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Question 20689: A Cover Alone Does Not Supply the Local Basis Axiom

On this page
  1. Original question and givens
  2. Definition and scope
  3. Step-by-step derivation
  4. Reverse checks
  5. Result
  6. This question review and source binding

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R0040-20689 · q-20261014-advance-20261107-r03-topological-basis-0089

Original question and givens

Use finite carrier Fin(5)={0,…,4} and the complete canonical candidate family basis=[[0, 1], [0, 2], [0, 3], [1, 2], [2, 3], [2, 4], [3, 4]]. Its union must cover the carrier; for every point in each pairwise intersection a candidate element containing that point must lie inside the intersection. Complete finite data, not sampled intersections, supply this audit. Give (covers_carrier, local_refinement, is_basis, cover_alone_suffices_general_claim, boundary).

  • G001 — Use finite carrier Fin(5)={0,…,4} and the complete canonical candidate family basis=[[0, 1], [0, 2], [0, 3], [1, 2], [2, 3], [2, 4], [3, 4]]. Its union must cover the carrier; for every point in each pairwise intersection a candidate element containing that point must lie inside the intersection. Complete finite data, not sampled intersections, supply this audit. Give (covers_carrier, local_refinement, is_basis, cover_alone_suffices_general_claim, boundary).

Definition and scope

Use the original question domain, assumptions and cited MKO definitions. This item has second-context mathematical and bilingual review, with the model, answer and every arithmetic step independently recomputed from the original parameters; the source question remains a candidate.

Step-by-step derivation

S001 · implies

Before

source operation=topbasis-subbasis-boundary;original snapshot and mathematics retained

After

explicit ordered source output fields=[covers_carrier, local_refinement, is_basis, cover_alone_suffices_general_claim, boundary]

Conditions: none

Keep the historical interface marker in the original snapshot while listing source fields here without JavaScript or external lookup.Labels are not mathematical evidence,and No retains its source claim scope.

Step identifiers and dependencies

Rule: source_ordered_interface_explicit_without_javascript · Uses: G001

S002 · implies

Before

original basis candidate:ORIGINAL complete candidate family=[[0, 1], [0, 2], [0, 3], [1, 2], [2, 3], [2, 4], [3, 4]];carrier Fin(5)

After

within-set numeric normalization=[[0, 1], [0, 2], [0, 3], [1, 2], [2, 3], [2, 4], [3, 4]];no duplicate points or duplicate FAMILY elements;WHOLE original family already in required key order;empty family allowed as candidate

Conditions: none

A topological basis is a family of subsets,not a vector-space basis;its size may exceed n.Normalize point order within each set without changing original family indices or deleting duplicates to repair data.An empty candidate family can still be audited.

Step identifiers and dependencies

Rule: topbasis_canonical_candidate_family_not_semantic_reordering · Uses: G001, S001

S003 · implies

Before

COMPLETE original candidate basis=[[0, 1], [0, 2], [0, 3], [1, 2], [2, 3], [2, 4], [3, 4]];ALL carrier points0..4

After

ALL [point,containing_basis_indices] rows=[[0, [0, 1, 2]], [1, [0, 3]], [2, [1, 3, 4, 5]], [3, [2, 4, 6]], [4, [5, 6]]];covers WHOLE carrier=Yes

Conditions: none

Every carrier point needs a containing candidate element for coverage.This does not replace local intersection refinement.With n≥1 an empty family fails coverage rather than being discarded as an unauditable candidate.

Step identifiers and dependencies

Rule: topbasis_every_point_cover_separate_from_refinement · Uses: G001, S002

S004 · implies

Before

ORIGINAL ordered basis positions(left,right)=[0, 0];actual basis elements=[[0, 1], [0, 1]];FULL intersection=[0, 1]

After

ALL [intersection_point,ALL refining original basis indices] rows=[[0, [0]], [1, [0]]];empty intersection gives VACUOUS local obligation;first failure retained in source order

Conditions: none

For each intersection point x require some candidate basis element containing x and contained in the intersection.The entire intersection need not itself be a basis element.Check all refiners before declaring absence,and keep the first left/right/point failure in source order.

Step identifiers and dependencies

Rule: topbasis_full_local_point_refinement_not_intersection_membership · Uses: G001, S003

S005 · implies

Before

ORIGINAL ordered basis positions(left,right)=[0, 1];actual basis elements=[[0, 1], [0, 2]];FULL intersection=[0]

After

ALL [intersection_point,ALL refining original basis indices] rows=[[0, []]];empty intersection gives VACUOUS local obligation;first failure retained in source order

Conditions: none

For each intersection point x require some candidate basis element containing x and contained in the intersection.The entire intersection need not itself be a basis element.Check all refiners before declaring absence,and keep the first left/right/point failure in source order.

