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← Question derivation index · Original question batch
R0051-26022 · q-20261015-advance-20261115-r06-minimum-vertex-cover-0097
Use complete unique vertices=[v0, v1, v2, v3, v4, v5, v6, v7, v8, v9, v10, v11] and loop-free duplicate-free undirected edges=[[v0, v1], [v1, v2], [v2, v3], [v3, v4], [v4, v5], [v5, v6], [v6, v7], [v7, v8], [v8, v9], [v9, v10], [v10, v11], [v11, v0]]. 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, v5, v6, v7, v8, v9, v10, v11] and loop-free duplicate-free undirected edges=[[v0, v1], [v1, v2], [v2, v3], [v3, v4], [v4, v5], [v5, v6], [v6, v7], [v7, v8], [v8, v9], [v9, v10], [v10, v11], [v11, v0]]. 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, v5, v6, v7, v8, v9, v10, v11], [[v0, v1], [v1, v2], [v2, v3], [v3, v4], [v4, v5], [v5, v6], [v6, v7], [v7, v8], [v8, v9], [v9, v10], [v10, v11], [v11, v0]], null]
After
derived = [[v0, v1, v10, v11, v2, v3, v4, v5, v6, v7, v8, v9], [[v0, v1], [v0, v11], [v1, v2], [v10, v11], [v10, v9], [v2, v3], [v3, v4], [v4, v5], [v5, v6], [v6, v7], [v7, v8], [v8, v9]], 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, v10, v11, v2, v3, v4, v5, v6, v7, v8, v9], [[v0, v1], [v0, v11], [v1, v2], [v10, v11], [v10, v9], [v2, v3], [v3, v4], [v4, v5], [v5, v6], [v6, v7], [v7, v8], [v8, v9]], 4096]
After
derived = [[1365, 1367, 1373, 1374, 1375, 1383, 1390, 1391, 1397, 1399, 1405, 1406, 1407, 1447, 1454, 1455, 1461, 1463, 1469, 1470, 1471, 1493, 1495, 1501, 1502, 1503, 1511, 1518, 1519, 1525, 1527, 1533, 1534, 1535, 1703, 1710, 1711, 1717, 1719, 1725, 1726, 1727, 1749, 1751, 1757, 1758, 1759, 1767, 1774, 1775, 1781, 1783, 1789, 1790, 1791, 1877, 1879, 1885, 1886, 1887, 1895, 1902, 1903, 1909, 1911, 1917, 1918, 1919, 1959, 1966, 1967, 1973, 1975, 1981, 1982, 1983, 2005, 2007, 2013, 2014, 2015, 2023, 2030, 2031, 2037, 2039, 2045, 2046, 2047, 2727, 2730, 2731, 2734, 2735, 2741, 2743, 2745, 2746, 2747, 2749, 2750, 2751, 2773, 2775, 2777, 2778, 2779, 2781, 2782, 2783, 2791, 2794, 2795, 2798, 2799, 2805, 2807, 2809, 2810, 2811, 2813, 2814, 2815, 2901, 2903, 2905, 2906, 2907, 2909, 2910, 2911, 2919, 2922, 2923, 2926, 2927, 2933, 2935, 2937, 2938, 2939, 2941, 2942, 2943, 2983, 2986, 2987, 2990, 2991, 2997, 2999, 3001, 3002, 3003, 3005, 3006, 3007, 3029, 3031, 3033, 3034, 3035, 3037, 3038, 3039, 3047, 3050, 3051, 3054, 3055, 3061, 3063, 3065, 3066, 3067, 3069, 3070, 3071, 3413, 3415, 3417, 3418, 3419, 3421, 3422, 3423, 3431, 3434, 3435, 3438, 3439, 3445, 3447, 3449, 3450, 3451, 3453, 3454, 3455, 3495, 3498, 3499, 3502, 3503, 3509, 3511, 3513, 3514, 3515, 3517, 3518, 3519, 3541, 3543, 3545, 3546, 3547, 3549, 3550, 3551, 