# 第 20664 題：拓撲基底 [返回所在分頁 104](https://ocme.evemisslab.com/zh-Hant/questions/page/104/) [← 逐題詳解目錄](https://ocme.evemisslab.com/zh-Hant/questions/) · [原題批次](https://ocme.evemisslab.com/zh-Hant/practice/batch-20261014-advance-20261107-r03-topological-basis-foundations/#q-20261014-advance-20261107-r03-topological-basis-0039) 本頁是已完成第二對話 AI 數理與中英語義覆核的教學詳解，只支持本題明示的模型與前提。沒有人工審查或新 Lean 證明，也不提升原題或 MKO 的 Canonical 真值狀態。 R0040-20664 · q-20261014-advance-20261107-r03-topological-basis-0039 ## 原題與已知條件 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],[2,3],[3],[4]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。 - G001 — 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],[2,3],[3],[4]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。 ## 定義與適用範圍 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。 - [mko-topological-basis](https://ocme.evemisslab.com/zh-Hant/mko/mko-topological-basis/) ## 逐步推導 S001 · implies操作前 source operation=topbasis-generated-topology;original snapshot and mathematics retained 操作後 explicit ordered source output fields=[topology, open_count, topology_certificate, certificate] 條件：無 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。 步驟識別與依賴規則：source_ordered_interface_explicit_without_javascript · 依賴：G001 S002 · implies操作前 original basis candidate:ORIGINAL complete candidate family=[[0], [0, 1], [1], [1, 2], [2], [2, 3], [3], [4]];carrier Fin(5) 操作後 within-set numeric normalization=[[0], [0, 1], [1], [1, 2], [2], [2, 3], [3], [4]];no duplicate points or duplicate FAMILY elements;WHOLE original family already in required key order;empty family allowed as candidate 條件：無 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。 步驟識別與依賴規則：topbasis_canonical_candidate_family_not_semantic_reordering · 依賴：G001, S001 S003 · implies操作前 COMPLETE original candidate basis=[[0], [0, 1], [1], [1, 2], [2], [2, 3], [3], [4]];ALL carrier points0..4 操作後 ALL [point,containing_basis_indices] rows=[[0, [0, 1]], [1, [1, 2, 3]], [2, [3, 4, 5]], [3, [5, 6]], [4, [7]]];covers WHOLE carrier=是 條件：無 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。 步驟識別與依賴規則：topbasis_every_point_cover_separate_from_refinement · 依賴：G001, S002 S004 · implies操作前 ORIGINAL ordered basis positions(left,right)=[0, 0];actual basis elements=[[0], [0]];FULL intersection=[0] 操作後 ALL [intersection_point,ALL refining original basis indices] rows=[[0, [0]]];empty intersection gives VACUOUS local obligation;first failure retained in source order 條件：無 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。 步驟識別與依賴規則：topbasis_full_local_point_refinement_not_intersection_membership · 依賴：G001, S003 S005 · implies操作前 ORIGINAL ordered basis positions(left,right)=[0, 1];actual basis elements=[[0], [0, 1]];FULL intersection=[0] 操作後 ALL [intersection_point,ALL refining original basis indices] rows=[[0, [0]]];empty intersection gives VACUOUS local obligation;first failure retained in source order 條件：無 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。 步驟識別與依賴規則：topbasis_full_local_point_refinement_not_intersection_membership · 依賴：G001, S004 S006 · implies操作前 ORIGINAL ordered basis positions(left,right)=[0, 2];actual basis elements=[[0], [1]];FULL intersection=[] 操作後 ALL [intersection_point,ALL refining original basis indices] rows=[];empty intersection gives VACUOUS local obligation;first failure retained in source order 條件：無 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。 步驟識別與依賴規則：topbasis_full_local_point_refinement_not_intersection_membership · 依賴：G001, S005 S007 · implies操作前 ORIGINAL ordered basis positions(left,right)=[0, 3];actual basis elements=[[0], [1, 2]];FULL intersection=[] 操作後 ALL [intersection_point,ALL refining original basis indices] rows=[];empty intersection gives VACUOUS local obligation;first failure retained in source order 條件：無 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。 步驟識別與依賴規則：topbasis_full_local_point_refinement_not_intersection_membership · 依賴：G001, S006 S008 · implies操作前 ORIGINAL ordered basis positions(left,right)=[0, 4];actual basis elements=[[0], [2]];FULL intersection=[] 操作後 ALL [intersection_point,ALL refining original basis indices] rows=[];empty intersection gives VACUOUS local obligation;first failure retained in source order 條件：無 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。 步驟識別與依賴規則：topbasis_full_local_point_refinement_not_intersection_membership · 依賴：G001, S007 S009 · implies操作前 ORIGINAL ordered basis positions(left,right)=[0, 5];actual basis elements=[[0], [2, 3]];FULL intersection=[] 操作後 ALL [intersection_point,ALL refining original basis indices] rows=[];empty intersection gives VACUOUS local obligation;first failure retained in source order 條件：無 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。 步驟識別與依賴規則：topbasis_full_local_point_refinement_not_intersection_membership · 依賴：G001, S008 S010 · implies操作前 ORIGINAL ordered basis positions(left,right)=[0, 6];actual basis elements=[[0], [3]];FULL intersection=[] 操作後 ALL [intersection_point,ALL refining original basis indices] rows=[];empty intersection gives VACUOUS local obligation;first failure retained in source order 條件：無 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。 