Step identifiers and dependencies

Rule: topbasis_full_local_point_refinement_not_intersection_membership · Uses: G001, S004

S006 · implies

Before

ORIGINAL ordered basis positions(left,right)=[0, 2];actual basis elements=[[0, 1], [0, 3]];FULL intersection=[0]

After

ALL [intersection_point,ALL refining original basis indices] rows=[[0, []]];empty intersection gives VACUOUS local obligation;first failure retained in source order

Conditions: none

For each intersection point x require some candidate basis element containing x and contained in the intersection.The entire intersection need not itself be a basis element.Check all refiners before declaring absence,and keep the first left/right/point failure in source order.

Step identifiers and dependencies

Rule: topbasis_full_local_point_refinement_not_intersection_membership · Uses: G001, S005

S007 · implies

Before

ORIGINAL ordered basis positions(left,right)=[0, 3];actual basis elements=[[0, 1], [1, 2]];FULL intersection=[1]

After

ALL [intersection_point,ALL refining original basis indices] rows=[[1, []]];empty intersection gives VACUOUS local obligation;first failure retained in source order

Conditions: none

For each intersection point x require some candidate basis element containing x and contained in the intersection.The entire intersection need not itself be a basis element.Check all refiners before declaring absence,and keep the first left/right/point failure in source order.

Step identifiers and dependencies

Rule: topbasis_full_local_point_refinement_not_intersection_membership · Uses: G001, S006

S008 · implies

Before

ORIGINAL ordered basis positions(left,right)=[0, 4];actual basis elements=[[0, 1], [2, 3]];FULL intersection=[]

After

ALL [intersection_point,ALL refining original basis indices] rows=[];empty intersection gives VACUOUS local obligation;first failure retained in source order

Conditions: none

For each intersection point x require some candidate basis element containing x and contained in the intersection.The entire intersection need not itself be a basis element.Check all refiners before declaring absence,and keep the first left/right/point failure in source order.

Step identifiers and dependencies

Rule: topbasis_full_local_point_refinement_not_intersection_membership · Uses: G001, S007

S009 · implies

Before

ORIGINAL ordered basis positions(left,right)=[0, 5];actual basis elements=[[0, 1], [2, 4]];FULL intersection=[]

After

ALL [intersection_point,ALL refining original basis indices] rows=[];empty intersection gives VACUOUS local obligation;first failure retained in source order

Conditions: none

For each intersection point x require some candidate basis element containing x and contained in the intersection.The entire intersection need not itself be a basis element.Check all refiners before declaring absence,and keep the first left/right/point failure in source order.

Step identifiers and dependencies

Rule: topbasis_full_local_point_refinement_not_intersection_membership · Uses: G001, S008

S010 · implies

Before

ORIGINAL ordered basis positions(left,right)=[0, 6];actual basis elements=[[0, 1], [3, 4]];FULL intersection=[]

After

ALL [intersection_point,ALL refining original basis indices] rows=[];empty intersection gives VACUOUS local obligation;first failure retained in source order

Conditions: none

For each intersection point x require some candidate basis element containing x and contained in the intersection.The entire intersection need not itself be a basis element.Check all refiners before declaring absence,and keep the first left/right/point failure in source order.

Step identifiers and dependencies

Rule: topbasis_full_local_point_refinement_not_intersection_membership · Uses: G001, S009

S011 · implies

Before

ORIGINAL ordered basis positions(left,right)=[1, 0];actual basis elements=[[0, 2], [0, 1]];FULL intersection=[0]

After

ALL [intersection_point,ALL refining original basis indices] rows=[[0, []]];empty intersection gives VACUOUS local obligation;first failure retained in source order

Conditions: none

For each intersection point x require some candidate basis element containing x and contained in the intersection.The entire intersection need not itself be a basis element.Check all refiners before declaring absence,and keep the first left/right/point failure in source order.

Step identifiers and dependencies

Rule: topbasis_full_local_point_refinement_not_intersection_membership · Uses: G001, S010

S012 · implies

Before

ORIGINAL ordered basis positions(left,right)=[1, 1];actual basis elements=[[0, 2], [0, 2]];FULL intersection=[0, 2]

After

ALL [intersection_point,ALL refining original basis indices] rows=[[0, [1]], [2, [1]]];empty intersection gives VACUOUS local obligation;first failure retained in source order

Conditions: none

For each intersection point x require some candidate basis element containing x and contained in the intersection.The entire intersection need not itself be a basis element.Check all refiners before declaring absence,and keep the first left/right/point failure in source order.

