Step-by-step derivation
S001 · implies
Before
source operation=topbasis-generated-topology;original snapshot and mathematics retained
After
explicit ordered source output fields=[topology, open_count, topology_certificate, certificate]
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], [2], [2, 3], [3], [4]];carrier Fin(5)
After
within-set numeric normalization=[[0], [1], [2], [2, 3], [3], [4]];no duplicate points or duplicate FAMILY elements;WHOLE original family already in required key order;empty family allowed as candidate
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], [2], [2, 3], [3], [4]];ALL carrier points0..4
After
ALL [point,containing_basis_indices] rows=[[0, [0]], [1, [1]], [2, [2, 3]], [3, [3, 4]], [4, [5]]];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], [0]];FULL intersection=[0]
After
ALL [intersection_point,ALL refining original basis indices] rows=[[0, [0]]];empty intersection gives VACUOUS local obligation;first failure retained in source order
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]];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, S004
S006 · implies
Before
ORIGINAL ordered basis positions(left,right)=[0, 2];actual basis elements=[[0], [2]];FULL intersection=[]
After
ALL [intersection_point,ALL refining original basis indices] rows=[];empty intersection gives VACUOUS local obligation;first failure retained in source order
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], [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, S006
S008 · implies
Before
ORIGINAL ordered basis positions(left,right)=[0, 4];actual basis elements=[[0], [3]];FULL intersection=[]
After
ALL [intersection_point,ALL refining original basis indices] rows=[];empty intersection gives VACUOUS local obligation;first failure retained in source order
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], [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)=[1, 0];actual basis elements=[[1], [0]];FULL intersection=[]
After
ALL [intersection_point,ALL refining original basis indices] rows=[];empty intersection gives VACUOUS local obligation;first failure retained in source order
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, 1];actual basis elements=[[1], [1]];FULL intersection=[1]
After
ALL [intersection_point,ALL refining original basis indices] rows=[[1, [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, S010
S012 · implies
Before
ORIGINAL ordered basis positions(left,right)=[1, 2];actual basis elements=[[1], [2]];FULL intersection=[]
After
ALL [intersection_point,ALL refining original basis indices] rows=[];empty intersection gives VACUOUS local obligation;first failure retained in source order
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, 3];actual basis elements=[[1], [2, 3]];FULL intersection=[]
After
ALL [intersection_point,ALL refining original basis indices] rows=[];empty intersection gives VACUOUS local obligation;first failure retained in source order
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, 4];actual basis elements=[[1], [3]];FULL intersection=[]
After
ALL [intersection_point,ALL refining original basis indices] rows=[];empty intersection gives VACUOUS local obligation;first failure retained in source order
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, 5];actual basis elements=[[1], [4]];FULL intersection=[]
After
ALL [intersection_point,ALL refining original basis indices] rows=[];empty intersection gives VACUOUS local obligation;first failure retained in source order
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)=[2, 0];actual basis elements=[[2], [0]];FULL intersection=[]
After
ALL [intersection_point,ALL refining original basis indices] rows=[];empty intersection gives VACUOUS local obligation;first failure retained in source order
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)=[2, 1];actual basis elements=[[2], [1]];FULL intersection=[]
After
ALL [intersection_point,ALL refining original basis indices] rows=[];empty intersection gives VACUOUS local obligation;first failure retained in source order
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, 2];actual basis elements=[[2], [2]];FULL intersection=[2]
After
ALL [intersection_point,ALL refining original basis indices] rows=[[2, [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, S017
S019 · implies
Before
ORIGINAL ordered basis positions(left,right)=[2, 3];actual basis elements=[[2], [2, 3]];FULL intersection=[2]
After
ALL [intersection_point,ALL refining original basis indices] rows=[[2, [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, S018
S020 · implies
Before
ORIGINAL ordered basis positions(left,right)=[2, 4];actual basis elements=[[2], [3]];FULL intersection=[]
After
ALL [intersection_point,ALL refining original basis indices] rows=[];empty intersection gives VACUOUS local obligation;first failure retained in source order
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, 5];actual basis elements=[[2], [4]];FULL intersection=[]
After
ALL [intersection_point,ALL refining original basis indices] rows=[];empty intersection gives VACUOUS local obligation;first failure retained in source order
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)=[3, 0];actual basis elements=[[2, 3], [0]];FULL intersection=[]
After
ALL [intersection_point,ALL refining original basis indices] rows=[];empty intersection gives VACUOUS local obligation;first failure retained in source order
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)=[3, 1];actual basis elements=[[2, 3], [1]];FULL intersection=[]
After
ALL [intersection_point,ALL refining original basis indices] rows=[];empty intersection gives VACUOUS local obligation;first failure retained in source order
