basis。逐全部候選refiner才可證不存在，首失敗保left/right/point的LEX順序。",
        "en": "For each intersection point x require some candidate basis element containing x and contained in the intersection.The entire intersection need not itself be a basis element.Check all refiners before declaring absence,and keep the first left/right/point failure in source order."
      },
      "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 ordered basis positions(left,right)=[2, 5];actual basis elements=[[1], [3]];FULL intersection=[]",
      "after": "ALL [intersection_point,ALL refining original basis indices] rows=[];empty intersection gives VACUOUS local obligation;first failure retained in source order",
      "relation": "implies",
      "rule_id": "topbasis_full_local_point_refinement_not_intersection_membership",
      "conditions": [],
      "explanation": {
        "zh-Hant": "對交集內每個x要存在某basis元素含x且包含在交集中，不要求整個交集本身在basis。逐全部候選refiner才可證不存在，首失敗保left/right/point的LEX順序。",
        "en": "For each intersection point x require some candidate basis element containing x and contained in the intersection.The entire intersection need not itself be a basis element.Check all refiners before declaring absence,and keep the first left/right/point failure in source order."
      },
      "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 ordered basis positions(left,right)=[2, 6];actual basis elements=[[1], [3, 4]];FULL intersection=[]",
      "after": "ALL [intersection_point,ALL refining original basis indices] rows=[];empty intersection gives VACUOUS local obligation;first failure retained in source order",
      "relation": "implies",
      "rule_id": "topbasis_full_local_point_refinement_not_intersection_membership",
      "conditions": [],
      "explanation": {
        "zh-Hant": "對交集內每個x要存在某basis元素含x且包含在交集中，不要求整個交集本身在basis。逐全部候選refiner才可證不存在，首失敗保left/right/point的LEX順序。",
        "en": "For each intersection point x require some candidate basis element containing x and contained in the intersection.The entire intersection need not itself be a basis element.Check all refiners before declaring absence,and keep the first left/right/point failure in source order."
      },
      "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 ordered basis positions(left,right)=[2, 7];actual basis elements=[[1], [4]];FULL intersection=[]",
      "after": "ALL [intersection_point,ALL refining original basis indices] rows=[];empty intersection gives VACUOUS local obligation;first failure retained in source order",
      "relation": "implies",
      "rule_id": "topbasis_full_local_point_refinement_not_intersection_membership",
      "conditions": [],
      "explanation": {
        "zh-Hant": "對交集內每個x要存在某basis元素含x且包含在交集中，不要求整個交集本身在basis。逐全部候選refiner才可證不存在，首失敗保left/right/point的LEX順序。",
        "en": "For each intersection point x require some candidate basis element containing x and contained in the intersection.The entire intersection need not itself be a basis element.Check all refiners before declaring absence,and keep the first left/right/point failure in source order."
      },
      "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 ordered basis positions(left,right)=[3, 0];actual basis elements=[[1, 2], [0]];FULL intersection=[]",
      "after": "ALL [intersection_point,ALL refining original basis indices] rows=[];empty intersection gives VACUOUS local obligation;first failure retained in source order",
      "relation": "implies",
      "rule_id": "topbasis_full_local_point_refinement_not_intersection_membership",
      "conditions": [],
      "explanation": {
        "zh-Hant": "對交集內每個x要存在某basis元素含x且包含在交集中，不要求整個交集本身在basis。逐全部候選refiner才可證不存在，首失敗保left/right/point的LEX順序。",
        "en": "For each intersection point x require some candidate basis element containing x and contained in the intersection.The entire intersection need not itself be a basis element.Check all refiners before declaring absence,and keep the first left/right/point failure in source order."
      },
      "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": "S029",
      "uses": [
        "G001",
        "S028"
      ],
      "before": "ORIGINAL ordered basis positions(left,right)=[3, 1];actual basis elements=[[1, 2], [0, 1]];FULL intersection=[1]",
      "after": "ALL [intersection_point,ALL refining original basis indices] rows=[[1, [2]]];empty intersection gives VACUOUS local obligation;first failure retained in source order",
      "relation": "implies",
      "rule_id": "topbasis_full_local_point_refinement_not_intersection_membership",
      "conditions": [],
      "explanation": {
        "zh-Hant": "對交集內每個x要存在某basis元素含x且包含在交集中，不要求整個交集本身在basis。逐全部候選refiner才可證不存在，首失敗保left/right/point的LEX順序。",
        "en": "For each intersection point x require some candidate basis element containing x and contained in the intersection.The entire intersection need not itself be a basis element.Check all refiners before declaring absence,and keep the first left/right/point failure in source order."
      },
      "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": "S030",
      "uses": [
        "G001",
        "S029"
      ],
      "before": "ORIGINAL ordered basis positions(left,right)=[3, 2];actual basis elements=[[1, 2], [1]];FULL intersection=[1]",
      "after": "ALL [intersection_point,ALL refining original basis indices] rows=[[1, [2]]];empty intersection gives VACUOUS local obligation;first failure retained in source order",
      "relation": "implies",
      "rule_id": "topbasis_full_local_point_refinement_not_intersection_membership",
      "conditions": [],
      "explanation": {
        "zh-Hant": "對交集內每個x要存在某basis元素含x且包含在交集中，不要求整個交集本身在basis。逐全部候選refiner才可證不存在，首失敗保left/right/point的LEX順序。",
        "en": "For each intersection point x require some candidate basis element containing x and contained in the intersection.The entire intersection need not itself be a basis element.Check all refiners before declaring absence,and keep the first left/right/point failure in source order."
      },
      "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": "S031",
      "uses": [
        "G001",
        "S030"
      ],
      "before": "ORIGINAL ordered basis positions(left,right)=[3, 3];actual basis elements=[[1, 2], [1, 2]];FULL intersection=[1, 2]",
      "after": "ALL [intersection_point,ALL refining original basis indices] rows=[[1, [2, 3]], [2, [3, 4]]];empty intersection gives VACUOUS local obligation;first failure retained in source order",
      "relation": "implies",
      "rule_id": "topbasis_full_local_point_refinement_not_intersection_membership",
      "conditions": [],
      "explanation": {
        "zh-Hant": "對交集內每個x要存在某basis元素含x且包含在交集中，不要求整個交集本身在basis。逐全部候選refiner才可證不存在，首失敗保left/right/point的LEX順序。",
        "en": "For each intersection point x require some candidate basis element containing x and contained in the intersection.The entire intersection need not itself be a basis element.Check all refiners before declaring absence,and keep the first left/right/point failure in source order."
      },
      "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": "S032",
      "uses": [
        "G001",
        "S031"
      ],
      "before": "ORIGINAL ordered basis positions(left,right)=[3, 4];actual basis elements=[[1, 2], [2]];FULL intersection=[2]",
      "after": "ALL [intersection_point,ALL refining original basis indices] rows=[[2, [4]]];empty intersection gives VACUOUS local obligation;first failure retained in source order",
      "relation": "implies",
      "rule_id": "topbasis_full_local_point_refinement_not_intersection_membership",
      "conditions": [],
      "explanation": {
        "zh-Hant": "對交集內每個x要存在某basis元素含x且包含在交集中，不要求整個交集本身在basis。逐全部候選refiner才可證不存在，首失敗保left/right/point的LEX順序。",
        "en": "For each intersection point x require some candidate basis element containing x and contained in the intersection.The entire intersection need not itself be a basis element.Check all refiners before declaring absence,and keep the first left/right/point failure in source order."
      },
      "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": "S033",
      "uses": [
        "G001",
        "S032"
      ],
      "before": "ORIGINAL ordered basis positions(left,right)=[3, 5];actual basis elements=[[1, 2], [3]];FULL intersection=[]",
      "after": "ALL [intersection_point,ALL refining original basis indices] rows=[];empty intersection gives VACUOUS local obligation;first failure retained in source order",
      "relation": "implies",
      "rule_id": "topbasis_full_local_point_refinement_not_intersection_membership",
      "conditions": [],
      "explanation": {
        "zh-Hant": "對交集內每個x要存在某basis元素含x且包含在交集中，不要求整個交集本身在basis。逐全部候選refiner才可證不存在，首失敗保left/right/point的LEX順序。",
        "en": "For each intersection point x require some candidate basis element containing x and contained in the intersection.The entire intersection need not itself be a basis element.Check all refiners before declaring absence,and keep the first left/right/point failure in source order."