Step identifiers and dependencies

Rule: topbasis_full_local_point_refinement_not_intersection_membership · Uses: G001, S011

S013 · implies

Before

ORIGINAL ordered basis positions(left,right)=[1, 2];actual basis elements=[[0, 2], [0, 3]];FULL intersection=[0]

After

ALL [intersection_point,ALL refining original basis indices] rows=[[0, []]];empty intersection gives VACUOUS local obligation;first failure retained in source order

Conditions: none

For each intersection point x require some candidate basis element containing x and contained in the intersection.The entire intersection need not itself be a basis element.Check all refiners before declaring absence,and keep the first left/right/point failure in source order.

Step identifiers and dependencies

Rule: topbasis_full_local_point_refinement_not_intersection_membership · Uses: G001, S012

S014 · implies

Before

ORIGINAL ordered basis positions(left,right)=[1, 3];actual basis elements=[[0, 2], [1, 2]];FULL intersection=[2]

After

ALL [intersection_point,ALL refining original basis indices] rows=[[2, []]];empty intersection gives VACUOUS local obligation;first failure retained in source order

Conditions: none

For each intersection point x require some candidate basis element containing x and contained in the intersection.The entire intersection need not itself be a basis element.Check all refiners before declaring absence,and keep the first left/right/point failure in source order.

Step identifiers and dependencies

Rule: topbasis_full_local_point_refinement_not_intersection_membership · Uses: G001, S013

S015 · implies

Before

ORIGINAL ordered basis positions(left,right)=[1, 4];actual basis elements=[[0, 2], [2, 3]];FULL intersection=[2]

After

ALL [intersection_point,ALL refining original basis indices] rows=[[2, []]];empty intersection gives VACUOUS local obligation;first failure retained in source order

Conditions: none

For each intersection point x require some candidate basis element containing x and contained in the intersection.The entire intersection need not itself be a basis element.Check all refiners before declaring absence,and keep the first left/right/point failure in source order.

Step identifiers and dependencies

Rule: topbasis_full_local_point_refinement_not_intersection_membership · Uses: G001, S014

S016 · implies

Before

ORIGINAL ordered basis positions(left,right)=[1, 5];actual basis elements=[[0, 2], [2, 4]];FULL intersection=[2]

After

ALL [intersection_point,ALL refining original basis indices] rows=[[2, []]];empty intersection gives VACUOUS local obligation;first failure retained in source order

Conditions: none

For each intersection point x require some candidate basis element containing x and contained in the intersection.The entire intersection need not itself be a basis element.Check all refiners before declaring absence,and keep the first left/right/point failure in source order.

Step identifiers and dependencies

Rule: topbasis_full_local_point_refinement_not_intersection_membership · Uses: G001, S015

S017 · implies

Before

ORIGINAL ordered basis positions(left,right)=[1, 6];actual basis elements=[[0, 2], [3, 4]];FULL intersection=[]

After

ALL [intersection_point,ALL refining original basis indices] rows=[];empty intersection gives VACUOUS local obligation;first failure retained in source order

Conditions: none

For each intersection point x require some candidate basis element containing x and contained in the intersection.The entire intersection need not itself be a basis element.Check all refiners before declaring absence,and keep the first left/right/point failure in source order.

Step identifiers and dependencies

Rule: topbasis_full_local_point_refinement_not_intersection_membership · Uses: G001, S016

S018 · implies

Before

ORIGINAL ordered basis positions(left,right)=[2, 0];actual basis elements=[[0, 3], [0, 1]];FULL intersection=[0]

After

ALL [intersection_point,ALL refining original basis indices] rows=[[0, []]];empty intersection gives VACUOUS local obligation;first failure retained in source order

Conditions: none

For each intersection point x require some candidate basis element containing x and contained in the intersection.The entire intersection need not itself be a basis element.Check all refiners before declaring absence,and keep the first left/right/point failure in source order.

Step identifiers and dependencies

Rule: topbasis_full_local_point_refinement_not_intersection_membership · Uses: G001, S017

S019 · implies

Before

ORIGINAL ordered basis positions(left,right)=[2, 1];actual basis elements=[[0, 3], [0, 2]];FULL intersection=[0]

After

ALL [intersection_point,ALL refining original basis indices] rows=[[0, []]];empty intersection gives VACUOUS local obligation;first failure retained in source order

Conditions: none

For each intersection point x require some candidate basis element containing x and contained in the intersection.The entire intersection need not itself be a basis element.Check all refiners before declaring absence,and keep the first left/right/point failure in source order.

Step identifiers and dependencies

Rule: topbasis_full_local_point_refinement_not_intersection_membership · Uses: G001, S018

S020 · implies

Before

ORIGINAL ordered basis positions(left,right)=[2, 2];actual basis elements=[[0, 3], [0, 3]];FULL intersection=[0, 3]

After

ALL [intersection_point,ALL refining original basis indices] rows=[[0, [2]], [3, [2]]];empty intersection gives VACUOUS local obligation;first failure retained in source order

Conditions: none

For each intersection point x require some candidate basis element containing x and contained in the intersection.The entire intersection need not itself be a basis element.Check all refiners before declaring absence,and keep the first left/right/point failure in source order.

