第 20672 題：拓撲基底

本頁導覽

原題與已知條件
定義與適用範圍
逐步推導
逐步反向核對
結論
本題覆核與來源綁定

R0040-20672 · q-20261014-advance-20261107-r03-topological-basis-0047

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],[3],[3,4],[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],[3],[3,4],[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

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], [3], [3, 4], [4]];carrier Fin(5)

操作後

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

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], [3], [3, 4], [4]];ALL carrier points0..4

操作後

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

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], [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, 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, 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], [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, 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, 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], [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, 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, 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], [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, S032

S034 · implies

操作前

ORIGINAL ordered basis positions(left,right)=[3, 6];actual basis elements=[[1, 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, 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], [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, S040

S042 · implies

操作前

ORIGINAL ordered basis positions(left,right)=[4, 6];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, 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=[[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=[[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=[[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=[[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, S046

S048 · implies

操作前

ORIGINAL ordered basis positions(left,right)=[5, 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, S047

S049 · implies

操作前

ORIGINAL ordered basis positions(left,right)=[5, 5];actual basis elements=[[3], [3]];FULL intersection=[3]

操作後

ALL [intersection_point,ALL refining original basis indices] rows=[[3, [5]]];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=[[3], [3, 4]];FULL intersection=[3]

操作後

ALL [intersection_point,ALL refining original basis indices] rows=[[3, [5]]];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=[[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, 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, S051

S053 · implies

操作前

ORIGINAL ordered basis positions(left,right)=[6, 1];actual basis elements=[[3, 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, S052

S054 · implies

操作前

ORIGINAL ordered basis positions(left,right)=[6, 2];actual basis elements=[[3, 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, S053

S055 · implies

操作前

ORIGINAL ordered basis positions(left,right)=[6, 3];actual basis elements=[[3, 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, S054

S056 · implies

操作前

ORIGINAL ordered basis positions(left,right)=[6, 4];actual basis elements=[[3, 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, S055

S057 · implies

操作前

ORIGINAL ordered basis positions(left,right)=[6, 5];actual basis elements=[[3, 4], [3]];FULL intersection=[3]

操作後

ALL [intersection_point,ALL refining original basis indices] rows=[[3, [5]]];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, 4], [3, 4]];FULL intersection=[3, 4]

操作後

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

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], [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, 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], [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, 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, 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], [3], [3, 4], [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

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

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

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

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=[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=[[[], [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]]

條件：無

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, 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=[[[], [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]]

條件：無

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], [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]]

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], [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]]

條件：無

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], [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]]

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: 538c459c383aea485a572051040c7e0c854134eb9b7a31b552e2f9e592dfe87a
candidate record SHA-256: d04d5f074b79c03a6d31ca01fae8b8f6275b6042a689aa9dd0ea3d64e0b91519
semantic review receipt SHA-256: c6483ed1229af9183fe5c704a2627b1558ad9a31bfbb0d2ee2d5c1a1c299ab4d
question-first witness SHA-256: e81f9a560d630fcb23441146f008c4feca7aceab3ebe4de2cf1997c061375ef7
Lean: NOT_RUN · human review: NOT_PERFORMED

同源完整 JSON