步驟識別與依賴規則：topbasis_full_local_point_refinement_not_intersection_membership · 依賴：G001, S009 S011 · implies操作前 ORIGINAL ordered basis positions(left,right)=[0, 7];actual basis elements=[[0], [4]];FULL intersection=[] 操作後 ALL [intersection_point,ALL refining original basis indices] rows=[];empty intersection gives VACUOUS local obligation;first failure retained in source order 條件：無 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。 步驟識別與依賴規則：topbasis_full_local_point_refinement_not_intersection_membership · 依賴：G001, S010 S012 · implies操作前 ORIGINAL ordered basis positions(left,right)=[1, 0];actual basis elements=[[0, 1], [0]];FULL intersection=[0] 操作後 ALL [intersection_point,ALL refining original basis indices] rows=[[0, [0]]];empty intersection gives VACUOUS local obligation;first failure retained in source order 條件：無 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。 步驟識別與依賴規則：topbasis_full_local_point_refinement_not_intersection_membership · 依賴：G001, S011 S013 · implies操作前 ORIGINAL ordered basis positions(left,right)=[1, 1];actual basis elements=[[0, 1], [0, 1]];FULL intersection=[0, 1] 操作後 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 條件：無 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。 步驟識別與依賴規則：topbasis_full_local_point_refinement_not_intersection_membership · 依賴：G001, S012 S014 · implies操作前 ORIGINAL ordered basis positions(left,right)=[1, 2];actual basis elements=[[0, 1], [1]];FULL intersection=[1] 操作後 ALL [intersection_point,ALL refining original basis indices] rows=[[1, [2]]];empty intersection gives VACUOUS local obligation;first failure retained in source order 條件：無 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。 步驟識別與依賴規則：topbasis_full_local_point_refinement_not_intersection_membership · 依賴：G001, S013 S015 · implies操作前 ORIGINAL ordered basis positions(left,right)=[1, 3];actual basis elements=[[0, 1], [1, 2]];FULL intersection=[1] 操作後 ALL [intersection_point,ALL refining original basis indices] rows=[[1, [2]]];empty intersection gives VACUOUS local obligation;first failure retained in source order 條件：無 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。 步驟識別與依賴規則：topbasis_full_local_point_refinement_not_intersection_membership · 依賴：G001, S014 S016 · implies操作前 ORIGINAL ordered basis positions(left,right)=[1, 4];actual basis elements=[[0, 1], [2]];FULL intersection=[] 操作後 ALL [intersection_point,ALL refining original basis indices] rows=[];empty intersection gives VACUOUS local obligation;first failure retained in source order 條件：無 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。 步驟識別與依賴規則：topbasis_full_local_point_refinement_not_intersection_membership · 依賴：G001, S015 S017 · implies操作前 ORIGINAL ordered basis positions(left,right)=[1, 5];actual basis elements=[[0, 1], [2, 3]];FULL intersection=[] 操作後 ALL [intersection_point,ALL refining original basis indices] rows=[];empty intersection gives VACUOUS local obligation;first failure retained in source order 條件：無 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。 步驟識別與依賴規則：topbasis_full_local_point_refinement_not_intersection_membership · 依賴：G001, S016 S018 · implies操作前 ORIGINAL ordered basis positions(left,right)=[1, 6];actual basis elements=[[0, 1], [3]];FULL intersection=[] 操作後 ALL [intersection_point,ALL refining original basis indices] rows=[];empty intersection gives VACUOUS local obligation;first failure retained in source order 條件：無 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。 步驟識別與依賴規則：topbasis_full_local_point_refinement_not_intersection_membership · 依賴：G001, S017 S019 · implies操作前 ORIGINAL ordered basis positions(left,right)=[1, 7];actual basis elements=[[0, 1], [4]];FULL intersection=[] 操作後 ALL [intersection_point,ALL refining original basis indices] rows=[];empty intersection gives VACUOUS local obligation;first failure retained in source order 條件：無 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。 步驟識別與依賴規則：topbasis_full_local_point_refinement_not_intersection_membership · 依賴：G001, S018 S020 · implies操作前 ORIGINAL ordered basis positions(left,right)=[2, 0];actual basis elements=[[1], [0]];FULL intersection=[] 操作後 ALL [intersection_point,ALL refining original basis indices] rows=[];empty intersection gives VACUOUS local obligation;first failure retained in source order 條件：無 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。 步驟識別與依賴規則：topbasis_full_local_point_refinement_not_intersection_membership · 依賴：G001, S019 S021 · implies操作前 ORIGINAL ordered basis positions(left,right)=[2, 1];actual basis elements=[[1], [0, 1]];FULL intersection=[1] 操作後 ALL [intersection_point,ALL refining original basis indices] rows=[[1, [2]]];empty intersection gives VACUOUS local obligation;first failure retained in source order 條件：無 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。 步驟識別與依賴規則：topbasis_full_local_point_refinement_not_intersection_membership · 依賴：G001, S020 S022 · implies操作前 ORIGINAL ordered basis positions(left,right)=[2, 2];actual basis elements=[[1], [1]];FULL intersection=[1] 操作後 ALL [intersection_point,ALL refining original basis indices] rows=[[1, [2]]];empty intersection gives VACUOUS local obligation;first failure retained in source order 條件：無 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。 