Step identifiers and dependencies

Rule: topbasis_full_local_point_refinement_not_intersection_membership · Uses: G001, S019

S021 · implies

Before

ORIGINAL ordered basis positions(left,right)=[2, 3];actual basis elements=[[0, 3], [1, 2]];FULL intersection=[]

After

ALL [intersection_point,ALL refining original basis indices] rows=[];empty intersection gives VACUOUS local obligation;first failure retained in source order

Conditions: none

For each intersection point x require some candidate basis element containing x and contained in the intersection.The entire intersection need not itself be a basis element.Check all refiners before declaring absence,and keep the first left/right/point failure in source order.

Step identifiers and dependencies

Rule: topbasis_full_local_point_refinement_not_intersection_membership · Uses: G001, S020

S022 · implies

Before

ORIGINAL ordered basis positions(left,right)=[2, 4];actual basis elements=[[0, 3], [2, 3]];FULL intersection=[3]

After

ALL [intersection_point,ALL refining original basis indices] rows=[[3, []]];empty intersection gives VACUOUS local obligation;first failure retained in source order

Conditions: none

For each intersection point x require some candidate basis element containing x and contained in the intersection.The entire intersection need not itself be a basis element.Check all refiners before declaring absence,and keep the first left/right/point failure in source order.

Step identifiers and dependencies

Rule: topbasis_full_local_point_refinement_not_intersection_membership · Uses: G001, S021

S023 · implies

Before

ORIGINAL ordered basis positions(left,right)=[2, 5];actual basis elements=[[0, 3], [2, 4]];FULL intersection=[]

After

ALL [intersection_point,ALL refining original basis indices] rows=[];empty intersection gives VACUOUS local obligation;first failure retained in source order

Conditions: none

For each intersection point x require some candidate basis element containing x and contained in the intersection.The entire intersection need not itself be a basis element.Check all refiners before declaring absence,and keep the first left/right/point failure in source order.

Step identifiers and dependencies

Rule: topbasis_full_local_point_refinement_not_intersection_membership · Uses: G001, S022

S024 · implies

Before

ORIGINAL ordered basis positions(left,right)=[2, 6];actual basis elements=[[0, 3], [3, 4]];FULL intersection=[3]

After

ALL [intersection_point,ALL refining original basis indices] rows=[[3, []]];empty intersection gives VACUOUS local obligation;first failure retained in source order

Conditions: none

For each intersection point x require some candidate basis element containing x and contained in the intersection.The entire intersection need not itself be a basis element.Check all refiners before declaring absence,and keep the first left/right/point failure in source order.

Step identifiers and dependencies

Rule: topbasis_full_local_point_refinement_not_intersection_membership · Uses: G001, S023

S025 · implies

Before

ORIGINAL ordered basis positions(left,right)=[3, 0];actual basis elements=[[1, 2], [0, 1]];FULL intersection=[1]

After

ALL [intersection_point,ALL refining original basis indices] rows=[[1, []]];empty intersection gives VACUOUS local obligation;first failure retained in source order

Conditions: none

For each intersection point x require some candidate basis element containing x and contained in the intersection.The entire intersection need not itself be a basis element.Check all refiners before declaring absence,and keep the first left/right/point failure in source order.

Step identifiers and dependencies

Rule: topbasis_full_local_point_refinement_not_intersection_membership · Uses: G001, S024

S026 · implies

Before

ORIGINAL ordered basis positions(left,right)=[3, 1];actual basis elements=[[1, 2], [0, 2]];FULL intersection=[2]

After

ALL [intersection_point,ALL refining original basis indices] rows=[[2, []]];empty intersection gives VACUOUS local obligation;first failure retained in source order

Conditions: none

For each intersection point x require some candidate basis element containing x and contained in the intersection.The entire intersection need not itself be a basis element.Check all refiners before declaring absence,and keep the first left/right/point failure in source order.

Step identifiers and dependencies

Rule: topbasis_full_local_point_refinement_not_intersection_membership · Uses: G001, S025

S027 · implies

Before

ORIGINAL ordered basis positions(left,right)=[3, 2];actual basis elements=[[1, 2], [0, 3]];FULL intersection=[]

After

ALL [intersection_point,ALL refining original basis indices] rows=[];empty intersection gives VACUOUS local obligation;first failure retained in source order

Conditions: none

For each intersection point x require some candidate basis element containing x and contained in the intersection.The entire intersection need not itself be a basis element.Check all refiners before declaring absence,and keep the first left/right/point failure in source order.