3559, 3562, 3563, 3566, 3567, 3573, 3575, 3577, 3578, 3579, 3581, 3582, 3583, 3751, 3754, 3755, 3758, 3759, 3765, 3767, 3769, 3770, 3771, 3773, 3774, 3775, 3797, 3799, 3801, 3802, 3803, 3805, 3806, 3807, 3815, 3818, 3819, 3822, 3823, 3829, 3831, 3833, 3834, 3835, 3837, 3838, 3839, 3925, 3927, 3929, 3930, 3931, 3933, 3934, 3935, 3943, 3946, 3947, 3950, 3951, 3957, 3959, 3961, 3962, 3963, 3965, 3966, 3967, 4007, 4010, 4011, 4014, 4015, 4021, 4023, 4025, 4026, 4027, 4029, 4030, 4031, 4053, 4055, 4057, 4058, 4059, 4061, 4062, 4063, 4071, 4074, 4075, 4078, 4079, 4085, 4087, 4089, 4090, 4091, 4093, 4094, 4095], [[0, 0], [1, 2], [2, 1], [3, 3], [4, 0], [5, 2], [6, 1], [7, 5], [8, 0], [9, 2], [10, 4], [11, 4], [12, 0], [13, 2], [14, 5], [15, 5], [16, 0], [17, 3], [18, 1], [19, 3], [20, 0], [21, 6], [22, 1], [23, 6], [24, 0], [25, 4], [26, 4], [27, 4], [28, 0], [29, 6], [30, 6], [31, 6], [32, 0], [33, 2], [34, 1], [35, 3], [36, 0], [37, 2], [38, 1], [39, 7], [40, 0], [41, 2], [42, 4], [43, 4], [44, 0], [45, 2], [46, 7], [47, 7], [48, 0], [49, 3], [50, 1], [51, 3], [52, 0], [53, 7], [54, 1], [55, 7], [56, 0], [57, 4], [58, 4], [59, 4], [60, 0], [61, 7], [62, 7], [63, 7], [64, 0], [65, 2], [66, 1], [67, 3], [68, 0], [69, 2], [70, 1], [71, 5], [72, 0], [73, 2], [74, 4], [75, 4], [76, 0], [77, 2], [78, 5], [79, 5], [80, 0], [81, 3], [82, 1], [83, 3], [84, 0], [85, 8], [86, 1], [87, 8], [88, 0], [89, 4], [90, 4], [91, 4], [92, 0], [93, 8], [94, 8], [95, 8], [96, 0], [97, 2], [98, 1], [99, 3], [100, 0], [101, 2], [102, 1], [103, 8], [104, 0], [105, 2], [106, 4], [107, 4], [108, 0], [109, 2], [110, 8], [111, 8], [112, 0], [113, 3], [114, 1], [115, 3], [116, 0], [117, 8], [118, 1], [119, 8], [120, 0], [121, 4], [122, 4], [123, 4], [124, 0], [125, 8], [126, 8], [127, 8], [128, 0], [129, 2], [130, 1], [131, 3], [132, 0], [133, 2], [134, 1], [135, 5], [136, 0], [137, 2], [138, 4], [139, 4], [140, 0], [141, 2], [142, 5], [143, 5], [144, 0], [145, 3], [146, 1], [147, 3], [148, 0], [149, 6], [150, 1], [151, 6], [152, 0], [153, 4], [154, 4], [155, 4], [156, 0], [157, 6], [158, 6], [159, 6], [160, 0], [161, 2], [162, 1], [163, 3], [164, 0], [165, 2], [166, 1], [167, 9], [168, 0], [169, 2], [170, 4], [171, 4], [172, 0], [173, 2], [174, 9], [175, 9], [176, 0], [177, 3], [178, 1], [179, 3], [180, 0], [181, 9], [182, 1], [183, 9], [184, 0], [185, 4], [186, 4], [187, 4], [188, 0], [189, 9], [190, 9], [191, 9], [192, 0], [193, 2], [194, 1], [195, 3], [196, 0], [197, 2], [198, 1], [199, 5], [200, 0], [201, 2], [202, 4], [203, 4], [204, 0], [205, 2], [206, 5], [207, 5], [208, 0], [209, 3], [210, 1], [211, 3], [212, 0], [213, 9], [214, 1], [215, 9], [216, 