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R0051-26024 · q-20261015-advance-20261115-r06-minimum-vertex-cover-0099
Use complete unique vertices=[v0, v1, v2, v3, v4] and loop-free duplicate-free undirected edges=[[v0, v1], [v0, v2], [v0, v3], [v0, v4], [v1, v2], [v1, v3], [v1, v4], [v2, v3], [v2, v4], [v3, v4]]. An edge needs at least one selected endpoint. Preserve source canonical subset order. Mode=multiple_minima. Give (mode, edges, minimum_size, minimum_covers, mode_certificate, boundary).
G001 — Use complete unique vertices=[v0, v1, v2, v3, v4] and loop-free duplicate-free undirected edges=[[v0, v1], [v0, v2], [v0, v3], [v0, v4], [v1, v2], [v1, v3], [v1, v4], [v2, v3], [v2, v4], [v3, v4]]. An edge needs at least one selected endpoint. Preserve source canonical subset order. Mode=multiple_minima. Give (mode, edges, minimum_size, minimum_covers, mode_certificate, 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 = [mincover-empty-multiple-boundary]
After
derived = [[mode, edges, minimum_size, minimum_covers, mode_certificate, boundary]]
Conditions: none
List the full source field order without JavaScript.Original modes select contract6/4/6/6 and IVT8/7/8 roles,without padding or answer-shape inference.
Rule: source_ordered_fields_without_JS · Uses: G001
Before
source roles = [[v0, v1, v2, v3, v4], [[v0, v1], [v0, v2], [v0, v3], [v0, v4], [v1, v2], [v1, v3], [v1, v4], [v2, v3], [v2, v4], [v3, v4]], null]
After
derived = [[v0, v1, v2, v3, v4], [[v0, v1], [v0, v2], [v0, v3], [v0, v4], [v1, v2], [v1, v3], [v1, v4], [v2, v3], [v2, v4], [v3, v4]], null]
Conditions: none
The carrier is nonempty with at most12 vertices;sort endpoints by UTF16 then edge families in English.Reject loops/reversed duplicates.An omitted candidate stays null,not an invented empty candidate.
Rule: Mincover_complete_simple_graph_UTF16_English_and_optional_candidate · Uses: G001, S001
Before
source roles = [[v0, v1, v2, v3, v4], [[v0, v1], [v0, v2], [v0, v3], [v0, v4], [v1, v2], [v1, v3], [v1, v4], [v2, v3], [v2, v4], [v3, v4]], 32]
After
derived = [[15, 23, 27, 29, 30, 31], [[0, 0], [1, 4], [2, 1], [3, 7], [4, 0], [5, 5], [6, 2], [7, 9], [8, 0], [9, 4], [10, 1], [11, 8], [12, 0], [13, 6], [14, 3], [16, 0], [17, 4], [18, 1], [19, 7], [20, 0], [21, 5], [22, 2], [24, 0], [25, 4], [26, 1], [28, 0]], [[v0, v1, v2, v3], [v0, v1, v2, v4], [v0, v1, v3, v4], [v0, v2, v3, v4], [v1, v2, v3, v4], [v0, v1, v2, v3, v4]], [0, 0, 0, 0, 5, 1], 4, [[v0, v1, v2, v3], [v0, v1, v2, v4], [v0, v1, v3, v4], [v0, v2, v3, v4], [v1, v2, v3, v4]]]
Conditions: none
Audit every mask against every edge,retaining the first uncovered-edge refutation for each failure and every accepted set in source order.Zero smaller-size counts give a lower bound and actual minimum covers attain it.Keep all ties in numeric-mask order,not English/size order.
Rule: Mincover_ALL_masks_acceptance_refutations_ALLcovers_ALLminimizers · Uses: G001, S002
Before
source roles = [multiple_minima, [v0, v1, v2, v3, v4], [[v0, v1], [v0, v2], [v0, v3], [v0, v4], [v1, v2], [v1, v3], [v1, v4], [v2, v3], [v2, v4], [v3, v4]]]
After
derived = [4, [[v0, v1, v2, v3], [v0, v1, v2, v4], [v0, v1, v3, v4], [v0, v2, v3, v4], [v1, v2, v3, v4]], Yes]
Conditions: none
empty_graph means edgeless,not an empty vertex carrier;the unique minimum is empty.multiple_minima retains all of at least two solutions,not one witness as uniqueness.No Konig,weighted or directed extension.
Rule: Mincover_original_empty_OR_multiple_boundary_not_uniqueness_guess · Uses: G001, S003
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.(multiple_minima, [[v0, v1], [v0, v2], [v0, v3], [v0, v4], [v1, v2], [v1, v3], [v1, v4], [v2, v3], [v2, v4], [v3, v4]], 4, [[v0, v1, v2, v3], [v0, v1, v2, v4], [v0, v1, v3, v4], [v0, v2, v3, v4], [v1, v2, v3, v4]], Yes, empty_graph_has_unique_empty_minimum_cover_while_other_graphs_may_have_multiple_minima_without_Konig_or_weighted_claims)
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: 00c03639e070dfd572c99a4ee881fcd657f5a2bafec41f0aa8e5226b87b17ddbcandidate record SHA-256: 0ef8efb3778830ed5233cb20ef8df83cc84c3baa73cb422e5de9b9d39ac5ec2esemantic review receipt SHA-256: 90df1e889242e9afdf01e068788ee8500ef366ffcfc0496e30a6800de32631c3question-first witness SHA-256: ae1e849608590ca856c9f3b6911ba58e3bf53c608eb7369606192a770f48bd5cLean: NOT_RUN · human review: NOT_PERFORMED