Step identifiers and dependencies

Rule: topbasis_full_local_point_refinement_not_intersection_membership · Uses: G001, S026

S028 · implies

Before

ORIGINAL ordered basis positions(left,right)=[3, 3];actual basis elements=[[1, 2], [1, 2]];FULL intersection=[1, 2]

After

ALL [intersection_point,ALL refining original basis indices] rows=[[1, [3]], [2, [3]]];empty intersection gives VACUOUS local obligation;first failure retained in source order

Conditions: none

For each intersection point x require some candidate basis element containing x and contained in the intersection.The entire intersection need not itself be a basis element.Check all refiners before declaring absence,and keep the first left/right/point failure in source order.

Step identifiers and dependencies

Rule: topbasis_full_local_point_refinement_not_intersection_membership · Uses: G001, S027

S029 · implies

Before

ORIGINAL ordered basis positions(left,right)=[3, 4];actual basis elements=[[1, 2], [2, 3]];FULL intersection=[2]

After

ALL [intersection_point,ALL refining original basis indices] rows=[[2, []]];empty intersection gives VACUOUS local obligation;first failure retained in source order

Conditions: none

For each intersection point x require some candidate basis element containing x and contained in the intersection.The entire intersection need not itself be a basis element.Check all refiners before declaring absence,and keep the first left/right/point failure in source order.

Step identifiers and dependencies

Rule: topbasis_full_local_point_refinement_not_intersection_membership · Uses: G001, S028

S030 · implies

Before

ORIGINAL ordered basis positions(left,right)=[3, 5];actual basis elements=[[1, 2], [2, 4]];FULL intersection=[2]

After

ALL [intersection_point,ALL refining original basis indices] rows=[[2, []]];empty intersection gives VACUOUS local obligation;first failure retained in source order

Conditions: none

For each intersection point x require some candidate basis element containing x and contained in the intersection.The entire intersection need not itself be a basis element.Check all refiners before declaring absence,and keep the first left/right/point failure in source order.

Step identifiers and dependencies

Rule: topbasis_full_local_point_refinement_not_intersection_membership · Uses: G001, S029

S031 · implies

Before

ORIGINAL ordered basis positions(left,right)=[3, 6];actual basis elements=[[1, 2], [3, 4]];FULL intersection=[]

After

ALL [intersection_point,ALL refining original basis indices] rows=[];empty intersection gives VACUOUS local obligation;first failure retained in source order

Conditions: none

For each intersection point x require some candidate basis element containing x and contained in the intersection.The entire intersection need not itself be a basis element.Check all refiners before declaring absence,and keep the first left/right/point failure in source order.

Step identifiers and dependencies

Rule: topbasis_full_local_point_refinement_not_intersection_membership · Uses: G001, S030

S032 · implies

Before

ORIGINAL ordered basis positions(left,right)=[4, 0];actual basis elements=[[2, 3], [0, 1]];FULL intersection=[]

After

ALL [intersection_point,ALL refining original basis indices] rows=[];empty intersection gives VACUOUS local obligation;first failure retained in source order

Conditions: none

For each intersection point x require some candidate basis element containing x and contained in the intersection.The entire intersection need not itself be a basis element.Check all refiners before declaring absence,and keep the first left/right/point failure in source order.

Step identifiers and dependencies

Rule: topbasis_full_local_point_refinement_not_intersection_membership · Uses: G001, S031

S033 · implies

Before

ORIGINAL ordered basis positions(left,right)=[4, 1];actual basis elements=[[2, 3], [0, 2]];FULL intersection=[2]

After

ALL [intersection_point,ALL refining original basis indices] rows=[[2, []]];empty intersection gives VACUOUS local obligation;first failure retained in source order

Conditions: none

For each intersection point x require some candidate basis element containing x and contained in the intersection.The entire intersection need not itself be a basis element.Check all refiners before declaring absence,and keep the first left/right/point failure in source order.

Step identifiers and dependencies

Rule: topbasis_full_local_point_refinement_not_intersection_membership · Uses: G001, S032

S034 · implies

Before

ORIGINAL ordered basis positions(left,right)=[4, 2];actual basis elements=[[2, 3], [0, 3]];FULL intersection=[3]

After

ALL [intersection_point,ALL refining original basis indices] rows=[[3, []]];empty intersection gives VACUOUS local obligation;first failure retained in source order

Conditions: none

For each intersection point x require some candidate basis element containing x and contained in the intersection.The entire intersection need not itself be a basis element.Check all refiners before declaring absence,and keep the first left/right/point failure in source order.