步驟識別與依賴規則：topbasis_full_local_point_refinement_not_intersection_membership · 依賴：G001, S021 S023 · implies操作前 ORIGINAL ordered basis positions(left,right)=[2, 3];actual basis elements=[[1], [1, 2]];FULL intersection=[1] 操作後 ALL [intersection_point,ALL refining original basis indices] rows=[[1, [2]]];empty intersection gives VACUOUS local obligation;first failure retained in source order 條件：無 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。 步驟識別與依賴規則：topbasis_full_local_point_refinement_not_intersection_membership · 依賴：G001, S022 S024 · implies操作前 ORIGINAL ordered basis positions(left,right)=[2, 4];actual basis elements=[[1], [2]];FULL intersection=[] 操作後 ALL [intersection_point,ALL refining original basis indices] rows=[];empty intersection gives VACUOUS local obligation;first failure retained in source order 條件：無 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。 步驟識別與依賴規則：topbasis_full_local_point_refinement_not_intersection_membership · 依賴：G001, S023 S025 · implies操作前 ORIGINAL ordered basis positions(left,right)=[2, 5];actual basis elements=[[1], [2, 3]];FULL intersection=[] 操作後 ALL [intersection_point,ALL refining original basis indices] rows=[];empty intersection gives VACUOUS local obligation;first failure retained in source order 條件：無 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。 步驟識別與依賴規則：topbasis_full_local_point_refinement_not_intersection_membership · 依賴：G001, S024 S026 · implies操作前 ORIGINAL ordered basis positions(left,right)=[2, 6];actual basis elements=[[1], [3]];FULL intersection=[] 操作後 ALL [intersection_point,ALL refining original basis indices] rows=[];empty intersection gives VACUOUS local obligation;first failure retained in source order 條件：無 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。 步驟識別與依賴規則：topbasis_full_local_point_refinement_not_intersection_membership · 依賴：G001, S025 S027 · implies操作前 ORIGINAL ordered basis positions(left,right)=[2, 7];actual basis elements=[[1], [4]];FULL intersection=[] 操作後 ALL [intersection_point,ALL refining original basis indices] rows=[];empty intersection gives VACUOUS local obligation;first failure retained in source order 條件：無 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。 步驟識別與依賴規則：topbasis_full_local_point_refinement_not_intersection_membership · 依賴：G001, S026 S028 · implies操作前 ORIGINAL ordered basis positions(left,right)=[3, 0];actual basis elements=[[1, 2], [0]];FULL intersection=[] 操作後 ALL [intersection_point,ALL refining original basis indices] rows=[];empty intersection gives VACUOUS local obligation;first failure retained in source order 條件：無 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。 步驟識別與依賴規則：topbasis_full_local_point_refinement_not_intersection_membership · 依賴：G001, S027 S029 · implies操作前 ORIGINAL ordered basis positions(left,right)=[3, 1];actual basis elements=[[1, 2], [0, 1]];FULL intersection=[1] 操作後 ALL [intersection_point,ALL refining original basis indices] rows=[[1, [2]]];empty intersection gives VACUOUS local obligation;first failure retained in source order 條件：無 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。 步驟識別與依賴規則：topbasis_full_local_point_refinement_not_intersection_membership · 依賴：G001, S028 S030 · implies操作前 ORIGINAL ordered basis positions(left,right)=[3, 2];actual basis elements=[[1, 2], [1]];FULL intersection=[1] 操作後 ALL [intersection_point,ALL refining original basis indices] rows=[[1, [2]]];empty intersection gives VACUOUS local obligation;first failure retained in source order 條件：無 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。 步驟識別與依賴規則：topbasis_full_local_point_refinement_not_intersection_membership · 依賴：G001, S029 S031 · implies操作前 ORIGINAL ordered basis positions(left,right)=[3, 3];actual basis elements=[[1, 2], [1, 2]];FULL intersection=[1, 2] 操作後 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 條件：無 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。 步驟識別與依賴規則：topbasis_full_local_point_refinement_not_intersection_membership · 依賴：G001, S030 S032 · implies操作前 ORIGINAL ordered basis positions(left,right)=[3, 4];actual basis elements=[[1, 2], [2]];FULL intersection=[2] 操作後 ALL [intersection_point,ALL refining original basis indices] rows=[[2, [4]]];empty intersection gives VACUOUS local obligation;first failure retained in source order 條件：無 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。 步驟識別與依賴規則：topbasis_full_local_point_refinement_not_intersection_membership · 依賴：G001, S031 S033 · implies操作前 ORIGINAL ordered basis positions(left,right)=[3, 5];actual basis elements=[[1, 2], [2, 3]];FULL intersection=[2] 操作後 ALL [intersection_point,ALL refining original basis indices] rows=[[2, [4]]];empty intersection gives VACUOUS local obligation;first failure retained in source order 條件：無 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。 步驟識別與依賴規則：topbasis_full_local_point_refinement_not_intersection_membership · 依賴：G001, S032 S034 · implies操作前 ORIGINAL ordered basis positions(left,right)=[3, 6];actual basis elements=[[1, 2], [3]];FULL intersection=[] 操作後 ALL [intersection_point,ALL refining original basis indices] rows=[];empty intersection gives VACUOUS local obligation;first failure retained in source order 條件：無 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。 