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R0052-26512 · q-20261015-advance-20261117-r07-topological-connected-component-0087
Use mode=cardinality_not_definition and complete model={"points":[0,1,2,3,4],"topology":[[],[0,1,2,3,4],[0],[1,2,3,4]],"basepoint":0}. Give (mode, basepoint, own_component, larger_other_component, own_cardinality, other_cardinality, global_cardinality_definition_claim, boundary).
G001 — Use mode=cardinality_not_definition and complete model={"points":[0,1,2,3,4],"topology":[[],[0,1,2,3,4],[0],[1,2,3,4]],"basepoint":0}. Give (mode, basepoint, own_component, larger_other_component, own_cardinality, other_cardinality, global_cardinality_definition_claim, boundary).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.
Before
source roles = [component-boundary]
After
derived = [[mode, basepoint, own_component, larger_component, own_size, larger_size, global_maximum_definition, certificate]]
Conditions: none
List this operation/mode exact source field order without JavaScript.Keep Boolean,string and null roles distinct;do not pad fields or infer the interface from answer shape.
Rule: source_ordered_fields_without_JS · Uses: G001
Before
source roles = [component ORIGINAL space, [0, 1, 2, 3, 4], [[], [0, 1, 2, 3, 4], [0], [1, 2, 3, 4]], No]
After
derived = [[0, 1, 2, 3, 4], [[], [0, 1, 2, 3, 4], [0], [1, 2, 3, 4]], [[0, 0, 0, 0, Yes, Yes], [0, 1, 31, 0, Yes, Yes], [0, 2, 1, 0, Yes, Yes], [0, 3, 30, 0, Yes, Yes], [1, 0, 31, 0, Yes, Yes], [1, 1, 31, 31, Yes, Yes], [1, 2, 31, 1, Yes, Yes], [1, 3, 31, 30, Yes, Yes], [2, 0, 1, 0, Yes, Yes], [2, 1, 31, 1, Yes, Yes], [2, 2, 1, 1, Yes, Yes], [2, 3, 31, 0, Yes, Yes], [3, 0, 30, 0, Yes, Yes], [3, 1, 31, 30, Yes, Yes], [3, 2, 31, 0, Yes, Yes], [3, 3, 30, 30, Yes, Yes]]]
Conditions: none
Retain JSON types and English source ordering of sets and open families.Check union/intersection for every open pair and empty/full sets;in a finite family binary closure gives arbitrary-union closure,not sampled qualification.
Rule: Topology_full_typed_carrier_and_ALL_pair_axioms · Uses: G001, S001
Before
source roles = [[0, 1, 2, 3, 4], [[], [0, 1, 2, 3, 4], [0], [1, 2, 3, 4]]]
After
derived = [[[1, [0], [[], [0]], null, Yes], [2, [1], [[], [1]], null, Yes], [3, [0, 1], [[], [0, 1], [0], [1]], [0], No], [4, [2], [[], [2]], null, Yes], [5, [0, 2], [[], [0, 2], [0], [2]], [0], No], [6, [1, 2], [[], [1, 2]], null, Yes], [7, [0, 1, 2], [[], [0, 1, 2], [0], [1, 2]], [0], No], [8, [3], [[], [3]], null, Yes], [9, [0, 3], [[], [0, 3], [0], [3]], [0], No], [10, [1, 3], [[], [1, 3]], null, Yes], [11, [0, 1, 3], [[], [0, 1, 3], [0], [1, 3]], [0], No], [12, [2, 3], [[], [2, 3]], null, Yes], [13, [0, 2, 3], [[], [0, 2, 3], [0], [2, 3]], [0], No], [14, [1, 2, 3], [[], [1, 2, 3]], null, Yes], [15, [0, 1, 2, 3], [[], [0, 1, 2, 3], [0], [1, 2, 3]], [0], No], [16, [4], [[], [4]], null, Yes], [17, [0, 4], [[], [0, 4], [0], [4]], [0], No], [18, [1, 4], [[], [1, 4]], null, Yes], [19, [0, 1, 4], [[], [0, 1, 4], [0], [1, 4]], [0], No], [20, [2, 4], [[], [2, 4]], null, Yes], [21, [0, 2, 4], [[], [0, 2, 4], [0], [2, 4]], [0], No], [22, [1, 2, 4], [[], [1, 2, 4]], null, Yes], [23, [0, 1, 2, 4], [[], [0, 1, 2, 4], [0], [1, 2, 4]], [0], No], [24, [3, 4], [[], [3, 4]], null, Yes], [25, [0, 3, 4], [[], [0, 3, 4], [0], [3, 4]], [0], No], [26, [1, 3, 4], [[], [1, 3, 4]], null, Yes], [27, [0, 1, 3, 4], [[], [0, 1, 3, 4], [0], [1, 3, 4]], [0], No], [28, [2, 3, 4], [[], [2, 3, 4]], null, Yes], [29, [0, 2, 3, 4], [[], [0, 2, 3, 4], [0], [2, 3, 4]], [0], No], [30, [1, 2, 3, 4], [[], [1, 2, 3, 4]], null, Yes], [31, [0, 1, 2, 3, 4], [[], [0, 1, 2, 3, 4], [0], [1, 2, 3, 4]], [0], No]], [[0], [1], [2], [1, 2], [3], [1, 3], [2, 3], [1, 2, 3], [4], [1, 4], [2, 4], [1, 2, 4], [3, 4], [1, 3, 4], [2, 3, 4], [1, 2, 3, 4]]]
Conditions: none
For every nonempty subset generate all subspace opens O intersect A,then test every proper nonempty part and its relative complement for an open separation.Absence qualifies connectedness;retain numeric-mask order,including singletons.