Step identifiers and dependencies

Rule: topbasis_full_local_point_refinement_not_intersection_membership · Uses: G001, S033

S035 · implies

Before

ORIGINAL ordered basis positions(left,right)=[4, 3];actual basis elements=[[2, 3], [1, 2]];FULL intersection=[2]

After

ALL [intersection_point,ALL refining original basis indices] rows=[[2, []]];empty intersection gives VACUOUS local obligation;first failure retained in source order

Conditions: none

For each intersection point x require some candidate basis element containing x and contained in the intersection.The entire intersection need not itself be a basis element.Check all refiners before declaring absence,and keep the first left/right/point failure in source order.

Step identifiers and dependencies

Rule: topbasis_full_local_point_refinement_not_intersection_membership · Uses: G001, S034

S036 · implies

Before

ORIGINAL ordered basis positions(left,right)=[4, 4];actual basis elements=[[2, 3], [2, 3]];FULL intersection=[2, 3]

After

ALL [intersection_point,ALL refining original basis indices] rows=[[2, [4]], [3, [4]]];empty intersection gives VACUOUS local obligation;first failure retained in source order

Conditions: none

For each intersection point x require some candidate basis element containing x and contained in the intersection.The entire intersection need not itself be a basis element.Check all refiners before declaring absence,and keep the first left/right/point failure in source order.

Step identifiers and dependencies

Rule: topbasis_full_local_point_refinement_not_intersection_membership · Uses: G001, S035

S037 · implies

Before

ORIGINAL ordered basis positions(left,right)=[4, 5];actual basis elements=[[2, 3], [2, 4]];FULL intersection=[2]

After

ALL [intersection_point,ALL refining original basis indices] rows=[[2, []]];empty intersection gives VACUOUS local obligation;first failure retained in source order

Conditions: none

For each intersection point x require some candidate basis element containing x and contained in the intersection.The entire intersection need not itself be a basis element.Check all refiners before declaring absence,and keep the first left/right/point failure in source order.

Step identifiers and dependencies

Rule: topbasis_full_local_point_refinement_not_intersection_membership · Uses: G001, S036

S038 · implies

Before

ORIGINAL ordered basis positions(left,right)=[4, 6];actual basis elements=[[2, 3], [3, 4]];FULL intersection=[3]

After

ALL [intersection_point,ALL refining original basis indices] rows=[[3, []]];empty intersection gives VACUOUS local obligation;first failure retained in source order

Conditions: none

For each intersection point x require some candidate basis element containing x and contained in the intersection.The entire intersection need not itself be a basis element.Check all refiners before declaring absence,and keep the first left/right/point failure in source order.

Step identifiers and dependencies

Rule: topbasis_full_local_point_refinement_not_intersection_membership · Uses: G001, S037

S039 · implies

Before

ORIGINAL ordered basis positions(left,right)=[5, 0];actual basis elements=[[2, 4], [0, 1]];FULL intersection=[]

After

ALL [intersection_point,ALL refining original basis indices] rows=[];empty intersection gives VACUOUS local obligation;first failure retained in source order

Conditions: none

For each intersection point x require some candidate basis element containing x and contained in the intersection.The entire intersection need not itself be a basis element.Check all refiners before declaring absence,and keep the first left/right/point failure in source order.

Step identifiers and dependencies

Rule: topbasis_full_local_point_refinement_not_intersection_membership · Uses: G001, S038

S040 · implies

Before

ORIGINAL ordered basis positions(left,right)=[5, 1];actual basis elements=[[2, 4], [0, 2]];FULL intersection=[2]

After

ALL [intersection_point,ALL refining original basis indices] rows=[[2, []]];empty intersection gives VACUOUS local obligation;first failure retained in source order

Conditions: none

For each intersection point x require some candidate basis element containing x and contained in the intersection.The entire intersection need not itself be a basis element.Check all refiners before declaring absence,and keep the first left/right/point failure in source order.

Step identifiers and dependencies

Rule: topbasis_full_local_point_refinement_not_intersection_membership · Uses: G001, S039

S041 · implies

Before

ORIGINAL ordered basis positions(left,right)=[5, 2];actual basis elements=[[2, 4], [0, 3]];FULL intersection=[]

After

ALL [intersection_point,ALL refining original basis indices] rows=[];empty intersection gives VACUOUS local obligation;first failure retained in source order

Conditions: none

For each intersection point x require some candidate basis element containing x and contained in the intersection.The entire intersection need not itself be a basis element.Check all refiners before declaring absence,and keep the first left/right/point failure in source order.

Step identifiers and dependencies

Rule: topbasis_full_local_point_refinement_not_intersection_membership · Uses: G001, S040

S042 · implies

Before

ORIGINAL ordered basis positions(left,right)=[5, 3];actual basis elements=[[2, 4], [1, 2]];FULL intersection=[2]

After

ALL [intersection_point,ALL refining original basis indices] rows=[[2, []]];empty intersection gives VACUOUS local obligation;first failure retained in source order

Conditions: none

For each intersection point x require some candidate basis element containing x and contained in the intersection.The entire intersection need not itself be a basis element.Check all refiners before declaring absence,and keep the first left/right/point failure in source order.

