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R0011-06097 · q-20261009-advance-20261013-r06-event-0072
S=Fin(8)={0,…,7}; F consists of unions of complete disjoint nonempty blocks=[[0, 3, 4], [1], [2], [5], [6], [7]]. Candidate E=[0]⊆S. Give (is E a union of whole blocks?, partially hit block-member lists, smallest F set containing E). Sort inside each block and order blocks numerically lexicographically.
G001 — S=Fin(8)={0,…,7}; F consists of unions of complete disjoint nonempty blocks=[[0, 3, 4], [1], [2], [5], [6], [7]]. Candidate E=[0]⊆S. Give (is E a union of whole blocks?, partially hit block-member lists, smallest F set containing E). Sort inside each block and order blocks numerically lexicographically.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
S=Fin(8)=[0, 1, 2, 3, 4, 5, 6, 7];declared event model=partition_unions
After
finite S alone does not specify its event family or a probability law
Conditions: none
A finite sample space does not automatically choose F=P(S).Use the declared event family;set data alone give no probabilities,equiprobability or independence.
Rule: declared_finite_event_family · Uses: G001
Before
ordered blocks=[[0, 3, 4], [1], [2], [5], [6], [7]];S=[0, 1, 2, 3, 4, 5, 6, 7]
After
nonempty,pairwise disjoint blocks cover S exactly
Conditions: none
Verify nonempty disjoint blocks covering all S;an arbitrary family of subsets is not automatically a partition.
Rule: complete_event_partition_premise · Uses: G001, S001
Before
block=[0, 3, 4];E selects=[0]
After
hit=Yes;whole block selected=No
Conditions: none
A member of F must include or omit each entire block.A partial hit is still a subset of S but is not in this declared F.
Rule: partition_event_block_audit · Uses: G001, S002
Before
block=[1];E selects=[]
After
hit=No;whole block selected=No
Conditions: none
A member of F must include or omit each entire block.A partial hit is still a subset of S but is not in this declared F.
Rule: partition_event_block_audit · Uses: G001, S003
Before
block=[2];E selects=[]
After
hit=No;whole block selected=No
Conditions: none
A member of F must include or omit each entire block.A partial hit is still a subset of S but is not in this declared F.
Rule: partition_event_block_audit · Uses: G001, S004
Before
block=[5];E selects=[]
After
hit=No;whole block selected=No
Conditions: none
A member of F must include or omit each entire block.A partial hit is still a subset of S but is not in this declared F.
Rule: partition_event_block_audit · Uses: G001, S005
Before
block=[6];E selects=[]
After
hit=No;whole block selected=No
Conditions: none
A member of F must include or omit each entire block.A partial hit is still a subset of S but is not in this declared F.
Rule: partition_event_block_audit · Uses: G001, S006
Before
block=[7];E selects=[]
After
hit=No;whole block selected=No
Conditions: none
A member of F must include or omit each entire block.A partial hit is still a subset of S but is not in this declared F.
Rule: partition_event_block_audit · Uses: G001, S007
Before
partial blocks=[[0, 3, 4]];all hit blocks=[[0, 3, 4]]
After
qualified event=No;smallest containing F-set=[0, 3, 4]
Conditions: none
Any block-union containing E must include every hit block.Their union itself belongs to F,so it is the smallest containing F-set,not a closure in an unspecified topology.
Rule: partition_union_minimal_event_hull · Uses: G001, S008
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.(No, [[0, 3, 4]], [0, 3, 4])
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: b6f65c9045b98fa2cc7c00070086881cff5ba2c61f701308f6b83b87e6df6f7bcandidate record SHA-256: 8af681924ca1be4080a1841bec3359967f223265f0ab7cc6e0b7e320e0a6f15bsemantic review receipt SHA-256: b9bbc1c947f477a9c4bb5a579368ac5265f2da4572ce612c30f3633ecaf4ae80question-first witness SHA-256: 93f720df3ab3af13a792a3e6b443728e3c2a56276df839911f36569333236c55Lean: NOT_RUN · human review: NOT_PERFORMED