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)=[3, 2];actual basis elements=[[2, 3], [2]];FULL intersection=[2]
After
ALL [intersection_point,ALL refining original basis indices] rows=[[2, [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, S023
S025 · implies
Before
ORIGINAL ordered basis positions(left,right)=[3, 3];actual basis elements=[[2, 3], [2, 3]];FULL intersection=[2, 3]
After
ALL [intersection_point,ALL refining original basis indices] rows=[[2, [2, 3]], [3, [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, S024
S026 · implies
Before
ORIGINAL ordered basis positions(left,right)=[3, 4];actual basis elements=[[2, 3], [3]];FULL intersection=[3]
After
ALL [intersection_point,ALL refining original basis indices] rows=[[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, S025
S027 · implies
Before
ORIGINAL ordered basis positions(left,right)=[3, 5];actual basis elements=[[2, 3], [4]];FULL intersection=[]
After
ALL [intersection_point,ALL refining original basis indices] rows=[];empty intersection gives VACUOUS local obligation;first failure retained in source order
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)=[4, 0];actual basis elements=[[3], [0]];FULL intersection=[]
After
ALL [intersection_point,ALL refining original basis indices] rows=[];empty intersection gives VACUOUS local obligation;first failure retained in source order
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)=[4, 1];actual basis elements=[[3], [1]];FULL intersection=[]
After
ALL [intersection_point,ALL refining original basis indices] rows=[];empty intersection gives VACUOUS local obligation;first failure retained in source order
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)=[4, 2];actual basis elements=[[3], [2]];FULL intersection=[]
After
ALL [intersection_point,ALL refining original basis indices] rows=[];empty intersection gives VACUOUS local obligation;first failure retained in source order
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)=[4, 3];actual basis elements=[[3], [2, 3]];FULL intersection=[3]
After
ALL [intersection_point,ALL refining original basis indices] rows=[[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, S030
S032 · implies
Before
ORIGINAL ordered basis positions(left,right)=[4, 4];actual basis elements=[[3], [3]];FULL intersection=[3]
After
ALL [intersection_point,ALL refining original basis indices] rows=[[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, S031
S033 · implies
Before
ORIGINAL ordered basis positions(left,right)=[4, 5];actual basis elements=[[3], [4]];FULL intersection=[]
After
ALL [intersection_point,ALL refining original basis indices] rows=[];empty intersection gives VACUOUS local obligation;first failure retained in source order
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)=[5, 0];actual basis elements=[[4], [0]];FULL intersection=[]
After
ALL [intersection_point,ALL refining original basis indices] rows=[];empty intersection gives VACUOUS local obligation;first failure retained in source order
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)=[5, 1];actual basis elements=[[4], [1]];FULL intersection=[]
After
ALL [intersection_point,ALL refining original basis indices] rows=[];empty intersection gives VACUOUS local obligation;first failure retained in source order
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)=[5, 2];actual basis elements=[[4], [2]];FULL intersection=[]
After
ALL [intersection_point,ALL refining original basis indices] rows=[];empty intersection gives VACUOUS local obligation;first failure retained in source order
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)=[5, 3];actual basis elements=[[4], [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, S036
S038 · implies
Before
ORIGINAL ordered basis positions(left,right)=[5, 4];actual basis elements=[[4], [3]];FULL intersection=[]
After
ALL [intersection_point,ALL refining original basis indices] rows=[];empty intersection gives VACUOUS local obligation;first failure retained in source order
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, 5];actual basis elements=[[4], [4]];FULL intersection=[4]
After
ALL [intersection_point,ALL refining original basis indices] rows=[[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, S038
S040 · implies
Before
ACTUAL cover=Yes;ALL local point obligations refined=Yes;first failure[left,right,point,intersection]=null
After
abstract basis validity=Yes 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, S039
S041 · implies
Before
ORIGINAL complete candidate basis in fixed family-index order=[[0], [1], [2], [2, 3], [3], [4]];carrier Fin(5)
After
prefix0 has EXACTLY the empty-subfamily union:empty set,with empty index witness;this mathematical union exists even when basis axioms fail
Conditions: none
The empty subfamily always has empty union,without requiring the empty set to be a basis element.This step constructs unions only and does not call an invalid basis's unions a topology.
Step identifiers and dependencies
Rule: topbasis_empty_union_initial_state_not_missing_empty_set · Uses: G001, S040
S042 · implies
Before
next ORIGINAL basis index=0;basis set=[0];ALL previous-prefix unions in numeric POINT-mask order=[[]]
After
ALL [previous_union,union_with_new_basis,index_witness] rows=[[[], [0], [0]]];keep both omit/include choices,then deduplicate EQUAL UNIONS;new canonical union family=[[], [0]]
Conditions: none
Any subfamily of the new prefix either omits the new element or includes it with a union from the previous prefix.These choices are complete and every state has an actual index witness.Deduplicate equal unions,not original basis positions.There are at most2^n distinct states,not a sample of the2^basisCount subfamilies.