      },
      "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": "S034",
      "uses": [
        "G001",
        "S033"
      ],
      "before": "ORIGINAL ordered basis positions(left,right)=[3, 6];actual basis elements=[[1, 2], [3, 4]];FULL intersection=[]",
      "after": "ALL [intersection_point,ALL refining original basis indices] rows=[];empty intersection gives VACUOUS local obligation;first failure retained in source order",
      "relation": "implies",
      "rule_id": "topbasis_full_local_point_refinement_not_intersection_membership",
      "conditions": [],
      "explanation": {
        "zh-Hant": "對交集內每個x要存在某basis元素含x且包含在交集中，不要求整個交集本身在basis。逐全部候選refiner才可證不存在，首失敗保left/right/point的LEX順序。",
        "en": "For each intersection point x require some candidate basis element containing x and contained in the intersection.The entire intersection need not itself be a basis element.Check all refiners before declaring absence,and keep the first left/right/point failure in source order."
      },
      "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": "S035",
      "uses": [
        "G001",
        "S034"
      ],
      "before": "ORIGINAL ordered basis positions(left,right)=[3, 7];actual basis elements=[[1, 2], [4]];FULL intersection=[]",
      "after": "ALL [intersection_point,ALL refining original basis indices] rows=[];empty intersection gives VACUOUS local obligation;first failure retained in source order",
      "relation": "implies",
      "rule_id": "topbasis_full_local_point_refinement_not_intersection_membership",
      "conditions": [],
      "explanation": {
        "zh-Hant": "對交集內每個x要存在某basis元素含x且包含在交集中，不要求整個交集本身在basis。逐全部候選refiner才可證不存在，首失敗保left/right/point的LEX順序。",
        "en": "For each intersection point x require some candidate basis element containing x and contained in the intersection.The entire intersection need not itself be a basis element.Check all refiners before declaring absence,and keep the first left/right/point failure in source order."
      },
      "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": "S036",
      "uses": [
        "G001",
        "S035"
      ],
      "before": "ORIGINAL ordered basis positions(left,right)=[4, 0];actual basis elements=[[2], [0]];FULL intersection=[]",
      "after": "ALL [intersection_point,ALL refining original basis indices] rows=[];empty intersection gives VACUOUS local obligation;first failure retained in source order",
      "relation": "implies",
      "rule_id": "topbasis_full_local_point_refinement_not_intersection_membership",
      "conditions": [],
      "explanation": {
        "zh-Hant": "對交集內每個x要存在某basis元素含x且包含在交集中，不要求整個交集本身在basis。逐全部候選refiner才可證不存在，首失敗保left/right/point的LEX順序。",
        "en": "For each intersection point x require some candidate basis element containing x and contained in the intersection.The entire intersection need not itself be a basis element.Check all refiners before declaring absence,and keep the first left/right/point failure in source order."
      },
      "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": "S037",
      "uses": [
        "G001",
        "S036"
      ],
      "before": "ORIGINAL ordered basis positions(left,right)=[4, 1];actual basis elements=[[2], [0, 1]];FULL intersection=[]",
      "after": "ALL [intersection_point,ALL refining original basis indices] rows=[];empty intersection gives VACUOUS local obligation;first failure retained in source order",
      "relation": "implies",
      "rule_id": "topbasis_full_local_point_refinement_not_intersection_membership",
      "conditions": [],
      "explanation": {
        "zh-Hant": "對交集內每個x要存在某basis元素含x且包含在交集中，不要求整個交集本身在basis。逐全部候選refiner才可證不存在，首失敗保left/right/point的LEX順序。",
        "en": "For each intersection point x require some candidate basis element containing x and contained in the intersection.The entire intersection need not itself be a basis element.Check all refiners before declaring absence,and keep the first left/right/point failure in source order."
      },
      "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": "S038",
      "uses": [
        "G001",
        "S037"
      ],
      "before": "ORIGINAL ordered basis positions(left,right)=[4, 2];actual basis elements=[[2], [1]];FULL intersection=[]",
      "after": "ALL [intersection_point,ALL refining original basis indices] rows=[];empty intersection gives VACUOUS local obligation;first failure retained in source order",
      "relation": "implies",
      "rule_id": "topbasis_full_local_point_refinement_not_intersection_membership",
      "conditions": [],
      "explanation": {
        "zh-Hant": "對交集內每個x要存在某basis元素含x且包含在交集中，不要求整個交集本身在basis。逐全部候選refiner才可證不存在，首失敗保left/right/point的LEX順序。",
        "en": "For each intersection point x require some candidate basis element containing x and contained in the intersection.The entire intersection need not itself be a basis element.Check all refiners before declaring absence,and keep the first left/right/point failure in source order."
      },
      "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": "S039",
      "uses": [
        "G001",
        "S038"
      ],
      "before": "ORIGINAL ordered basis positions(left,right)=[4, 3];actual basis elements=[[2], [1, 2]];FULL intersection=[2]",
      "after": "ALL [intersection_point,ALL refining original basis indices] rows=[[2, [4]]];empty intersection gives VACUOUS local obligation;first failure retained in source order",
      "relation": "implies",
      "rule_id": "topbasis_full_local_point_refinement_not_intersection_membership",
      "conditions": [],
      "explanation": {
        "zh-Hant": "對交集內每個x要存在某basis元素含x且包含在交集中，不要求整個交集本身在basis。逐全部候選refiner才可證不存在，首失敗保left/right/point的LEX順序。",
        "en": "For each intersection point x require some candidate basis element containing x and contained in the intersection.The entire intersection need not itself be a basis element.Check all refiners before declaring absence,and keep the first left/right/point failure in source order."
      },
      "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": "S040",
      "uses": [
        "G001",
        "S039"
      ],
      "before": "ORIGINAL ordered basis positions(left,right)=[4, 4];actual basis elements=[[2], [2]];FULL intersection=[2]",
      "after": "ALL [intersection_point,ALL refining original basis indices] rows=[[2, [4]]];empty intersection gives VACUOUS local obligation;first failure retained in source order",
      "relation": "implies",
      "rule_id": "topbasis_full_local_point_refinement_not_intersection_membership",
      "conditions": [],
      "explanation": {
        "zh-Hant": "對交集內每個x要存在某basis元素含x且包含在交集中，不要求整個交集本身在basis。逐全部候選refiner才可證不存在，首失敗保left/right/point的LEX順序。",
        "en": "For each intersection point x require some candidate basis element containing x and contained in the intersection.The entire intersection need not itself be a basis element.Check all refiners before declaring absence,and keep the first left/right/point failure in source order."
      },
      "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": "S041",
      "uses": [
        "G001",
        "S040"
      ],
      "before": "ORIGINAL ordered basis positions(left,right)=[4, 5];actual basis elements=[[2], [3]];FULL intersection=[]",
      "after": "ALL [intersection_point,ALL refining original basis indices] rows=[];empty intersection gives VACUOUS local obligation;first failure retained in source order",
      "relation": "implies",
      "rule_id": "topbasis_full_local_point_refinement_not_intersection_membership",
      "conditions": [],
      "explanation": {
        "zh-Hant": "對交集內每個x要存在某basis元素含x且包含在交集中，不要求整個交集本身在basis。逐全部候選refiner才可證不存在，首失敗保left/right/point的LEX順序。",
        "en": "For each intersection point x require some candidate basis element containing x and contained in the intersection.The entire intersection need not itself be a basis element.Check all refiners before declaring absence,and keep the first left/right/point failure in source order."