Step identifiers and dependencies

Rule: topbasis_full_local_point_refinement_not_intersection_membership · Uses: G001, S041

S043 · implies

Before

ORIGINAL ordered basis positions(left,right)=[5, 4];actual basis elements=[[2, 4], [2, 3]];FULL intersection=[2]

After

ALL [intersection_point,ALL refining original basis indices] rows=[[2, []]];empty intersection gives VACUOUS local obligation;first failure retained in source order

Conditions: none

For each intersection point x require some candidate basis element containing x and contained in the intersection.The entire intersection need not itself be a basis element.Check all refiners before declaring absence,and keep the first left/right/point failure in source order.

Step identifiers and dependencies

Rule: topbasis_full_local_point_refinement_not_intersection_membership · Uses: G001, S042

S044 · implies

Before

ORIGINAL ordered basis positions(left,right)=[5, 5];actual basis elements=[[2, 4], [2, 4]];FULL intersection=[2, 4]

After

ALL [intersection_point,ALL refining original basis indices] rows=[[2, [5]], [4, [5]]];empty intersection gives VACUOUS local obligation;first failure retained in source order

Conditions: none

For each intersection point x require some candidate basis element containing x and contained in the intersection.The entire intersection need not itself be a basis element.Check all refiners before declaring absence,and keep the first left/right/point failure in source order.

Step identifiers and dependencies

Rule: topbasis_full_local_point_refinement_not_intersection_membership · Uses: G001, S043

S045 · implies

Before

ORIGINAL ordered basis positions(left,right)=[5, 6];actual basis elements=[[2, 4], [3, 4]];FULL intersection=[4]

After

ALL [intersection_point,ALL refining original basis indices] rows=[[4, []]];empty intersection gives VACUOUS local obligation;first failure retained in source order

Conditions: none

For each intersection point x require some candidate basis element containing x and contained in the intersection.The entire intersection need not itself be a basis element.Check all refiners before declaring absence,and keep the first left/right/point failure in source order.

Step identifiers and dependencies

Rule: topbasis_full_local_point_refinement_not_intersection_membership · Uses: G001, S044

S046 · implies

Before

ORIGINAL ordered basis positions(left,right)=[6, 0];actual basis elements=[[3, 4], [0, 1]];FULL intersection=[]

After

ALL [intersection_point,ALL refining original basis indices] rows=[];empty intersection gives VACUOUS local obligation;first failure retained in source order

Conditions: none

For each intersection point x require some candidate basis element containing x and contained in the intersection.The entire intersection need not itself be a basis element.Check all refiners before declaring absence,and keep the first left/right/point failure in source order.

Step identifiers and dependencies

Rule: topbasis_full_local_point_refinement_not_intersection_membership · Uses: G001, S045

S047 · implies

Before

ORIGINAL ordered basis positions(left,right)=[6, 1];actual basis elements=[[3, 4], [0, 2]];FULL intersection=[]

After

ALL [intersection_point,ALL refining original basis indices] rows=[];empty intersection gives VACUOUS local obligation;first failure retained in source order

Conditions: none

For each intersection point x require some candidate basis element containing x and contained in the intersection.The entire intersection need not itself be a basis element.Check all refiners before declaring absence,and keep the first left/right/point failure in source order.

Step identifiers and dependencies

Rule: topbasis_full_local_point_refinement_not_intersection_membership · Uses: G001, S046

S048 · implies

Before

ORIGINAL ordered basis positions(left,right)=[6, 2];actual basis elements=[[3, 4], [0, 3]];FULL intersection=[3]

After

ALL [intersection_point,ALL refining original basis indices] rows=[[3, []]];empty intersection gives VACUOUS local obligation;first failure retained in source order

Conditions: none

For each intersection point x require some candidate basis element containing x and contained in the intersection.The entire intersection need not itself be a basis element.Check all refiners before declaring absence,and keep the first left/right/point failure in source order.

Step identifiers and dependencies

Rule: topbasis_full_local_point_refinement_not_intersection_membership · Uses: G001, S047

S049 · implies

Before

ORIGINAL ordered basis positions(left,right)=[6, 3];actual basis elements=[[3, 4], [1, 2]];FULL intersection=[]

After

ALL [intersection_point,ALL refining original basis indices] rows=[];empty intersection gives VACUOUS local obligation;first failure retained in source order

Conditions: none

For each intersection point x require some candidate basis element containing x and contained in the intersection.The entire intersection need not itself be a basis element.Check all refiners before declaring absence,and keep the first left/right/point failure in source order.