Step identifiers and dependencies
Rule: topbasis_complete_prefix_union_DP_not_sampled_subfamilies · Uses: G001, S041
S043 · implies
Before
next ORIGINAL basis index=1;basis set=[1];ALL previous-prefix unions in numeric POINT-mask order=[[], [0]]
After
ALL [previous_union,union_with_new_basis,index_witness] rows=[[[], [1], [1]], [[0], [0, 1], [0, 1]]];keep both omit/include choices,then deduplicate EQUAL UNIONS;new canonical union family=[[], [0], [0, 1], [1]]
Conditions: none
Any subfamily of the new prefix either omits the new element or includes it with a union from the previous prefix.These choices are complete and every state has an actual index witness.Deduplicate equal unions,not original basis positions.There are at most2^n distinct states,not a sample of the2^basisCount subfamilies.
Step identifiers and dependencies
Rule: topbasis_complete_prefix_union_DP_not_sampled_subfamilies · Uses: G001, S042
S044 · implies
Before
next ORIGINAL basis index=2;basis set=[2];ALL previous-prefix unions in numeric POINT-mask order=[[], [0], [1], [0, 1]]
After
ALL [previous_union,union_with_new_basis,index_witness] rows=[[[], [2], [2]], [[0], [0, 2], [0, 2]], [[1], [1, 2], [1, 2]], [[0, 1], [0, 1, 2], [0, 1, 2]]];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]]
Conditions: none
Any subfamily of the new prefix either omits the new element or includes it with a union from the previous prefix.These choices are complete and every state has an actual index witness.Deduplicate equal unions,not original basis positions.There are at most2^n distinct states,not a sample of the2^basisCount subfamilies.
Step identifiers and dependencies
Rule: topbasis_complete_prefix_union_DP_not_sampled_subfamilies · Uses: G001, S043
S045 · implies
Before
next ORIGINAL basis index=3;basis set=[2, 3];ALL previous-prefix unions in numeric POINT-mask order=[[], [0], [1], [0, 1], [2], [0, 2], [1, 2], [0, 1, 2]]
After
ALL [previous_union,union_with_new_basis,index_witness] rows=[[[], [2, 3], [3]], [[0], [0, 2, 3], [0, 3]], [[1], [1, 2, 3], [1, 3]], [[0, 1], [0, 1, 2, 3], [0, 1, 3]], [[2], [2, 3], [2, 3]], [[0, 2], [0, 2, 3], [0, 2, 3]], [[1, 2], [1, 2, 3], [1, 2, 3]], [[0, 1, 2], [0, 1, 2, 3], [0, 1, 2, 3]]];keep both omit/include choices,then deduplicate EQUAL UNIONS;new canonical union family=[[], [0], [0, 1], [0, 1, 2], [0, 1, 2, 3], [0, 2], [0, 2, 3], [1], [1, 2], [1, 2, 3], [2], [2, 3]]
Conditions: none
Any subfamily of the new prefix either omits the new element or includes it with a union from the previous prefix.These choices are complete and every state has an actual index witness.Deduplicate equal unions,not original basis positions.There are at most2^n distinct states,not a sample of the2^basisCount subfamilies.
Step identifiers and dependencies
Rule: topbasis_complete_prefix_union_DP_not_sampled_subfamilies · Uses: G001, S044
S046 · implies
Before
next ORIGINAL basis index=4;basis set=[3];ALL previous-prefix unions in numeric POINT-mask order=[[], [0], [1], [0, 1], [2], [0, 2], [1, 2], [0, 1, 2], [2, 3], [0, 2, 3], [1, 2, 3], [0, 1, 2, 3]]
After
ALL [previous_union,union_with_new_basis,index_witness] rows=[[[], [3], [4]], [[0], [0, 3], [0, 4]], [[1], [1, 3], [1, 4]], [[0, 1], [0, 1, 3], [0, 1, 4]], [[2], [2, 3], [2, 4]], [[0, 2], [0, 2, 3], [0, 2, 4]], [[1, 2], [1, 2, 3], [1, 2, 4]], [[0, 1, 2], [0, 1, 2, 3], [0, 1, 2, 4]], [[2, 3], [2, 3], [3, 4]], [[0, 2, 3], [0, 2, 3], [0, 3, 4]], [[1, 2, 3], [1, 2, 3], [1, 3, 4]], [[0, 1, 2, 3], [0, 1, 2, 3], [0, 1, 3, 4]]];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]]
Conditions: none
Any subfamily of the new prefix either omits the new element or includes it with a union from the previous prefix.These choices are complete and every state has an actual index witness.Deduplicate equal unions,not original basis positions.There are at most2^n distinct states,not a sample of the2^basisCount subfamilies.