      },
      "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": "S042",
      "uses": [
        "G001",
        "S041"
      ],
      "before": "ORIGINAL ordered basis positions(left,right)=[4, 6];actual basis elements=[[2], [3, 4]];FULL intersection=[]",
      "after": "ALL [intersection_point,ALL refining original basis indices] rows=[];empty intersection gives VACUOUS local obligation;first failure retained in source order",
      "relation": "implies",
      "rule_id": "topbasis_full_local_point_refinement_not_intersection_membership",
      "conditions": [],
      "explanation": {
        "zh-Hant": "對交集內每個x要存在某basis元素含x且包含在交集中，不要求整個交集本身在basis。逐全部候選refiner才可證不存在，首失敗保left/right/point的LEX順序。",
        "en": "For each intersection point x require some candidate basis element containing x and contained in the intersection.The entire intersection need not itself be a basis element.Check all refiners before declaring absence,and keep the first left/right/point failure in source order."
      },
      "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": "S043",
      "uses": [
        "G001",
        "S042"
      ],
      "before": "ORIGINAL ordered basis positions(left,right)=[4, 7];actual basis elements=[[2], [4]];FULL intersection=[]",
      "after": "ALL [intersection_point,ALL refining original basis indices] rows=[];empty intersection gives VACUOUS local obligation;first failure retained in source order",
      "relation": "implies",
      "rule_id": "topbasis_full_local_point_refinement_not_intersection_membership",
      "conditions": [],
      "explanation": {
        "zh-Hant": "對交集內每個x要存在某basis元素含x且包含在交集中，不要求整個交集本身在basis。逐全部候選refiner才可證不存在，首失敗保left/right/point的LEX順序。",
        "en": "For each intersection point x require some candidate basis element containing x and contained in the intersection.The entire intersection need not itself be a basis element.Check all refiners before declaring absence,and keep the first left/right/point failure in source order."
      },
      "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": "S044",
      "uses": [
        "G001",
        "S043"
      ],
      "before": "ORIGINAL ordered basis positions(left,right)=[5, 0];actual basis elements=[[3], [0]];FULL intersection=[]",
      "after": "ALL [intersection_point,ALL refining original basis indices] rows=[];empty intersection gives VACUOUS local obligation;first failure retained in source order",
      "relation": "implies",
      "rule_id": "topbasis_full_local_point_refinement_not_intersection_membership",
      "conditions": [],
      "explanation": {
        "zh-Hant": "對交集內每個x要存在某basis元素含x且包含在交集中，不要求整個交集本身在basis。逐全部候選refiner才可證不存在，首失敗保left/right/point的LEX順序。",
        "en": "For each intersection point x require some candidate basis element containing x and contained in the intersection.The entire intersection need not itself be a basis element.Check all refiners before declaring absence,and keep the first left/right/point failure in source order."
      },
      "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": "S045",
      "uses": [
        "G001",
        "S044"
      ],
      "before": "ORIGINAL ordered basis positions(left,right)=[5, 1];actual basis elements=[[3], [0, 1]];FULL intersection=[]",
      "after": "ALL [intersection_point,ALL refining original basis indices] rows=[];empty intersection gives VACUOUS local obligation;first failure retained in source order",
      "relation": "implies",
      "rule_id": "topbasis_full_local_point_refinement_not_intersection_membership",
      "conditions": [],
      "explanation": {
        "zh-Hant": "對交集內每個x要存在某basis元素含x且包含在交集中，不要求整個交集本身在basis。逐全部候選refiner才可證不存在，首失敗保left/right/point的LEX順序。",
        "en": "For each intersection point x require some candidate basis element containing x and contained in the intersection.The entire intersection need not itself be a basis element.Check all refiners before declaring absence,and keep the first left/right/point failure in source order."
      },
      "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": "S046",
      "uses": [
        "G001",
        "S045"
      ],
      "before": "ORIGINAL ordered basis positions(left,right)=[5, 2];actual basis elements=[[3], [1]];FULL intersection=[]",
      "after": "ALL [intersection_point,ALL refining original basis indices] rows=[];empty intersection gives VACUOUS local obligation;first failure retained in source order",
      "relation": "implies",
      "rule_id": "topbasis_full_local_point_refinement_not_intersection_membership",
      "conditions": [],
      "explanation": {
        "zh-Hant": "對交集內每個x要存在某basis元素含x且包含在交集中，不要求整個交集本身在basis。逐全部候選refiner才可證不存在，首失敗保left/right/point的LEX順序。",
        "en": "For each intersection point x require some candidate basis element containing x and contained in the intersection.The entire intersection need not itself be a basis element.Check all refiners before declaring absence,and keep the first left/right/point failure in source order."
      },
      "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": "S047",
      "uses": [
        "G001",
        "S046"
      ],
      "before": "ORIGINAL ordered basis positions(left,right)=[5, 3];actual basis elements=[[3], [1, 2]];FULL intersection=[]",
      "after": "ALL [intersection_point,ALL refining original basis indices] rows=[];empty intersection gives VACUOUS local obligation;first failure retained in source order",
      "relation": "implies",
      "rule_id": "topbasis_full_local_point_refinement_not_intersection_membership",
      "conditions": [],
      "explanation": {
        "zh-Hant": "對交集內每個x要存在某basis元素含x且包含在交集中，不要求整個交集本身在basis。逐全部候選refiner才可證不存在，首失敗保left/right/point的LEX順序。",
        "en": "For each intersection point x require some candidate basis element containing x and contained in the intersection.The entire intersection need not itself be a basis element.Check all refiners before declaring absence,and keep the first left/right/point failure in source order."
      },
      "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": "S048",
      "uses": [
        "G001",
        "S047"
      ],
      "before": "ORIGINAL ordered basis positions(left,right)=[5, 4];actual basis elements=[[3], [2]];FULL intersection=[]",
      "after": "ALL [intersection_point,ALL refining original basis indices] rows=[];empty intersection gives VACUOUS local obligation;first failure retained in source order",
      "relation": "implies",
      "rule_id": "topbasis_full_local_point_refinement_not_intersection_membership",
      "conditions": [],
      "explanation": {
        "zh-Hant": "對交集內每個x要存在某basis元素含x且包含在交集中，不要求整個交集本身在basis。逐全部候選refiner才可證不存在，首失敗保left/right/point的LEX順序。",
        "en": "For each intersection point x require some candidate basis element containing x and contained in the intersection.The entire intersection need not itself be a basis element.Check all refiners before declaring absence,and keep the first left/right/point failure in source order."
      },
      "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": "S049",
      "uses": [
        "G001",
        "S048"
      ],
      "before": "ORIGINAL ordered basis positions(left,right)=[5, 5];actual basis elements=[[3], [3]];FULL intersection=[3]",
      "after": "ALL [intersection_point,ALL refining original basis indices] rows=[[3, [5]]];empty intersection gives VACUOUS local obligation;first failure retained in source order",
      "relation": "implies",
      "rule_id": "topbasis_full_local_point_refinement_not_intersection_membership",
      "conditions": [],
      "explanation": {
        "zh-Hant": "對交集內每個x要存在某basis元素含x且包含在交集中，不要求整個交集本身在basis。逐全部候選refiner才可證不存在，首失敗保left/right/point的LEX順序。",
        "en": "For each intersection point x require some candidate basis element containing x and contained in the intersection.The entire intersection need not itself be a basis element.Check all refiners before declaring absence,and keep the first left/right/point failure in source order."
      },
      "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": "S050",
      "uses": [
        "G001",
        "S049"
      ],
      "before": "ORIGINAL ordered basis positions(left,right)=[5, 6];actual basis elements=[[3], [3, 4]];FULL intersection=[3]",
      "after": "ALL [intersection_point,ALL refining original basis indices] rows=[[3, [5]]];empty intersection gives VACUOUS local obligation;first failure retained in source order",
      "relation": "implies",
      "rule_id": "topbasis_full_local_point_refinement_not_intersection_membership",
      "conditions": [],
      "explanation": {
        "zh-Hant": "對交集內每個x要存在某basis元素含x且包含在交集中，不要求整個交集本身在basis。逐全部候選refiner才可證不存在，首失敗保left/right/point的LEX順序。",
        "en": "For each intersection point x require some candidate basis element containing x and contained in the intersection.The entire intersection need not itself be a basis element.Check all refiners before declaring absence,and keep the first left/right/point failure in source order."