Rule: Component_ALL_nonempty_masks_induced_topology_no_clopen_separation · Uses: G001, S002
Before
source roles = [[[0], [1], [2], [1, 2], [3], [1, 3], [2, 3], [1, 2, 3], [4], [1, 4], [2, 4], [1, 2, 4], [3, 4], [1, 3, 4], [2, 3, 4], [1, 2, 3, 4]]]
After
derived = [[[0], [1, 2, 3, 4]], Yes, Yes, [Yes, Yes], [Yes, Yes]]
Conditions: none
Select inclusion-maximal members of the complete connected family,not maximum cardinality.Intersecting components have connected union and maximality makes them equal;singletons give coverage.The closure of a connected set remains connected,so components are closed;finitely many closed components make every complement closed,hence each is open here.
Rule: Component_inclusion_MAXIMAL_partition_finite_clopen · Uses: G001, S003
Before
source roles = [0, [[0]]]
After
derived = [[0], [0], Yes, Yes]
Conditions: none
All listed connected sets share the basepoint.A separation of their union puts every connected set on the side containing that point,leaving the other side empty,a contradiction.The union is connected and contains all such sets,hence is the unique inclusion-maximal component.
Rule: Component_basepoint_common_point_union_AND_maximality · Uses: G001, S004
Before
source roles = [0, [0], [[0], [1, 2, 3, 4]]]
After
derived = [[1, 2, 3, 4], 1, 4, No]
Conditions: none
Choose the first larger OTHER component in source canonical order.The own component admits no connected enlargement but can be smaller than another;maximal is not global maximum cardinality.
Rule: Component_FIRST_canonical_larger_OTHER_not_global_maximum · Uses: G001, S005
S006 · premise_and_result_audit — Reverse-check the complete original data,types,field order and premises;finite audits are not formal proofs of infinite theorems.S005 · premise_and_result_audit — Reverse-check the complete original data,types,field order and premises;finite audits are not formal proofs of infinite theorems.S004 · premise_and_result_audit — Reverse-check the complete original data,types,field order and premises;finite audits are not formal proofs of infinite theorems.S003 · premise_and_result_audit — Reverse-check the complete original data,types,field order and premises;finite audits are not formal proofs of infinite theorems.S002 · premise_and_result_audit — Reverse-check the complete original data,types,field order and premises;finite audits are not formal proofs of infinite theorems.S001 · premise_and_result_audit — Reverse-check the complete original data,types,field order and premises;finite audits are not formal proofs of infinite theorems.(cardinality_not_definition, 0, [0], [1, 2, 3, 4], 1, 4, No, component_is_inclusion_maximal_not_global_maximum_cardinality)
The full substitutions and premise checks give the original answer; source identity,format and status are unchanged.
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.
input commit: 0f7be61e2c22317a4075561dc822c87d881a50dboriginal question SHA-256: fd72b8ec419c1107789faa92e288f07354ee1540509b5a8211740867a9c1264ccandidate record SHA-256: 877943bfc1b9bdb7f8285cb692e9770e2527174f48ffd93c67f0fecec800d440semantic review receipt SHA-256: 729b25cbf6bf5cb1d07a21a511ae216ec5fdd54a4a7aa2f7c21156b81ff37252question-first witness SHA-256: d1ab788770eaad04f986986e65afb0d55791e8a8c0d747e9a2b499f79fe78df2Lean: NOT_RUN · human review: NOT_PERFORMED