步驟識別與依賴規則：topbasis_full_local_point_refinement_not_intersection_membership · 依賴：G001, S033 S035 · implies操作前 ORIGINAL ordered basis positions(left,right)=[3, 7];actual basis elements=[[1, 2], [4]];FULL intersection=[] 操作後 ALL [intersection_point,ALL refining original basis indices] rows=[];empty intersection gives VACUOUS local obligation;first failure retained in source order 條件：無 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。 步驟識別與依賴規則：topbasis_full_local_point_refinement_not_intersection_membership · 依賴：G001, S034 S036 · implies操作前 ORIGINAL ordered basis positions(left,right)=[4, 0];actual basis elements=[[2], [0]];FULL intersection=[] 操作後 ALL [intersection_point,ALL refining original basis indices] rows=[];empty intersection gives VACUOUS local obligation;first failure retained in source order 條件：無 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。 步驟識別與依賴規則：topbasis_full_local_point_refinement_not_intersection_membership · 依賴：G001, S035 S037 · implies操作前 ORIGINAL ordered basis positions(left,right)=[4, 1];actual basis elements=[[2], [0, 1]];FULL intersection=[] 操作後 ALL [intersection_point,ALL refining original basis indices] rows=[];empty intersection gives VACUOUS local obligation;first failure retained in source order 條件：無 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。 步驟識別與依賴規則：topbasis_full_local_point_refinement_not_intersection_membership · 依賴：G001, S036 S038 · implies操作前 ORIGINAL ordered basis positions(left,right)=[4, 2];actual basis elements=[[2], [1]];FULL intersection=[] 操作後 ALL [intersection_point,ALL refining original basis indices] rows=[];empty intersection gives VACUOUS local obligation;first failure retained in source order 條件：無 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。 步驟識別與依賴規則：topbasis_full_local_point_refinement_not_intersection_membership · 依賴：G001, S037 S039 · implies操作前 ORIGINAL ordered basis positions(left,right)=[4, 3];actual basis elements=[[2], [1, 2]];FULL intersection=[2] 操作後 ALL [intersection_point,ALL refining original basis indices] rows=[[2, [4]]];empty intersection gives VACUOUS local obligation;first failure retained in source order 條件：無 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。 步驟識別與依賴規則：topbasis_full_local_point_refinement_not_intersection_membership · 依賴：G001, S038 S040 · implies操作前 ORIGINAL ordered basis positions(left,right)=[4, 4];actual basis elements=[[2], [2]];FULL intersection=[2] 操作後 ALL [intersection_point,ALL refining original basis indices] rows=[[2, [4]]];empty intersection gives VACUOUS local obligation;first failure retained in source order 條件：無 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。 步驟識別與依賴規則：topbasis_full_local_point_refinement_not_intersection_membership · 依賴：G001, S039 S041 · implies操作前 ORIGINAL ordered basis positions(left,right)=[4, 5];actual basis elements=[[2], [2, 3]];FULL intersection=[2] 操作後 ALL [intersection_point,ALL refining original basis indices] rows=[[2, [4]]];empty intersection gives VACUOUS local obligation;first failure retained in source order 條件：無 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。 步驟識別與依賴規則：topbasis_full_local_point_refinement_not_intersection_membership · 依賴：G001, S040 S042 · implies操作前 ORIGINAL ordered basis positions(left,right)=[4, 6];actual basis elements=[[2], [3]];FULL intersection=[] 操作後 ALL [intersection_point,ALL refining original basis indices] rows=[];empty intersection gives VACUOUS local obligation;first failure retained in source order 條件：無 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。 步驟識別與依賴規則：topbasis_full_local_point_refinement_not_intersection_membership · 依賴：G001, S041 S043 · implies操作前 ORIGINAL ordered basis positions(left,right)=[4, 7];actual basis elements=[[2], [4]];FULL intersection=[] 操作後 ALL [intersection_point,ALL refining original basis indices] rows=[];empty intersection gives VACUOUS local obligation;first failure retained in source order 條件：無 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。 步驟識別與依賴規則：topbasis_full_local_point_refinement_not_intersection_membership · 依賴：G001, S042 S044 · implies操作前 ORIGINAL ordered basis positions(left,right)=[5, 0];actual basis elements=[[2, 3], [0]];FULL intersection=[] 操作後 ALL [intersection_point,ALL refining original basis indices] rows=[];empty intersection gives VACUOUS local obligation;first failure retained in source order 條件：無 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。 步驟識別與依賴規則：topbasis_full_local_point_refinement_not_intersection_membership · 依賴：G001, S043 S045 · implies操作前 ORIGINAL ordered basis positions(left,right)=[5, 1];actual basis elements=[[2, 3], [0, 1]];FULL intersection=[] 操作後 ALL [intersection_point,ALL refining original basis indices] rows=[];empty intersection gives VACUOUS local obligation;first failure retained in source order 條件：無 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。 