      },
      "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": "S051",
      "uses": [
        "G001",
        "S050"
      ],
      "before": "ORIGINAL ordered basis positions(left,right)=[5, 7];actual basis elements=[[3], [4]];FULL intersection=[]",
      "after": "ALL [intersection_point,ALL refining original basis indices] rows=[];empty intersection gives VACUOUS local obligation;first failure retained in source order",
      "relation": "implies",
      "rule_id": "topbasis_full_local_point_refinement_not_intersection_membership",
      "conditions": [],
      "explanation": {
        "zh-Hant": "對交集內每個x要存在某basis元素含x且包含在交集中，不要求整個交集本身在basis。逐全部候選refiner才可證不存在，首失敗保left/right/point的LEX順序。",
        "en": "For each intersection point x require some candidate basis element containing x and contained in the intersection.The entire intersection need not itself be a basis element.Check all refiners before declaring absence,and keep the first left/right/point failure in source order."
      },
      "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": "S052",
      "uses": [
        "G001",
        "S051"
      ],
      "before": "ORIGINAL ordered basis positions(left,right)=[6, 0];actual basis elements=[[3, 4], [0]];FULL intersection=[]",
      "after": "ALL [intersection_point,ALL refining original basis indices] rows=[];empty intersection gives VACUOUS local obligation;first failure retained in source order",
      "relation": "implies",
      "rule_id": "topbasis_full_local_point_refinement_not_intersection_membership",
      "conditions": [],
      "explanation": {
        "zh-Hant": "對交集內每個x要存在某basis元素含x且包含在交集中，不要求整個交集本身在basis。逐全部候選refiner才可證不存在，首失敗保left/right/point的LEX順序。",
        "en": "For each intersection point x require some candidate basis element containing x and contained in the intersection.The entire intersection need not itself be a basis element.Check all refiners before declaring absence,and keep the first left/right/point failure in source order."
      },
      "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": "S053",
      "uses": [
        "G001",
        "S052"
      ],
      "before": "ORIGINAL ordered basis positions(left,right)=[6, 1];actual basis elements=[[3, 4], [0, 1]];FULL intersection=[]",
      "after": "ALL [intersection_point,ALL refining original basis indices] rows=[];empty intersection gives VACUOUS local obligation;first failure retained in source order",
      "relation": "implies",
      "rule_id": "topbasis_full_local_point_refinement_not_intersection_membership",
      "conditions": [],
      "explanation": {
        "zh-Hant": "對交集內每個x要存在某basis元素含x且包含在交集中，不要求整個交集本身在basis。逐全部候選refiner才可證不存在，首失敗保left/right/point的LEX順序。",
        "en": "For each intersection point x require some candidate basis element containing x and contained in the intersection.The entire intersection need not itself be a basis element.Check all refiners before declaring absence,and keep the first left/right/point failure in source order."
      },
      "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": "S054",
      "uses": [
        "G001",
        "S053"
      ],
      "before": "ORIGINAL ordered basis positions(left,right)=[6, 2];actual basis elements=[[3, 4], [1]];FULL intersection=[]",
      "after": "ALL [intersection_point,ALL refining original basis indices] rows=[];empty intersection gives VACUOUS local obligation;first failure retained in source order",
      "relation": "implies",
      "rule_id": "topbasis_full_local_point_refinement_not_intersection_membership",
      "conditions": [],
      "explanation": {
        "zh-Hant": "對交集內每個x要存在某basis元素含x且包含在交集中，不要求整個交集本身在basis。逐全部候選refiner才可證不存在，首失敗保left/right/point的LEX順序。",
        "en": "For each intersection point x require some candidate basis element containing x and contained in the intersection.The entire intersection need not itself be a basis element.Check all refiners before declaring absence,and keep the first left/right/point failure in source order."
      },
      "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": "S055",
      "uses": [
        "G001",
        "S054"
      ],
      "before": "ORIGINAL ordered basis positions(left,right)=[6, 3];actual basis elements=[[3, 4], [1, 2]];FULL intersection=[]",
      "after": "ALL [intersection_point,ALL refining original basis indices] rows=[];empty intersection gives VACUOUS local obligation;first failure retained in source order",
      "relation": "implies",
      "rule_id": "topbasis_full_local_point_refinement_not_intersection_membership",
      "conditions": [],
      "explanation": {
        "zh-Hant": "對交集內每個x要存在某basis元素含x且包含在交集中，不要求整個交集本身在basis。逐全部候選refiner才可證不存在，首失敗保left/right/point的LEX順序。",
        "en": "For each intersection point x require some candidate basis element containing x and contained in the intersection.The entire intersection need not itself be a basis element.Check all refiners before declaring absence,and keep the first left/right/point failure in source order."
      },
      "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": "S056",
      "uses": [
        "G001",
        "S055"
      ],
      "before": "ORIGINAL ordered basis positions(left,right)=[6, 4];actual basis elements=[[3, 4], [2]];FULL intersection=[]",
      "after": "ALL [intersection_point,ALL refining original basis indices] rows=[];empty intersection gives VACUOUS local obligation;first failure retained in source order",
      "relation": "implies",
      "rule_id": "topbasis_full_local_point_refinement_not_intersection_membership",
      "conditions": [],
      "explanation": {
        "zh-Hant": "對交集內每個x要存在某basis元素含x且包含在交集中，不要求整個交集本身在basis。逐全部候選refiner才可證不存在，首失敗保left/right/point的LEX順序。",
        "en": "For each intersection point x require some candidate basis element containing x and contained in the intersection.The entire intersection need not itself be a basis element.Check all refiners before declaring absence,and keep the first left/right/point failure in source order."
      },
      "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": "S057",
      "uses": [
        "G001",
        "S056"
      ],
      "before": "ORIGINAL ordered basis positions(left,right)=[6, 5];actual basis elements=[[3, 4], [3]];FULL intersection=[3]",
      "after": "ALL [intersection_point,ALL refining original basis indices] rows=[[3, [5]]];empty intersection gives VACUOUS local obligation;first failure retained in source order",
      "relation": "implies",
      "rule_id": "topbasis_full_local_point_refinement_not_intersection_membership",
      "conditions": [],
      "explanation": {
        "zh-Hant": "對交集內每個x要存在某basis元素含x且包含在交集中，不要求整個交集本身在basis。逐全部候選refiner才可證不存在，首失敗保left/right/point的LEX順序。",
        "en": "For each intersection point x require some candidate basis element containing x and contained in the intersection.The entire intersection need not itself be a basis element.Check all refiners before declaring absence,and keep the first left/right/point failure in source order."
      },
      "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": "S058",
      "uses": [
        "G001",
        "S057"
      ],
      "before": "ORIGINAL ordered basis positions(left,right)=[6, 6];actual basis elements=[[3, 4], [3, 4]];FULL intersection=[3, 4]",
      "after": "ALL [intersection_point,ALL refining original basis indices] rows=[[3, [5, 6]], [4, [6, 7]]];empty intersection gives VACUOUS local obligation;first failure retained in source order",
      "relation": "implies",
      "rule_id": "topbasis_full_local_point_refinement_not_intersection_membership",
      "conditions": [],
      "explanation": {
        "zh-Hant": "對交集內每個x要存在某basis元素含x且包含在交集中，不要求整個交集本身在basis。逐全部候選refiner才可證不存在，首失敗保left/right/point的LEX順序。",
        "en": "For each intersection point x require some candidate basis element containing x and contained in the intersection.The entire intersection need not itself be a basis element.Check all refiners before declaring absence,and keep the first left/right/point failure in source order."
      },
      "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": "S059",
      "uses": [
        "G001",
        "S058"
      ],
      "before": "ORIGINAL ordered basis positions(left,right)=[6, 7];actual basis elements=[[3, 4], [4]];FULL intersection=[4]",
      "after": "ALL [intersection_point,ALL refining original basis indices] rows=[[4, [7]]];empty intersection gives VACUOUS local obligation;first failure retained in source order",
      "relation": "implies",
      "rule_id": "topbasis_full_local_point_refinement_not_intersection_membership",
      "conditions": [],
      "explanation": {
        "zh-Hant": "對交集內每個x要存在某basis元素含x且包含在交集中，不要求整個交集本身在basis。逐全部候選refiner才可證不存在，首失敗保left/right/point的LEX順序。",
        "en": "For each intersection point x require some candidate basis element containing x and contained in the intersection.The entire intersection need not itself be a basis element.Check all refiners before declaring absence,and keep the first left/right/point failure in source order."