0], [217, 4], [218, 4], [219, 4], [220, 0], [221, 9], [222, 9], [223, 9], [224, 0], [225, 2], [226, 1], [227, 3], [228, 0], [229, 2], [230, 1], [231, 9], [232, 0], [233, 2], [234, 4], [235, 4], [236, 0], [237, 2], [238, 9], [239, 9], [240, 0], [241, 3], [242, 1], [243, 3], [244, 0], [245, 9], [246, 1], [247, 9], [248, 0], [249, 4], [250, 4], [251, 4], [252, 0], [253, 9], [254, 9], [255, 9], [256, 0], [257, 2], [258, 1], [259, 3], [260, 0], [261, 2], [262, 1], [263, 5], [264, 0], [265, 2], [266, 4], [267, 4], [268, 0], [269, 2], [270, 5], [271, 5], [272, 0], [273, 3], [274, 1], [275, 3], [276, 0], [277, 6], [278, 1], [279, 6], [280, 0], [281, 4], [282, 4], [283, 4], [284, 0], [285, 6], [286, 6], [287, 6], [288, 0], [289, 2], [290, 1], [291, 3], [292, 0], [293, 2], [294, 1], [295, 7], [296, 0], [297, 2], [298, 4], [299, 4], [300, 0], [301, 2], [302, 7], [303, 7], [304, 0], [305, 3], [306, 1], [307, 3], [308, 0], [309, 7], [310, 1], [311, 7], [312, 0], [313, 4], [314, 4], [315, 4], [316, 0], [317, 7], [318, 7], [319, 7], [320, 0], [321, 2], [322, 1], [323, 3], [324, 0], [325, 2], [326, 1], [327, 5], [328, 0], [329, 2], [330, 4], [331, 4], [332, 0], [333, 2], [334, 5], [335, 5], [336, 0], [337, 3], [338, 1], [339, 3], [340, 0], [341, 10], [342, 1], [343, 10], [344, 0], [345, 4], [346, 4], [347, 4], [348, 0], [349, 10], [350, 10], [351, 10], [352, 0], [353, 2], [354, 1], [355, 3], [356, 0], [357, 2], [358, 1], [359, 10], [360, 0], [361, 2], [362, 4], [363, 4], [364, 0], [365, 2], [366, 10], [367, 10], [368, 0], [369, 3], [370, 1], [371, 3], [372, 0], [373, 10], [374, 1], [375, 10], [376, 0], [377, 4], [378, 4], [379, 4], [380, 0], [381, 10], [382, 10], [383, 10], [384, 0], [385, 2], [386, 1], [387, 3], [388, 0], [389, 2], [390, 1], [391, 5], [392, 0], [393, 2], [394, 4], [395, 4], [396, 0], [397, 2], [398, 5], [399, 5], [400, 0], [401, 3], [402, 1], [403, 3], [404, 0], [405, 6], [406, 1], [407, 6], [408, 0], [409, 4], [410, 4], [411, 4], [412, 0], [413, 6], [414, 6], [415, 6], [416, 0], [417, 2], [418, 1], [419, 3], [420, 0], [421, 2], [422, 1], [423, 10], [424, 0], [425, 2], [426, 4], [427, 4], [428, 0], [429, 2], [430, 10], [431, 10], [432, 0], [433, 3], [434, 1], [435, 3], [436, 0], [437, 10], [438, 1], [439, 10], [440, 0], [441, 4], [442, 4], [443, 4], [444, 0], [445, 10], [446, 10], [447, 10], [448, 0], [449, 2], [450, 1], [451, 3], [452, 0], [453, 2], [454, 1], [455, 5], [456, 0], [457, 2], [458, 4], [459, 4], [460, 0], [461, 2], [462, 5], [463, 5], [464, 0], [465, 3], [466, 1], [467, 3], [468, 0], [469, 10], [470, 1], [471, 10], [472, 0], [473, 4], [474, 4], [475, 4], [476, 0], [477, 10], [478, 10], [479, 10], [480, 