步驟識別與依賴規則：topbasis_full_local_point_refinement_not_intersection_membership · 依賴：G001, S044 S046 · implies操作前 ORIGINAL ordered basis positions(left,right)=[5, 2];actual basis elements=[[2, 3], [1]];FULL intersection=[] 操作後 ALL [intersection_point,ALL refining original basis indices] rows=[];empty intersection gives VACUOUS local obligation;first failure retained in source order 條件：無 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。 步驟識別與依賴規則：topbasis_full_local_point_refinement_not_intersection_membership · 依賴：G001, S045 S047 · implies操作前 ORIGINAL ordered basis positions(left,right)=[5, 3];actual basis elements=[[2, 3], [1, 2]];FULL intersection=[2] 操作後 ALL [intersection_point,ALL refining original basis indices] rows=[[2, [4]]];empty intersection gives VACUOUS local obligation;first failure retained in source order 條件：無 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。 步驟識別與依賴規則：topbasis_full_local_point_refinement_not_intersection_membership · 依賴：G001, S046 S048 · implies操作前 ORIGINAL ordered basis positions(left,right)=[5, 4];actual basis elements=[[2, 3], [2]];FULL intersection=[2] 操作後 ALL [intersection_point,ALL refining original basis indices] rows=[[2, [4]]];empty intersection gives VACUOUS local obligation;first failure retained in source order 條件：無 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。 步驟識別與依賴規則：topbasis_full_local_point_refinement_not_intersection_membership · 依賴：G001, S047 S049 · implies操作前 ORIGINAL ordered basis positions(left,right)=[5, 5];actual basis elements=[[2, 3], [2, 3]];FULL intersection=[2, 3] 操作後 ALL [intersection_point,ALL refining original basis indices] rows=[[2, [4, 5]], [3, [5, 6]]];empty intersection gives VACUOUS local obligation;first failure retained in source order 條件：無 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。 步驟識別與依賴規則：topbasis_full_local_point_refinement_not_intersection_membership · 依賴：G001, S048 S050 · implies操作前 ORIGINAL ordered basis positions(left,right)=[5, 6];actual basis elements=[[2, 3], [3]];FULL intersection=[3] 操作後 ALL [intersection_point,ALL refining original basis indices] rows=[[3, [6]]];empty intersection gives VACUOUS local obligation;first failure retained in source order 條件：無 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。 步驟識別與依賴規則：topbasis_full_local_point_refinement_not_intersection_membership · 依賴：G001, S049 S051 · implies操作前 ORIGINAL ordered basis positions(left,right)=[5, 7];actual basis elements=[[2, 3], [4]];FULL intersection=[] 操作後 ALL [intersection_point,ALL refining original basis indices] rows=[];empty intersection gives VACUOUS local obligation;first failure retained in source order 條件：無 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。 步驟識別與依賴規則：topbasis_full_local_point_refinement_not_intersection_membership · 依賴：G001, S050 S052 · implies操作前 ORIGINAL ordered basis positions(left,right)=[6, 0];actual basis elements=[[3], [0]];FULL intersection=[] 操作後 ALL [intersection_point,ALL refining original basis indices] rows=[];empty intersection gives VACUOUS local obligation;first failure retained in source order 條件：無 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。 步驟識別與依賴規則：topbasis_full_local_point_refinement_not_intersection_membership · 依賴：G001, S051 S053 · implies操作前 ORIGINAL ordered basis positions(left,right)=[6, 1];actual basis elements=[[3], [0, 1]];FULL intersection=[] 操作後 ALL [intersection_point,ALL refining original basis indices] rows=[];empty intersection gives VACUOUS local obligation;first failure retained in source order 條件：無 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。 步驟識別與依賴規則：topbasis_full_local_point_refinement_not_intersection_membership · 依賴：G001, S052 S054 · implies操作前 ORIGINAL ordered basis positions(left,right)=[6, 2];actual basis elements=[[3], [1]];FULL intersection=[] 操作後 ALL [intersection_point,ALL refining original basis indices] rows=[];empty intersection gives VACUOUS local obligation;first failure retained in source order 條件：無 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。 步驟識別與依賴規則：topbasis_full_local_point_refinement_not_intersection_membership · 依賴：G001, S053 S055 · implies操作前 ORIGINAL ordered basis positions(left,right)=[6, 3];actual basis elements=[[3], [1, 2]];FULL intersection=[] 操作後 ALL [intersection_point,ALL refining original basis indices] rows=[];empty intersection gives VACUOUS local obligation;first failure retained in source order 條件：無 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。 步驟識別與依賴規則：topbasis_full_local_point_refinement_not_intersection_membership · 依賴：G001, S054 S056 · implies操作前 ORIGINAL ordered basis positions(left,right)=[6, 4];actual basis elements=[[3], [2]];FULL intersection=[] 操作後 ALL [intersection_point,ALL refining original basis indices] rows=[];empty intersection gives VACUOUS local obligation;first failure retained in source order 條件：無 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。 步驟識別與依賴規則：topbasis_full_local_point_refinement_not_intersection_membership · 依賴：G001, S055 S057 · implies操作前 ORIGINAL ordered basis positions(left,right)=[6, 5];actual basis elements=[[3], [2, 3]];FULL intersection=[3] 操作後 ALL [intersection_point,ALL refining original basis indices] rows=[[3, [6]]];empty intersection gives VACUOUS local obligation;first failure retained in source order 條件：無 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。 