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R0051-26053 · q-20261015-advance-20261115-r06-minimum-vertex-cover-0053
Use complete unique vertices=[c0, l0_0, l0_1, l0_2, l0_3] and loop-free duplicate-free undirected edges=[[c0, l0_0], [c0, l0_1], [c0, l0_2], [c0, l0_3]], candidate cover=[l0_0, l0_1, l0_2, l0_3]. An edge needs at least one selected endpoint. Preserve source canonical subset order. Give (candidate, deletion_rows, inclusion_minimal, minimum_size, smaller_cover, minimum, certificate).
G001 — Use complete unique vertices=[c0, l0_0, l0_1, l0_2, l0_3] and loop-free duplicate-free undirected edges=[[c0, l0_0], [c0, l0_1], [c0, l0_2], [c0, l0_3]], candidate cover=[l0_0, l0_1, l0_2, l0_3]. An edge needs at least one selected endpoint. Preserve source canonical subset order. Give (candidate, deletion_rows, inclusion_minimal, minimum_size, smaller_cover, minimum, certificate).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-minimal-not-minimum]
After
derived = [[candidate, deletion_rows, inclusion_minimal, minimum_size, smaller_cover, minimum, certificate]]
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 = [[c0, l0_0, l0_1, l0_2, l0_3], [[c0, l0_0], [c0, l0_1], [c0, l0_2], [c0, l0_3]], [l0_0, l0_1, l0_2, l0_3]]
After
derived = [[c0, l0_0, l0_1, l0_2, l0_3], [[c0, l0_0], [c0, l0_1], [c0, l0_2], [c0, l0_3]], [l0_0, l0_1, l0_2, l0_3]]
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 = [[c0, l0_0, l0_1, l0_2, l0_3], [[c0, l0_0], [c0, l0_1], [c0, l0_2], [c0, l0_3]], 32]
After
derived = [[1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, 29, 30, 31], [[0, 0], [2, 1], [4, 0], [6, 2], [8, 0], [10, 1], [12, 0], [14, 3], [16, 0], [18, 1], [20, 0], [22, 2], [24, 0], [26, 1], [28, 0]], [[c0], [c0, l0_0], [c0, l0_1], [c0, l0_0, l0_1], [c0, l0_2], [c0, l0_0, l0_2], [c0, l0_1, l0_2], [c0, l0_0, l0_1, l0_2], [c0, l0_3], [c0, l0_0, l0_3], [c0, l0_1, l0_3], [c0, l0_0, l0_1, l0_3], [c0, l0_2, l0_3], [c0, l0_0, l0_2, l0_3], [c0, l0_1, l0_2, l0_3], [l0_0, l0_1, l0_2, l0_3], [c0, l0_0, l0_1, l0_2, l0_3]], [0, 1, 4, 6, 5, 1], 1, [[c0]]]
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 = [[l0_0, l0_1, l0_2, l0_3], [[c0, l0_0], [c0, l0_1], [c0, l0_2], [c0, l0_3]]]
After
derived = [[[[c0, l0_0], Yes], [[c0, l0_1], Yes], [[c0, l0_2], Yes], [[c0, l0_3], Yes]], 4, 1, [c0]]
Conditions: none
First qualify an actual candidate cover.The smaller witness is the first minimum from the complete source list,not an arbitrary small set.
Rule: Mincover_actual_candidate_complete_cover_and_smaller_witness · Uses: G001, S003
Before
source roles = [[l0_0, l0_1, l0_2, l0_3], [[l0_0, [l0_1, l0_2, l0_3], No], [l0_1, [l0_0, l0_2, l0_3], No], [l0_2, [l0_0, l0_1, l0_3], No], [l0_3, [l0_0, l0_1, l0_2], No]]]
After
derived = [Yes, 1, [c0], No]
Conditions: none
Every single deletion loses coverage,so the candidate is inclusion-minimal.But exhaustive search supplies a strictly smaller cover,so it is not minimum.Superset monotonicity makes deletion tests cover all proper-subset obligations,not maximum or optimal size.
Rule: Mincover_ALL_single_deletions_minimal_BUT_smaller_cover_exists · Uses: G001, S004
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.([l0_0, l0_1, l0_2, l0_3], [[l0_0, [l0_1, l0_2, l0_3], No], [l0_1, [l0_0, l0_2, l0_3], No], [l0_2, [l0_0, l0_1, l0_3], No], [l0_3, [l0_0, l0_1, l0_2], No]], Yes, 1, [c0], No, inclusion_minimal_is_not_minimum)
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: ece6911b30974946d66fa597e7a15d6a33088b3575a7e9959293bbd0d08a7124candidate record SHA-256: 0afbc49e9aa6e13a5d82717eff0b7e2c2c6520585df7c01838c7c2579d3cb47dsemantic review receipt SHA-256: 90df1e889242e9afdf01e068788ee8500ef366ffcfc0496e30a6800de32631c3question-first witness SHA-256: ae1e849608590ca856c9f3b6911ba58e3bf53c608eb7369606192a770f48bd5cLean: NOT_RUN · human review: NOT_PERFORMED