      },
      "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": "S060",
      "uses": [
        "G001",
        "S059"
      ],
      "before": "ORIGINAL ordered basis positions(left,right)=[7, 0];actual basis elements=[[4], [0]];FULL intersection=[]",
      "after": "ALL [intersection_point,ALL refining original basis indices] rows=[];empty intersection gives VACUOUS local obligation;first failure retained in source order",
      "relation": "implies",
      "rule_id": "topbasis_full_local_point_refinement_not_intersection_membership",
      "conditions": [],
      "explanation": {
        "zh-Hant": "對交集內每個x要存在某basis元素含x且包含在交集中，不要求整個交集本身在basis。逐全部候選refiner才可證不存在，首失敗保left/right/point的LEX順序。",
        "en": "For each intersection point x require some candidate basis element containing x and contained in the intersection.The entire intersection need not itself be a basis element.Check all refiners before declaring absence,and keep the first left/right/point failure in source order."
      },
      "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": "S061",
      "uses": [
        "G001",
        "S060"
      ],
      "before": "ORIGINAL ordered basis positions(left,right)=[7, 1];actual basis elements=[[4], [0, 1]];FULL intersection=[]",
      "after": "ALL [intersection_point,ALL refining original basis indices] rows=[];empty intersection gives VACUOUS local obligation;first failure retained in source order",
      "relation": "implies",
      "rule_id": "topbasis_full_local_point_refinement_not_intersection_membership",
      "conditions": [],
      "explanation": {
        "zh-Hant": "對交集內每個x要存在某basis元素含x且包含在交集中，不要求整個交集本身在basis。逐全部候選refiner才可證不存在，首失敗保left/right/point的LEX順序。",
        "en": "For each intersection point x require some candidate basis element containing x and contained in the intersection.The entire intersection need not itself be a basis element.Check all refiners before declaring absence,and keep the first left/right/point failure in source order."
      },
      "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": "S062",
      "uses": [
        "G001",
        "S061"
      ],
      "before": "ORIGINAL ordered basis positions(left,right)=[7, 2];actual basis elements=[[4], [1]];FULL intersection=[]",
      "after": "ALL [intersection_point,ALL refining original basis indices] rows=[];empty intersection gives VACUOUS local obligation;first failure retained in source order",
      "relation": "implies",
      "rule_id": "topbasis_full_local_point_refinement_not_intersection_membership",
      "conditions": [],
      "explanation": {
        "zh-Hant": "對交集內每個x要存在某basis元素含x且包含在交集中，不要求整個交集本身在basis。逐全部候選refiner才可證不存在，首失敗保left/right/point的LEX順序。",
        "en": "For each intersection point x require some candidate basis element containing x and contained in the intersection.The entire intersection need not itself be a basis element.Check all refiners before declaring absence,and keep the first left/right/point failure in source order."
      },
      "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": "S063",
      "uses": [
        "G001",
        "S062"
      ],
      "before": "ORIGINAL ordered basis positions(left,right)=[7, 3];actual basis elements=[[4], [1, 2]];FULL intersection=[]",
      "after": "ALL [intersection_point,ALL refining original basis indices] rows=[];empty intersection gives VACUOUS local obligation;first failure retained in source order",
      "relation": "implies",
      "rule_id": "topbasis_full_local_point_refinement_not_intersection_membership",
      "conditions": [],
      "explanation": {
        "zh-Hant": "對交集內每個x要存在某basis元素含x且包含在交集中，不要求整個交集本身在basis。逐全部候選refiner才可證不存在，首失敗保left/right/point的LEX順序。",
        "en": "For each intersection point x require some candidate basis element containing x and contained in the intersection.The entire intersection need not itself be a basis element.Check all refiners before declaring absence,and keep the first left/right/point failure in source order."
      },
      "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": "S064",
      "uses": [
        "G001",
        "S063"
      ],
      "before": "ORIGINAL ordered basis positions(left,right)=[7, 4];actual basis elements=[[4], [2]];FULL intersection=[]",
      "after": "ALL [intersection_point,ALL refining original basis indices] rows=[];empty intersection gives VACUOUS local obligation;first failure retained in source order",
      "relation": "implies",
      "rule_id": "topbasis_full_local_point_refinement_not_intersection_membership",
      "conditions": [],
      "explanation": {
        "zh-Hant": "對交集內每個x要存在某basis元素含x且包含在交集中，不要求整個交集本身在basis。逐全部候選refiner才可證不存在，首失敗保left/right/point的LEX順序。",
        "en": "For each intersection point x require some candidate basis element containing x and contained in the intersection.The entire intersection need not itself be a basis element.Check all refiners before declaring absence,and keep the first left/right/point failure in source order."
      },
      "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": "S065",
      "uses": [
        "G001",
        "S064"
      ],
      "before": "ORIGINAL ordered basis positions(left,right)=[7, 5];actual basis elements=[[4], [3]];FULL intersection=[]",
      "after": "ALL [intersection_point,ALL refining original basis indices] rows=[];empty intersection gives VACUOUS local obligation;first failure retained in source order",
      "relation": "implies",
      "rule_id": "topbasis_full_local_point_refinement_not_intersection_membership",
      "conditions": [],
      "explanation": {
        "zh-Hant": "對交集內每個x要存在某basis元素含x且包含在交集中，不要求整個交集本身在basis。逐全部候選refiner才可證不存在，首失敗保left/right/point的LEX順序。",
        "en": "For each intersection point x require some candidate basis element containing x and contained in the intersection.The entire intersection need not itself be a basis element.Check all refiners before declaring absence,and keep the first left/right/point failure in source order."
      },
      "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": "S066",
      "uses": [
        "G001",
        "S065"
      ],
      "before": "ORIGINAL ordered basis positions(left,right)=[7, 6];actual basis elements=[[4], [3, 4]];FULL intersection=[4]",
      "after": "ALL [intersection_point,ALL refining original basis indices] rows=[[4, [7]]];empty intersection gives VACUOUS local obligation;first failure retained in source order",
      "relation": "implies",
      "rule_id": "topbasis_full_local_point_refinement_not_intersection_membership",
      "conditions": [],
      "explanation": {
        "zh-Hant": "對交集內每個x要存在某basis元素含x且包含在交集中，不要求整個交集本身在basis。逐全部候選refiner才可證不存在，首失敗保left/right/point的LEX順序。",
        "en": "For each intersection point x require some candidate basis element containing x and contained in the intersection.The entire intersection need not itself be a basis element.Check all refiners before declaring absence,and keep the first left/right/point failure in source order."
      },
      "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": "S067",
      "uses": [
        "G001",
        "S066"
      ],
      "before": "ORIGINAL ordered basis positions(left,right)=[7, 7];actual basis elements=[[4], [4]];FULL intersection=[4]",
      "after": "ALL [intersection_point,ALL refining original basis indices] rows=[[4, [7]]];empty intersection gives VACUOUS local obligation;first failure retained in source order",
      "relation": "implies",
      "rule_id": "topbasis_full_local_point_refinement_not_intersection_membership",
      "conditions": [],
      "explanation": {
        "zh-Hant": "對交集內每個x要存在某basis元素含x且包含在交集中，不要求整個交集本身在basis。逐全部候選refiner才可證不存在，首失敗保left/right/point的LEX順序。",
        "en": "For each intersection point x require some candidate basis element containing x and contained in the intersection.The entire intersection need not itself be a basis element.Check all refiners before declaring absence,and keep the first left/right/point failure in source order."
      },
      "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": "S068",
      "uses": [
        "G001",
        "S067"
      ],
      "before": "ACTUAL cover=是;ALL local point obligations refined=是;first failure[left,right,point,intersection]=null",
      "after": "abstract basis validity=是 is cover AND LOCAL refinement;not closure of the family under intersections and not a given-topology certificate yet",
      "relation": "implies",
      "rule_id": "topbasis_cover_and_local_refinement_conjunction",
      "conditions": [],
      "explanation": {
        "zh-Hant": "兩項合取才給抽象basis；cover-only不夠，另一方面好basis不必對交運算封閉。相對已給topology還有開性與生成所有開集的其他責任。",
        "en": "Their conjunction gives an abstract basis.Coverage alone is insufficient,and a valid basis need not be intersection-closed.A basis for a separately supplied topology also needs openness and generation of every given open."