步驟識別與依賴規則：topbasis_full_local_point_refinement_not_intersection_membership · 依賴：G001, S056 S058 · implies操作前 ORIGINAL ordered basis positions(left,right)=[6, 6];actual basis elements=[[3], [3]];FULL intersection=[3] 操作後 ALL [intersection_point,ALL refining original basis indices] rows=[[3, [6]]];empty intersection gives VACUOUS local obligation;first failure retained in source order 條件：無 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。 步驟識別與依賴規則：topbasis_full_local_point_refinement_not_intersection_membership · 依賴：G001, S057 S059 · implies操作前 ORIGINAL ordered basis positions(left,right)=[6, 7];actual basis elements=[[3], [4]];FULL intersection=[] 操作後 ALL [intersection_point,ALL refining original basis indices] rows=[];empty intersection gives VACUOUS local obligation;first failure retained in source order 條件：無 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。 步驟識別與依賴規則：topbasis_full_local_point_refinement_not_intersection_membership · 依賴：G001, S058 S060 · implies操作前 ORIGINAL ordered basis positions(left,right)=[7, 0];actual basis elements=[[4], [0]];FULL intersection=[] 操作後 ALL [intersection_point,ALL refining original basis indices] rows=[];empty intersection gives VACUOUS local obligation;first failure retained in source order 條件：無 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。 步驟識別與依賴規則：topbasis_full_local_point_refinement_not_intersection_membership · 依賴：G001, S059 S061 · implies操作前 ORIGINAL ordered basis positions(left,right)=[7, 1];actual basis elements=[[4], [0, 1]];FULL intersection=[] 操作後 ALL [intersection_point,ALL refining original basis indices] rows=[];empty intersection gives VACUOUS local obligation;first failure retained in source order 條件：無 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。 步驟識別與依賴規則：topbasis_full_local_point_refinement_not_intersection_membership · 依賴：G001, S060 S062 · implies操作前 ORIGINAL ordered basis positions(left,right)=[7, 2];actual basis elements=[[4], [1]];FULL intersection=[] 操作後 ALL [intersection_point,ALL refining original basis indices] rows=[];empty intersection gives VACUOUS local obligation;first failure retained in source order 條件：無 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。 步驟識別與依賴規則：topbasis_full_local_point_refinement_not_intersection_membership · 依賴：G001, S061 S063 · implies操作前 ORIGINAL ordered basis positions(left,right)=[7, 3];actual basis elements=[[4], [1, 2]];FULL intersection=[] 操作後 ALL [intersection_point,ALL refining original basis indices] rows=[];empty intersection gives VACUOUS local obligation;first failure retained in source order 條件：無 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。 步驟識別與依賴規則：topbasis_full_local_point_refinement_not_intersection_membership · 依賴：G001, S062 S064 · implies操作前 ORIGINAL ordered basis positions(left,right)=[7, 4];actual basis elements=[[4], [2]];FULL intersection=[] 操作後 ALL [intersection_point,ALL refining original basis indices] rows=[];empty intersection gives VACUOUS local obligation;first failure retained in source order 條件：無 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。 步驟識別與依賴規則：topbasis_full_local_point_refinement_not_intersection_membership · 依賴：G001, S063 S065 · implies操作前 ORIGINAL ordered basis positions(left,right)=[7, 5];actual basis elements=[[4], [2, 3]];FULL intersection=[] 操作後 ALL [intersection_point,ALL refining original basis indices] rows=[];empty intersection gives VACUOUS local obligation;first failure retained in source order 條件：無 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。 步驟識別與依賴規則：topbasis_full_local_point_refinement_not_intersection_membership · 依賴：G001, S064 S066 · implies操作前 ORIGINAL ordered basis positions(left,right)=[7, 6];actual basis elements=[[4], [3]];FULL intersection=[] 操作後 ALL [intersection_point,ALL refining original basis indices] rows=[];empty intersection gives VACUOUS local obligation;first failure retained in source order 條件：無 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。 步驟識別與依賴規則：topbasis_full_local_point_refinement_not_intersection_membership · 依賴：G001, S065 S067 · implies操作前 ORIGINAL ordered basis positions(left,right)=[7, 7];actual basis elements=[[4], [4]];FULL intersection=[4] 操作後 ALL [intersection_point,ALL refining original basis indices] rows=[[4, [7]]];empty intersection gives VACUOUS local obligation;first failure retained in source order 條件：無 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。 步驟識別與依賴規則：topbasis_full_local_point_refinement_not_intersection_membership · 依賴：G001, S066 S068 · implies操作前 ACTUAL cover=是;ALL local point obligations refined=是;first failure[left,right,point,intersection]=null 操作後 abstract basis validity=是 is cover AND LOCAL refinement;not closure of the family under intersections and not a given-topology certificate yet 條件：無 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。 步驟識別與依賴規則：topbasis_cover_and_local_refinement_conjunction · 依賴：G001, S067 S069 · implies操作前 ORIGINAL complete candidate basis in fixed family-index order=[[0], [0, 1], [1], [1, 2], [2], [2, 3], [3], [4]];carrier Fin(5) 操作後 prefix0 has EXACTLY the empty-subfamily union:empty set,with empty index witness;this mathematical union exists even when basis axioms fail 條件：無 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。 