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R0049-25499 · q-20261015-advance-20261114-r07-vertex-cover-0049
Use complete finite unique vertices=[v0, v1, v2, v3, v4, v5, v6, v7, v8], loop-free simple undirected edges=[[v0, v1], [v0, v2], [v0, v3], [v0, v4], [v0, v5], [v0, v6], [v0, v7], [v0, v8]] and unique candidate subset=[]. Edges normalize to sorted unordered pairs and are duplicate-free. A cover selects at least one endpoint of every edge; isolated vertices need not be selected. Give (candidate, edge_coverage_rows, uncovered_edge, cover, boundary).
G001 — Use complete finite unique vertices=[v0, v1, v2, v3, v4, v5, v6, v7, v8], loop-free simple undirected edges=[[v0, v1], [v0, v2], [v0, v3], [v0, v4], [v0, v5], [v0, v6], [v0, v7], [v0, v8]] and unique candidate subset=[]. Edges normalize to sorted unordered pairs and are duplicate-free. A cover selects at least one endpoint of every edge; isolated vertices need not be selected. Give (candidate, edge_coverage_rows, uncovered_edge, cover, 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 = [vcover-uncovered-edge-witness]
After
derived = [[candidate, edge_coverage_rows, uncovered_edge, cover, boundary]]
Conditions: none
List the full source field order without JavaScript.LP boundary modes all have6 fields but select different roles from the original mode,not answer shape.
Rule: source_ordered_fields_without_JS · Uses: G001
Before
source roles = [[v0, v1, v2, v3, v4, v5, v6, v7, v8], [[v0, v1], [v0, v2], [v0, v3], [v0, v4], [v0, v5], [v0, v6], [v0, v7], [v0, v8]], []]
After
derived = [[v0, v1, v2, v3, v4, v5, v6, v7, v8], [[v0, v1], [v0, v2], [v0, v3], [v0, v4], [v0, v5], [v0, v6], [v0, v7], [v0, v8]], []]
Conditions: none
The carrier is nonempty with at most12 vertices.Normalize endpoints by JS UTF16<,then source English canonical edge order.Reject loops/reversed duplicate edges;this is not a directed graph or edge cover.
Rule: Vcover_complete_simple_undirected_graph_UTF16_then_English_order · Uses: G001, S001
Before
source roles = [0, [v0, v1], []]
After
derived = [[No, No], No]
Conditions: none
This edge is covered iff at least one endpoint is selected.Check the entire edge table,not vertex count.
Rule: Vcover_EVERY_edge_selected_endpoint_disjunction · Uses: G001, S002
Before
source roles = [1, [v0, v2], []]
After
derived = [[No, No], No]
Conditions: none
This edge is covered iff at least one endpoint is selected.Check the entire edge table,not vertex count.
Rule: Vcover_EVERY_edge_selected_endpoint_disjunction · Uses: G001, S003
Before
source roles = [2, [v0, v3], []]
After
derived = [[No, No], No]
Conditions: none
This edge is covered iff at least one endpoint is selected.Check the entire edge table,not vertex count.
Rule: Vcover_EVERY_edge_selected_endpoint_disjunction · Uses: G001, S004
Before
source roles = [3, [v0, v4], []]
After
derived = [[No, No], No]
Conditions: none
This edge is covered iff at least one endpoint is selected.Check the entire edge table,not vertex count.
Rule: Vcover_EVERY_edge_selected_endpoint_disjunction · Uses: G001, S005
Before
source roles = [4, [v0, v5], []]
After
derived = [[No, No], No]
Conditions: none
This edge is covered iff at least one endpoint is selected.Check the entire edge table,not vertex count.
Rule: Vcover_EVERY_edge_selected_endpoint_disjunction · Uses: G001, S006
Before
source roles = [5, [v0, v6], []]
After
derived = [[No, No], No]
Conditions: none
This edge is covered iff at least one endpoint is selected.Check the entire edge table,not vertex count.
Rule: Vcover_EVERY_edge_selected_endpoint_disjunction · Uses: G001, S007
Before
source roles = [6, [v0, v7], []]
After
derived = [[No, No], No]
Conditions: none
This edge is covered iff at least one endpoint is selected.Check the entire edge table,not vertex count.
Rule: Vcover_EVERY_edge_selected_endpoint_disjunction · Uses: G001, S008
Before
source roles = [7, [v0, v8], []]
After
derived = [[No, No], No]
Conditions: none
This edge is covered iff at least one endpoint is selected.Check the entire edge table,not vertex count.
Rule: Vcover_EVERY_edge_selected_endpoint_disjunction · Uses: G001, S009
Before
source roles = [[[[v0, v1], No], [[v0, v2], No], [[v0, v3], No], [[v0, v4], No], [[v0, v5], No], [[v0, v6], No], [[v0, v7], No], [[v0, v8], No]]]
After
derived = [No, [v0, v1], []]
Conditions: none
Conjoin all edges and retain the first uncovered edge in source order,or null,notFalse.Isolated vertices have no incident coverage obligation and are optional.
Rule: Vcover_COMPLETE_edge_conjunction_FIRST_uncovered_or_NULL · Uses: G001, S010
Before
source roles = [[v0, v1], []]
After
derived = [No]
Conditions: none
Both endpoints of the first uncovered edge lie outside C,refuting the universal cover condition.An out-of-carrier or duplicate candidate is a type failure,not this well-typed false witness.
Rule: Vcover_actual_uncovered_edge_refutes_universal · Uses: G001, S011
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.([], [[[v0, v1], No], [[v0, v2], No], [[v0, v3], No], [[v0, v4], No], [[v0, v5], No], [[v0, v6], No], [[v0, v7], No], [[v0, v8], No]], [v0, v1], No, one_edge_with_both_endpoints_outside_C_refutes_vertex_cover)
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: 3c433abe0b070e2a5499cfcfd8db4ef25a2067c4229c9101bcf535da29252026candidate record SHA-256: fc54a0abf39fd051761dbd96f14d68fdc91d114e69706c73338db0546cf5186bsemantic review receipt SHA-256: 6212b8b0b3b273f3a514e26ebd17d7e81219c56a8f47e40047ee2b6bf01d1ba3question-first witness SHA-256: f32a23d5e1b66c7794c60def58741bcdfacff24f3cdfc9bc48109c8f71543cf3Lean: NOT_RUN · human review: NOT_PERFORMED