      },
      "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": "S069",
      "uses": [
        "G001",
        "S068"
      ],
      "before": "ORIGINAL complete candidate basis in fixed family-index order=[[0], [0, 1], [1], [1, 2], [2], [3], [3, 4], [4]];carrier Fin(5)",
      "after": "prefix0 has EXACTLY the empty-subfamily union:empty set,with empty index witness;this mathematical union exists even when basis axioms fail",
      "relation": "implies",
      "rule_id": "topbasis_empty_union_initial_state_not_missing_empty_set",
      "conditions": [],
      "explanation": {
        "zh-Hant": "空subfamily的union固定為空集合，毋須空集合本身是basis元素。此步只談union構造，不提前把不合格basis的unions稱topology。",
        "en": "The empty subfamily always has empty union,without requiring the empty set to be a basis element.This step constructs unions only and does not call an invalid basis's unions a topology."
      },
      "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": "S070",
      "uses": [
        "G001",
        "S069"
      ],
      "before": "next ORIGINAL basis index=0;basis set=[0];ALL previous-prefix unions in numeric POINT-mask order=[[]]",
      "after": "ALL [previous_union,union_with_new_basis,index_witness] rows=[[[], [0], [0]]];keep both omit/include choices,then deduplicate EQUAL UNIONS;new canonical union family=[[], [0]]",
      "relation": "implies",
      "rule_id": "topbasis_complete_prefix_union_DP_not_sampled_subfamilies",
      "conditions": [],
      "explanation": {
        "zh-Hant": "任意新prefix子族不是不選新basis元素，就是選它並與前prefix的某union合併，因此兩類選擇完整且每一狀態有真index見證。只合併相同union，不去掉原basis位置；狀態最多2^n，不需假裝抽樣了2^basisCount個族。",
        "en": "Any subfamily of the new prefix either omits the new element or includes it with a union from the previous prefix.These choices are complete and every state has an actual index witness.Deduplicate equal unions,not original basis positions.There are at most2^n distinct states,not a sample of the2^basisCount subfamilies."
      },
      "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": "S071",
      "uses": [
        "G001",
        "S070"
      ],
      "before": "next ORIGINAL basis index=1;basis set=[0, 1];ALL previous-prefix unions in numeric POINT-mask order=[[], [0]]",
      "after": "ALL [previous_union,union_with_new_basis,index_witness] rows=[[[], [0, 1], [1]], [[0], [0, 1], [0, 1]]];keep both omit/include choices,then deduplicate EQUAL UNIONS;new canonical union family=[[], [0], [0, 1]]",
      "relation": "implies",
      "rule_id": "topbasis_complete_prefix_union_DP_not_sampled_subfamilies",
      "conditions": [],
      "explanation": {
        "zh-Hant": "任意新prefix子族不是不選新basis元素，就是選它並與前prefix的某union合併，因此兩類選擇完整且每一狀態有真index見證。只合併相同union，不去掉原basis位置；狀態最多2^n，不需假裝抽樣了2^basisCount個族。",
        "en": "Any subfamily of the new prefix either omits the new element or includes it with a union from the previous prefix.These choices are complete and every state has an actual index witness.Deduplicate equal unions,not original basis positions.There are at most2^n distinct states,not a sample of the2^basisCount subfamilies."
      },
      "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": "S072",
      "uses": [
        "G001",
        "S071"
      ],
      "before": "next ORIGINAL basis index=2;basis set=[1];ALL previous-prefix unions in numeric POINT-mask order=[[], [0], [0, 1]]",
      "after": "ALL [previous_union,union_with_new_basis,index_witness] rows=[[[], [1], [2]], [[0], [0, 1], [0, 2]], [[0, 1], [0, 1], [1, 2]]];keep both omit/include choices,then deduplicate EQUAL UNIONS;new canonical union family=[[], [0], [0, 1], [1]]",
      "relation": "implies",
      "rule_id": "topbasis_complete_prefix_union_DP_not_sampled_subfamilies",
      "conditions": [],
      "explanation": {
        "zh-Hant": "任意新prefix子族不是不選新basis元素，就是選它並與前prefix的某union合併，因此兩類選擇完整且每一狀態有真index見證。只合併相同union，不去掉原basis位置；狀態最多2^n，不需假裝抽樣了2^basisCount個族。",
        "en": "Any subfamily of the new prefix either omits the new element or includes it with a union from the previous prefix.These choices are complete and every state has an actual index witness.Deduplicate equal unions,not original basis positions.There are at most2^n distinct states,not a sample of the2^basisCount subfamilies."
      },
      "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": "S073",
      "uses": [
        "G001",
        "S072"
      ],
      "before": "next ORIGINAL basis index=3;basis set=[1, 2];ALL previous-prefix unions in numeric POINT-mask order=[[], [0], [1], [0, 1]]",
      "after": "ALL [previous_union,union_with_new_basis,index_witness] rows=[[[], [1, 2], [3]], [[0], [0, 1, 2], [0, 3]], [[1], [1, 2], [2, 3]], [[0, 1], [0, 1, 2], [1, 3]]];keep both omit/include choices,then deduplicate EQUAL UNIONS;new canonical union family=[[], [0], [0, 1], [0, 1, 2], [1], [1, 2]]",
      "relation": "implies",
      "rule_id": "topbasis_complete_prefix_union_DP_not_sampled_subfamilies",
      "conditions": [],
      "explanation": {
        "zh-Hant": "任意新prefix子族不是不選新basis元素，就是選它並與前prefix的某union合併，因此兩類選擇完整且每一狀態有真index見證。只合併相同union，不去掉原basis位置；狀態最多2^n，不需假裝抽樣了2^basisCount個族。",
        "en": "Any subfamily of the new prefix either omits the new element or includes it with a union from the previous prefix.These choices are complete and every state has an actual index witness.Deduplicate equal unions,not original basis positions.There are at most2^n distinct states,not a sample of the2^basisCount subfamilies."
      },
      "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": "S074",
      "uses": [
        "G001",
        "S073"
      ],
      "before": "next ORIGINAL basis index=4;basis set=[2];ALL previous-prefix unions in numeric POINT-mask order=[[], [0], [1], [0, 1], [1, 2], [0, 1, 2]]",
      "after": "ALL [previous_union,union_with_new_basis,index_witness] rows=[[[], [2], [4]], [[0], [0, 2], [0, 4]], [[1], [1, 2], [2, 4]], [[0, 1], [0, 1, 2], [1, 4]], [[1, 2], [1, 2], [3, 4]], [[0, 1, 2], [0, 1, 2], [0, 3, 4]]];keep both omit/include choices,then deduplicate EQUAL UNIONS;new canonical union family=[[], [0], [0, 1], [0, 1, 2], [0, 2], [1], [1, 2], [2]]",
      "relation": "implies",
      "rule_id": "topbasis_complete_prefix_union_DP_not_sampled_subfamilies",
      "conditions": [],
      "explanation": {
        "zh-Hant": "任意新prefix子族不是不選新basis元素，就是選它並與前prefix的某union合併，因此兩類選擇完整且每一狀態有真index見證。只合併相同union，不去掉原basis位置；狀態最多2^n，不需假裝抽樣了2^basisCount個族。",
        "en": "Any subfamily of the new prefix either omits the new element or includes it with a union from the previous prefix.These choices are complete and every state has an actual index witness.Deduplicate equal unions,not original basis positions.There are at most2^n distinct states,not a sample of the2^basisCount subfamilies."