步驟識別與依賴規則：topbasis_empty_union_initial_state_not_missing_empty_set · 依賴：G001, S068 S070 · implies操作前 next ORIGINAL basis index=0;basis set=[0];ALL previous-prefix unions in numeric POINT-mask order=[[]] 操作後 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]] 條件：無 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。 步驟識別與依賴規則：topbasis_complete_prefix_union_DP_not_sampled_subfamilies · 依賴：G001, S069 S071 · implies操作前 next ORIGINAL basis index=1;basis set=[0, 1];ALL previous-prefix unions in numeric POINT-mask order=[[], [0]] 操作後 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]] 條件：無 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。 步驟識別與依賴規則：topbasis_complete_prefix_union_DP_not_sampled_subfamilies · 依賴：G001, S070 S072 · implies操作前 next ORIGINAL basis index=2;basis set=[1];ALL previous-prefix unions in numeric POINT-mask order=[[], [0], [0, 1]] 操作後 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]] 條件：無 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。 步驟識別與依賴規則：topbasis_complete_prefix_union_DP_not_sampled_subfamilies · 依賴：G001, S071 S073 · implies操作前 next ORIGINAL basis index=3;basis set=[1, 2];ALL previous-prefix unions in numeric POINT-mask order=[[], [0], [1], [0, 1]] 操作後 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]] 條件：無 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。 步驟識別與依賴規則：topbasis_complete_prefix_union_DP_not_sampled_subfamilies · 依賴：G001, S072 S074 · implies操作前 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]] 操作後 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]] 條件：無 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。 步驟識別與依賴規則：topbasis_complete_prefix_union_DP_not_sampled_subfamilies · 依賴：G001, S073 S075 · implies操作前 next ORIGINAL basis index=5;basis set=[2, 3];ALL previous-prefix unions in numeric POINT-mask order=[[], [0], [1], [0, 1], [2], [0, 2], [1, 2], [0, 1, 2]] 操作後 ALL [previous_union,union_with_new_basis,index_witness] rows=[[[], [2, 3], [5]], [[0], [0, 2, 3], [0, 5]], [[1], [1, 2, 3], [2, 5]], [[0, 1], [0, 1, 2, 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, 2], [0, 2, 3], [1], [1, 2], [1, 2, 3], [2], [2, 3]] 條件：無 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。 步驟識別與依賴規則：topbasis_complete_prefix_union_DP_not_sampled_subfamilies · 依賴：G001, S074 S076 · implies操作前 next ORIGINAL basis index=6;basis set=[3];ALL previous-prefix unions in numeric POINT-mask order=[[], [0], [1], [0, 1], [2], [0, 2], [1, 2], [0, 1, 2], [2, 3], [0, 2, 3], [1, 2, 3], [0, 1, 2, 3]] 操作後 ALL [previous_union,union_with_new_basis,index_witness] rows=[[[], [3], [6]], [[0], [0, 3], [0, 6]], [[1], [1, 3], [2, 6]], [[0, 1], [0, 1, 3], [1, 6]], [[2], [2, 3], [4, 6]], [[0, 2], [0, 2, 3], [0, 4, 6]], [[1, 2], [1, 2, 3], [3, 6]], [[0, 1, 2], [0, 1, 2, 3], [0, 3, 6]], [[2, 3], [2, 3], [5, 6]], [[0, 2, 3], [0, 2, 3], [0, 5, 6]], [[1, 2, 3], [1, 2, 3], [2, 5, 6]], [[0, 1, 2, 3], [0, 1, 2, 3], [1, 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, 3], [0, 2], [0, 2, 3], [0, 3], [1], [1, 2], [1, 2, 3], [1, 3], [2], [2, 3], [3]] 條件：無 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。 步驟識別與依賴規則：topbasis_complete_prefix_union_DP_not_sampled_subfamilies · 依賴：G001, S075 S077 · implies操作前 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]] 操作後 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], [6, 7]], [[0, 3], [0, 3, 4], [0, 6, 7]], [[1, 3], [1, 3, 4], [2, 6, 7]], [[0, 1, 3], [0, 1, 3, 4], [1, 6, 7]], [[2, 3], [2, 3, 4], [5, 7]], [[0, 2, 3], [0, 2, 3, 4], [0, 5, 7]], [[1, 2, 3], [1, 2, 3, 4], [2, 5, 7]], [[0, 1, 2, 3], [0, 1, 2, 3, 4], [1, 5, 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]] 條件：無 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。 步驟識別與依賴規則：topbasis_complete_prefix_union_DP_not_sampled_subfamilies · 依賴：G001, S076 S078 · implies操作前 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]] 操作後 prefix induction covers EVERY original-index subfamily and every state is its actual union;mathematical all-union family COMPLETE including empty union 條件：無 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。 步驟識別與依賴規則：topbasis_all_subfamily_unions_complete_not_exponential_enumeration_claim · 依賴：G001, S077 S079 · implies操作前 JUST-VERIFIED cover and ALL local refinement for ORIGINAL basis=[[0], [0, 1], [1], [1, 2], [2], [2, 3], [3], [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]] 操作後 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) 條件：無 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。 步驟識別與依賴規則：topbasis_basis_to_topology_arbitrary_union_and_local_intersection_proof · 依賴：G001, S078 ## 逐步反向核對 - S079 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。 - S078 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。 - S077 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。 - S076 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。 - S075 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。 - S074 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。 - S073 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。 - S072 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。 - S071 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。 - S070 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。 - S069 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。 - S068 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。 - S067 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。 - S066 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。 - S065 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。 - S064 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。 - S063 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。 - S062 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。 - S061 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。 - S060 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。 - S059 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。 - S058 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。 - S057 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。 - S056 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。 - S055 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。 - S054 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。 - S053 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。 - S052 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。 - S051 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。 - S050 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。 - S049 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。 - S048 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。 - S047 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。 - S046 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。 - S045 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。 - S044 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。 - S043 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。 - S042 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。 - S041 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。 - S040 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。 - S039 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。 - S038 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。 - S037 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。 - S036 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。 - S035 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。 - S034 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。 - S033 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。 - S032 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。 - S031 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。 - S030 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。 - S029 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。 - S028 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。 - S027 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。 - S026 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。 - S025 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。 - S024 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。 - S023 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。 - S022 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。 - S021 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。 - S020 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。 - S019 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。 - S018 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。 - S017 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。 - S016 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。 - S015 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。 - S014 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。 - S013 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。 - S012 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。 - S011 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。 - S010 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。 - S009 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。 - S008 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。 - S007 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。 - S006 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。 - S005 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。 - S004 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。 - S003 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。 - S002 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。 - S001 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。 ## 結論 ([[], [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。 ## 本題覆核與來源綁定 覆核日期：2026-10-26 原題、答案、步驟表達式、數值 AST 與既有歷史保持原樣；原來的候選狀態屬於來源歷史，不覆寫為形式化證明。 來源雜湊與覆核憑證 - input commit: 0f7be61e2c22317a4075561dc822c87d881a50db - original question SHA-256: 6d897fc3997b6305fb817a16e219f86546d92bcfbbd9c8a96a14e3d0b909c933 - candidate record SHA-256: 8832ddd097f33a9ef8c37541950c1ec023b7cfb532f3077f7bb74946c566fd84 - semantic review receipt SHA-256: c6483ed1229af9183fe5c704a2627b1558ad9a31bfbb0d2ee2d5c1a1c299ab4d - question-first witness SHA-256: e81f9a560d630fcb23441146f008c4feca7aceab3ebe4de2cf1997c061375ef7 - Lean: NOT_RUN · human review: NOT_PERFORMED [同源完整 JSON](https://ocme.evemisslab.com/data/question-derivations/questions/q-20261014-advance-20261107-r03-topological-basis-0039.json)