Step identifiers and dependencies

Rule: topbasis_full_local_point_refinement_not_intersection_membership · Uses: G001, S048

S050 · implies

Before

ORIGINAL ordered basis positions(left,right)=[6, 4];actual basis elements=[[3, 4], [2, 3]];FULL intersection=[3]

After

ALL [intersection_point,ALL refining original basis indices] rows=[[3, []]];empty intersection gives VACUOUS local obligation;first failure retained in source order

Conditions: none

For each intersection point x require some candidate basis element containing x and contained in the intersection.The entire intersection need not itself be a basis element.Check all refiners before declaring absence,and keep the first left/right/point failure in source order.

Step identifiers and dependencies

Rule: topbasis_full_local_point_refinement_not_intersection_membership · Uses: G001, S049

S051 · implies

Before

ORIGINAL ordered basis positions(left,right)=[6, 5];actual basis elements=[[3, 4], [2, 4]];FULL intersection=[4]

After

ALL [intersection_point,ALL refining original basis indices] rows=[[4, []]];empty intersection gives VACUOUS local obligation;first failure retained in source order

Conditions: none

For each intersection point x require some candidate basis element containing x and contained in the intersection.The entire intersection need not itself be a basis element.Check all refiners before declaring absence,and keep the first left/right/point failure in source order.

Step identifiers and dependencies

Rule: topbasis_full_local_point_refinement_not_intersection_membership · Uses: G001, S050

S052 · implies

Before

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

After

ALL [intersection_point,ALL refining original basis indices] rows=[[3, [6]], [4, [6]]];empty intersection gives VACUOUS local obligation;first failure retained in source order

Conditions: none

For each intersection point x require some candidate basis element containing x and contained in the intersection.The entire intersection need not itself be a basis element.Check all refiners before declaring absence,and keep the first left/right/point failure in source order.

Step identifiers and dependencies

Rule: topbasis_full_local_point_refinement_not_intersection_membership · Uses: G001, S051

S053 · implies

Before

ACTUAL cover=Yes;ALL local point obligations refined=No;first failure[left,right,point,intersection]=[0, 1, 0, [0]]

After

abstract basis validity=No is cover AND LOCAL refinement;not closure of the family under intersections and not a given-topology certificate yet

Conditions: none

Their conjunction gives an abstract basis.Coverage alone is insufficient,and a valid basis need not be intersection-closed.A basis for a separately supplied topology also needs openness and generation of every given open.

Step identifiers and dependencies

Rule: topbasis_cover_and_local_refinement_conjunction · Uses: G001, S052

S054 · implies

Before

ACTUAL complete candidate cover=Yes,local refinement=No,is_basis=No

After

cover-alone SUFFICES GENERAL CLAIM=No;not a negation of actual is_basis and not a test proving subbasis status false

Conditions: none

The final No rejects the general cover-only basis inference without negating the independently computed is_basis.A valid basis may still be true.This field does not audit subbasis status,whose usual generation also takes finite intersections before unions.

Step identifiers and dependencies

Rule: topbasis_cover_alone_No_not_actual_basis_or_subbasis_negation · Uses: G001, S053

Reverse checks

  • S054 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S053 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S052 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S051 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S050 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S049 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S048 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S047 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S046 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S045 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S044 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S043 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S042 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S041 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S040 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S039 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S038 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S037 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S036 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S035 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S034 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S033 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S032 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S031 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S030 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S029 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S028 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S027 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S026 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S025 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S024 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S023 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S022 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S021 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S020 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S019 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S018 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S017 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S016 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S015 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S014 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S013 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S012 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S011 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S010 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S009 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S008 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S007 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S006 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S005 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S004 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S003 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S002 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S001 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.

Result

(Yes, No, No, No, cover_without_local_intersection_refinement_is_not_a_basis_and_may_only_be_a_subbasis)

The full substitutions and premise checks give the original answer; source identity,format and status are unchanged.

This question review and source binding

Review date: 2026-10-26

Original questions, answers, step expressions, numeric ASTs and historical records are retained. Source candidate labels remain historical and are not overwritten as formal proof.

Source hashes and review receipts
  • input commit: 0f7be61e2c22317a4075561dc822c87d881a50db
  • original question SHA-256: fa9529a9d411a59e384ae2da0bf2e90f09cf42b90be86957c3ee171de60b3ea1
  • candidate record SHA-256: 771601f7d580c15101df4b084dacd416e4697c5aa40e73948ca95b7a01bc0b7d
  • semantic review receipt SHA-256: c6483ed1229af9183fe5c704a2627b1558ad9a31bfbb0d2ee2d5c1a1c299ab4d
  • question-first witness SHA-256: e81f9a560d630fcb23441146f008c4feca7aceab3ebe4de2cf1997c061375ef7
  • Lean: NOT_RUN · human review: NOT_PERFORMED
Same-source full JSON