      },
      "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": "S075",
      "uses": [
        "G001",
        "S074"
      ],
      "before": "next ORIGINAL basis index=5;basis set=[3];ALL previous-prefix unions in numeric POINT-mask order=[[], [0], [1], [0, 1], [2], [0, 2], [1, 2], [0, 1, 2]]",
      "after": "ALL [previous_union,union_with_new_basis,index_witness] rows=[[[], [3], [5]], [[0], [0, 3], [0, 5]], [[1], [1, 3], [2, 5]], [[0, 1], [0, 1, 3], [1, 5]], [[2], [2, 3], [4, 5]], [[0, 2], [0, 2, 3], [0, 4, 5]], [[1, 2], [1, 2, 3], [3, 5]], [[0, 1, 2], [0, 1, 2, 3], [0, 3, 5]]];keep both omit/include choices,then deduplicate EQUAL UNIONS;new canonical union family=[[], [0], [0, 1], [0, 1, 2], [0, 1, 2, 3], [0, 1, 3], [0, 2], [0, 2, 3], [0, 3], [1], [1, 2], [1, 2, 3], [1, 3], [2], [2, 3], [3]]",
      "relation": "implies",
      "rule_id": "topbasis_complete_prefix_union_DP_not_sampled_subfamilies",
      "conditions": [],
      "explanation": {
        "zh-Hant": "任意新prefix子族不是不選新basis元素，就是選它並與前prefix的某union合併，因此兩類選擇完整且每一狀態有真index見證。只合併相同union，不去掉原basis位置；狀態最多2^n，不需假裝抽樣了2^basisCount個族。",
        "en": "Any subfamily of the new prefix either omits the new element or includes it with a union from the previous prefix.These choices are complete and every state has an actual index witness.Deduplicate equal unions,not original basis positions.There are at most2^n distinct states,not a sample of the2^basisCount subfamilies."
      },
      "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": "S076",
      "uses": [
        "G001",
        "S075"
      ],
      "before": "next ORIGINAL basis index=6;basis set=[3, 4];ALL previous-prefix unions in numeric POINT-mask order=[[], [0], [1], [0, 1], [2], [0, 2], [1, 2], [0, 1, 2], [3], [0, 3], [1, 3], [0, 1, 3], [2, 3], [0, 2, 3], [1, 2, 3], [0, 1, 2, 3]]",
      "after": "ALL [previous_union,union_with_new_basis,index_witness] rows=[[[], [3, 4], [6]], [[0], [0, 3, 4], [0, 6]], [[1], [1, 3, 4], [2, 6]], [[0, 1], [0, 1, 3, 4], [1, 6]], [[2], [2, 3, 4], [4, 6]], [[0, 2], [0, 2, 3, 4], [0, 4, 6]], [[1, 2], [1, 2, 3, 4], [3, 6]], [[0, 1, 2], [0, 1, 2, 3, 4], [0, 3, 6]], [[3], [3, 4], [5, 6]], [[0, 3], [0, 3, 4], [0, 5, 6]], [[1, 3], [1, 3, 4], [2, 5, 6]], [[0, 1, 3], [0, 1, 3, 4], [1, 5, 6]], [[2, 3], [2, 3, 4], [4, 5, 6]], [[0, 2, 3], [0, 2, 3, 4], [0, 4, 5, 6]], [[1, 2, 3], [1, 2, 3, 4], [3, 5, 6]], [[0, 1, 2, 3], [0, 1, 2, 3, 4], [0, 3, 5, 6]]];keep both omit/include choices,then deduplicate EQUAL UNIONS;new canonical union family=[[], [0], [0, 1], [0, 1, 2], [0, 1, 2, 3], [0, 1, 2, 3, 4], [0, 1, 3], [0, 1, 3, 4], [0, 2], [0, 2, 3], [0, 2, 3, 4], [0, 3], [0, 3, 4], [1], [1, 2], [1, 2, 3], [1, 2, 3, 4], [1, 3], [1, 3, 4], [2], [2, 3], [2, 3, 4], [3], [3, 4]]",
      "relation": "implies",
      "rule_id": "topbasis_complete_prefix_union_DP_not_sampled_subfamilies",
      "conditions": [],
      "explanation": {
        "zh-Hant": "任意新prefix子族不是不選新basis元素，就是選它並與前prefix的某union合併，因此兩類選擇完整且每一狀態有真index見證。只合併相同union，不去掉原basis位置；狀態最多2^n，不需假裝抽樣了2^basisCount個族。",
        "en": "Any subfamily of the new prefix either omits the new element or includes it with a union from the previous prefix.These choices are complete and every state has an actual index witness.Deduplicate equal unions,not original basis positions.There are at most2^n distinct states,not a sample of the2^basisCount subfamilies."
      },
      "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": "S077",
      "uses": [
        "G001",
        "S076"
      ],
      "before": "next ORIGINAL basis index=7;basis set=[4];ALL previous-prefix unions in numeric POINT-mask order=[[], [0], [1], [0, 1], [2], [0, 2], [1, 2], [0, 1, 2], [3], [0, 3], [1, 3], [0, 1, 3], [2, 3], [0, 2, 3], [1, 2, 3], [0, 1, 2, 3], [3, 4], [0, 3, 4], [1, 3, 4], [0, 1, 3, 4], [2, 3, 4], [0, 2, 3, 4], [1, 2, 3, 4], [0, 1, 2, 3, 4]]",
      "after": "ALL [previous_union,union_with_new_basis,index_witness] rows=[[[], [4], [7]], [[0], [0, 4], [0, 7]], [[1], [1, 4], [2, 7]], [[0, 1], [0, 1, 4], [1, 7]], [[2], [2, 4], [4, 7]], [[0, 2], [0, 2, 4], [0, 4, 7]], [[1, 2], [1, 2, 4], [3, 7]], [[0, 1, 2], [0, 1, 2, 4], [0, 3, 7]], [[3], [3, 4], [5, 7]], [[0, 3], [0, 3, 4], [0, 5, 7]], [[1, 3], [1, 3, 4], [2, 5, 7]], [[0, 1, 3], [0, 1, 3, 4], [1, 5, 7]], [[2, 3], [2, 3, 4], [4, 5, 7]], [[0, 2, 3], [0, 2, 3, 4], [0, 4, 5, 7]], [[1, 2, 3], [1, 2, 3, 4], [3, 5, 7]], [[0, 1, 2, 3], [0, 1, 2, 3, 4], [0, 3, 5, 7]], [[3, 4], [3, 4], [6, 7]], [[0, 3, 4], [0, 3, 4], [0, 6, 7]], [[1, 3, 4], [1, 3, 4], [2, 6, 7]], [[0, 1, 3, 4], [0, 1, 3, 4], [1, 6, 7]], [[2, 3, 4], [2, 3, 4], [4, 6, 7]], [[0, 2, 3, 4], [0, 2, 3, 4], [0, 4, 6, 7]], [[1, 2, 3, 4], [1, 2, 3, 4], [3, 6, 7]], [[0, 1, 2, 3, 4], [0, 1, 2, 3, 4], [0, 3, 6, 7]]];keep both omit/include choices,then deduplicate EQUAL UNIONS;new canonical union family=[[], [0], [0, 1], [0, 1, 2], [0, 1, 2, 3], [0, 1, 2, 3, 4], [0, 1, 2, 4], [0, 1, 3], [0, 1, 3, 4], [0, 1, 4], [0, 2], [0, 2, 3], [0, 2, 3, 4], [0, 2, 4], [0, 3], [0, 3, 4], [0, 4], [1], [1, 2], [1, 2, 3], [1, 2, 3, 4], [1, 2, 4], [1, 3], [1, 3, 4], [1, 4], [2], [2, 3], [2, 3, 4], [2, 4], [3], [3, 4], [4]]",
      "relation": "implies",
      "rule_id": "topbasis_complete_prefix_union_DP_not_sampled_subfamilies",
      "conditions": [],
      "explanation": {
        "zh-Hant": "任意新prefix子族不是不選新basis元素，就是選它並與前prefix的某union合併，因此兩類選擇完整且每一狀態有真index見證。只合併相同union，不去掉原basis位置；狀態最多2^n，不需假裝抽樣了2^basisCount個族。",
        "en": "Any subfamily of the new prefix either omits the new element or includes it with a union from the previous prefix.These choices are complete and every state has an actual index witness.Deduplicate equal unions,not original basis positions.There are at most2^n distinct states,not a sample of the2^basisCount subfamilies."