Step identifiers and dependencies
Rule: topbasis_complete_prefix_union_DP_not_sampled_subfamilies · Uses: G001, S045
S047 · implies
Before
next ORIGINAL basis index=5;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]]
After
ALL [previous_union,union_with_new_basis,index_witness] rows=[[[], [4], [5]], [[0], [0, 4], [0, 5]], [[1], [1, 4], [1, 5]], [[0, 1], [0, 1, 4], [0, 1, 5]], [[2], [2, 4], [2, 5]], [[0, 2], [0, 2, 4], [0, 2, 5]], [[1, 2], [1, 2, 4], [1, 2, 5]], [[0, 1, 2], [0, 1, 2, 4], [0, 1, 2, 5]], [[3], [3, 4], [4, 5]], [[0, 3], [0, 3, 4], [0, 4, 5]], [[1, 3], [1, 3, 4], [1, 4, 5]], [[0, 1, 3], [0, 1, 3, 4], [0, 1, 4, 5]], [[2, 3], [2, 3, 4], [3, 5]], [[0, 2, 3], [0, 2, 3, 4], [0, 3, 5]], [[1, 2, 3], [1, 2, 3, 4], [1, 3, 5]], [[0, 1, 2, 3], [0, 1, 2, 3, 4], [0, 1, 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, 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]]
Conditions: none
Any subfamily of the new prefix either omits the new element or includes it with a union from the previous prefix.These choices are complete and every state has an actual index witness.Deduplicate equal unions,not original basis positions.There are at most2^n distinct states,not a sample of the2^basisCount subfamilies.
Step identifiers and dependencies
Rule: topbasis_complete_prefix_union_DP_not_sampled_subfamilies · Uses: G001, S046
S048 · implies
Before
ALL original basis positions processed;distinct union states=32<=2^5;canonical ALL-union family=[[], [0], [0, 1], [0, 1, 2], [0, 1, 2, 3], [0, 1, 2, 3, 4], [0, 1, 2, 4], [0, 1, 3], [0, 1, 3, 4], [0, 1, 4], [0, 2], [0, 2, 3], [0, 2, 3, 4], [0, 2, 4], [0, 3], [0, 3, 4], [0, 4], [1], [1, 2], [1, 2, 3], [1, 2, 3, 4], [1, 2, 4], [1, 3], [1, 3, 4], [1, 4], [2], [2, 3], [2, 3, 4], [2, 4], [3], [3, 4], [4]]
After
prefix induction covers EVERY original-index subfamily and every state is its actual union;mathematical all-union family COMPLETE including empty union
Conditions: none
Omit/include induction puts every subfamily union into the states,and every state has an actual subfamily witness,proving both directions.Do not misreport the smaller distinct-state count as the number of subfamilies explicitly enumerated.
Step identifiers and dependencies
Rule: topbasis_all_subfamily_unions_complete_not_exponential_enumeration_claim · Uses: G001, S047
S049 · implies
Before
JUST-VERIFIED cover and ALL local refinement for ORIGINAL basis=[[0], [1], [2], [2, 3], [3], [4]];COMPLETE all-union family=[[], [0], [0, 1], [0, 1, 2], [0, 1, 2, 3], [0, 1, 2, 3, 4], [0, 1, 2, 4], [0, 1, 3], [0, 1, 3, 4], [0, 1, 4], [0, 2], [0, 2, 3], [0, 2, 3, 4], [0, 2, 4], [0, 3], [0, 3, 4], [0, 4], [1], [1, 2], [1, 2, 3], [1, 2, 3, 4], [1, 2, 4], [1, 3], [1, 3, 4], [1, 4], [2], [2, 3], [2, 3, 4], [2, 4], [3], [3, 4], [4]]
After
empty/full carrier present;ANY union of basis-unions flattens to a basis-union;for x in U intersection V choose basis B,C around x then its verified local refiner inside B intersection C,so U intersection V is itself a basis-union;this family IS a topology on Fin(5)
Conditions: none
Local refinement supplies a basis element around each point inside U∩V,and their union equals that intersection.This proves intersection closure without requiring the basis family itself intersection-closed.Arbitrary-union flattening,coverage of the full carrier and the empty union complete the topology axioms.
Step identifiers and dependencies
Rule: topbasis_basis_to_topology_arbitrary_union_and_local_intersection_proof · Uses: G001, S048