      },
      "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": "S078",
      "uses": [
        "G001",
        "S077"
      ],
      "before": "ALL original basis positions processed;distinct union states=32<=2^5;canonical ALL-union family=[[], [0], [0, 1], [0, 1, 2], [0, 1, 2, 3], [0, 1, 2, 3, 4], [0, 1, 2, 4], [0, 1, 3], [0, 1, 3, 4], [0, 1, 4], [0, 2], [0, 2, 3], [0, 2, 3, 4], [0, 2, 4], [0, 3], [0, 3, 4], [0, 4], [1], [1, 2], [1, 2, 3], [1, 2, 3, 4], [1, 2, 4], [1, 3], [1, 3, 4], [1, 4], [2], [2, 3], [2, 3, 4], [2, 4], [3], [3, 4], [4]]",
      "after": "prefix induction covers EVERY original-index subfamily and every state is its actual union;mathematical all-union family COMPLETE including empty union",
      "relation": "implies",
      "rule_id": "topbasis_all_subfamily_unions_complete_not_exponential_enumeration_claim",
      "conditions": [],
      "explanation": {
        "zh-Hant": "以每步的omit/include歸納證所有子族的union都在狀態中，反向每個狀態有實際子族見證，所以雙向完備。沒有把較少狀態數說成逐次枚舉的subfamily次數。",
        "en": "Omit/include induction puts every subfamily union into the states,and every state has an actual subfamily witness,proving both directions.Do not misreport the smaller distinct-state count as the number of subfamilies explicitly enumerated."
      },
      "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": "S079",
      "uses": [
        "G001",
        "S078"
      ],
      "before": "JUST-VERIFIED cover and ALL local refinement for ORIGINAL basis=[[0], [0, 1], [1], [1, 2], [2], [3], [3, 4], [4]];COMPLETE all-union family=[[], [0], [0, 1], [0, 1, 2], [0, 1, 2, 3], [0, 1, 2, 3, 4], [0, 1, 2, 4], [0, 1, 3], [0, 1, 3, 4], [0, 1, 4], [0, 2], [0, 2, 3], [0, 2, 3, 4], [0, 2, 4], [0, 3], [0, 3, 4], [0, 4], [1], [1, 2], [1, 2, 3], [1, 2, 3, 4], [1, 2, 4], [1, 3], [1, 3, 4], [1, 4], [2], [2, 3], [2, 3, 4], [2, 4], [3], [3, 4], [4]]",
      "after": "empty/full carrier present;ANY union of basis-unions flattens to a basis-union;for x in U intersection V choose basis B,C around x then its verified local refiner inside B intersection C,so U intersection V is itself a basis-union;this family IS a topology on Fin(5)",
      "relation": "implies",
      "rule_id": "topbasis_basis_to_topology_arbitrary_union_and_local_intersection_proof",
      "conditions": [],
      "explanation": {
        "zh-Hant": "局部細化讓每個交集點都有落在U∩V中的basis元素，全部這些basis的union恰為U∩V；這才證交閉包，不要求basis族本身交封閉。加上任意union flatten、cover給全集及空union給空集，得到完整topology公理。",
        "en": "Local refinement supplies a basis element around each point inside U∩V,and their union equals that intersection.This proves intersection closure without requiring the basis family itself intersection-closed.Arbitrary-union flattening,coverage of the full carrier and the empty union complete the topology axioms."
      },
      "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
        ],
        [
          0,
          1
        ],
        [
          0,
          1,
          2
        ],
        [
          0,
          1,
          2,
          3
        ],
        [
          0,
          1,
          2,
          3,
          4
        ],
        [
          0,
          1,
          2,
          4
        ],
        [
          0,
          1,
          3
        ],
        [
          0,
          1,
          3,
          4
        ],
        [
          0,
          1,
          4
        ],
        [
          0,
          2
        ],
        [
          0,
          2,
          3
        ],
        [
          0,
          2,
          3,
          4
        ],
        [
          0,
          2,
          4
        ],
        [
          0,
          3
        ],
        [
          0,
          3,
          4
        ],
        [
          0,
          4
        ],
        [
          1
        ],
        [
          1,
          2
        ],
        [
          1,
          2,
          3
        ],
        [
          1,
          2,
          3,
          4
        ],
        [
          1,
          2,
          4
        ],
        [
          1,
          3
        ],
        [
          1,
          3,
          4
        ],
        [
          1,
          4
        ],
        [
          2
        ],
        [
          2,
          3
        ],
        [
          2,
          3,
          4
        ],
        [
          2,
          4
        ],
        [
          3
        ],
        [
          3,
          4
        ],
        [
          4
        ]
      ],
      32,
      true,
      "all_unions_of_basis_elements"
    ]
  },
  "conclusion": {
    "canonical_value": "([[], [0], [0, 1], [0, 1, 2], [0, 1, 2, 3], [0, 1, 2, 3, 4], [0, 1, 2, 4], [0, 1, 3], [0, 1, 3, 4], [0, 1, 4], [0, 2], [0, 2, 3], [0, 2, 3, 4], [0, 2, 4], [0, 3], [0, 3, 4], [0, 4], [1], [1, 2], [1, 2, 3], [1, 2, 3, 4], [1, 2, 4], [1, 3], [1, 3, 4], [1, 4], [2], [2, 3], [2, 3, 4], [2, 4], [3], [3, 4], [4]], 32, 是, all_unions_of_basis_elements)",
    "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-20261107-r03-topological-basis-foundations",
      "batch_path": "public/data/questions/batches/batch-20261014-advance-20261107-r03-topological-basis-foundations/questions.jsonl",
      "batch_sha256": "4a8c90498a6f9221eaa3d997bdd3089e61bef060c07eb741f8c657a093ffecd0",
      "original_line": 47,
      "original_line_sha256": "747961e5b31cad29adfe6a6c6f30257c896d809eccfecc2ed81c3b201d5cf39a",
      "question_compact_sha256": "538c459c383aea485a572051040c7e0c854134eb9b7a31b552e2f9e592dfe87a",
      "english_row_compact_sha256": "f640dc0cf9a9a2f7c21afbdb3c1b5efef82c7eaf07ee10e1de4aa283611a5499",
      "english_manifest_path": "public/data/i18n/editions/batch-20261014-advance-20261107-r03-topological-basis-foundations.en.json",
      "english_rows_path": "public/data/i18n/questions/batch-20261014-advance-20261107-r03-topological-basis-foundations.en.jsonl"
    },
    "mko_bindings": {
      "mko-topological-basis": {
        "sha256": "3aecebae96f5302a84e437770eaaec38b7992e61082720fa491302177ad43804",
        "english_sha256": "afbfa91d01de39a5536ec84bb9365435ac7fc714af5ea3f46ff7714bb03d0c78",
        "original_path": "public/data/mko/topological-basis.json",
        "english_path": "public/data/i18n/editions/mko-topological-basis.en.json"
      }
    },
    "selected_manifest_sha256": "5233edba9654a6cffd8b6b2d83c932f82ded15f9c0b7c733bd049943e672acff",
    "selected_part_sha256": "f1ed3239c9185524aa4f0839ab4cfbe8f7d74200b283ea09e850a792f0178a3b",
    "original_candidate_sha256": "d04d5f074b79c03a6d31ca01fae8b8f6275b6042a689aa9dd0ea3d64e0b91519",
    "original_candidate_line_sha256": "d04d5f074b79c03a6d31ca01fae8b8f6275b6042a689aa9dd0ea3d64e0b91519"
  },
  "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-20261026-008-p41",
    "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": "c6483ed1229af9183fe5c704a2627b1558ad9a31bfbb0d2ee2d5c1a1c299ab4d",
      "question_first_witness_sha256": "e81f9a560d630fcb23441146f008c4feca7aceab3ebe4de2cf1997c061375ef7"
    },
    "status": "second_context_reviewed",
    "mathematical_semantics": true,
    "bilingual_semantics": true,
    "steps_reviewed": 79,
    "method": "family_template_semantic_review_plus_independent_per_instance_reconstruction",
    "exact_question_first_check": {
      "model_sha256": "99c20013a071d9433472aa65256eaf8c2dce610d73529e97cca547befcbec81f",
      "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."
  }
}