0], [481, 2], [482, 1], [483, 3], [484, 0], [485, 2], [486, 1], [487, 10], [488, 0], [489, 2], [490, 4], [491, 4], [492, 0], [493, 2], [494, 10], [495, 10], [496, 0], [497, 3], [498, 1], [499, 3], [500, 0], [501, 10], [502, 1], [503, 10], [504, 0], [505, 4], [506, 4], [507, 4], [508, 0], [509, 10], [510, 10], [511, 10], [512, 0], [513, 2], [514, 1], [515, 3], [516, 0], [517, 2], [518, 1], [519, 5], [520, 0], [521, 2], [522, 4], [523, 4], [524, 0], [525, 2], [526, 5], [527, 5], [528, 0], [529, 3], [530, 1], [531, 3], [532, 0], [533, 6], [534, 1], [535, 6], [536, 0], [537, 4], [538, 4], [539, 4], [540, 0], [541, 6], [542, 6], [543, 6], [544, 0], [545, 2], [546, 1], [547, 3], [548, 0], [549, 2], [550, 1], [551, 7], [552, 0], [553, 2], [554, 4], [555, 4], [556, 0], [557, 2], [558, 7], [559, 7], [560, 0], [561, 3], [562, 1], [563, 3], [564, 0], [565, 7], [566, 1], [567, 7], [568, 0], [569, 4], [570, 4], [571, 4], [572, 0], [573, 7], [574, 7], [575, 7], [576, 0], [577, 2], [578, 1], [579, 3], [580, 0], [581, 2], [582, 1], [583, 5], [584, 0], [585, 2], [586, 4], [587, 4], [588, 0], [589, 2], [590, 5], [591, 5], [592, 0], [593, 3], [594, 1], [595, 3], [596, 0], [597, 8], [598, 1], [599, 8], [600, 0], [601, 4], [602, 4], [603, 4], [604, 0], [605, 8], [606, 8], [607, 8], [608, 0], [609, 2], [610, 1], [611, 3], [612, 0], [613, 2], [614, 1], [615, 8], [616, 0], [617, 2], [618, 4], [619, 4], [620, 0], [621, 2], [622, 8], [623, 8], [624, 0], [625, 3], [626, 1], [627, 3], [628, 0], [629, 8], [630, 1], [631, 8], [632, 0], [633, 4], [634, 4], [635, 4], [636, 0], [637, 8], [638, 8], [639, 8], [640, 0], [641, 2], [642, 1], [643, 3], [644, 0], [645, 2], [646, 1], [647, 5], [648, 0], [649, 2], [650, 4], [651, 4], [652, 0], [653, 2], [654, 5], [655, 5], [656, 0], [657, 3], [658, 1], [659, 3], [660, 0], [661, 6], [662, 1], [663, 6], [664, 0], [665, 4], [666, 4], [667, 4], [668, 0], [669, 6], [670, 6], [671, 6], [672, 0], [673, 2], [674, 1], [675, 3], [676, 0], [677, 2], [678, 1], [679, 11], [680, 0], [681, 2], [682, 4], [683, 4], [684, 0], [685, 2], [686, 11], [687, 11], [688, 0], [689, 3], [690, 1], [691, 3], [692, 0], [693, 11], [694, 1], [695, 11], [696, 0], [697, 4], [698, 4], [699, 4], [700, 0], [701, 11], [702, 11], [703, 11], [704, 0], [705, 2], [706, 1], [707, 3], [708, 0], [709, 2], [710, 1], [711, 5], [712, 0], [713, 2], [714, 4], [715, 4], [716, 0], [717, 2], [718, 5], [719, 5], [720, 0], [721, 3], [722, 1], [723, 3], [724, 0], [725, 11], [726, 1], [727, 11], [728, 0], [729, 4], [730, 4], [731, 4], [732, 0], [733, 11], [734, 11], [735, 11], [736, 0], [737, 2], [738, 1], [739, 3], [740, 0], [741, 2], [742, 1], [743, 11], [744, 0], [745, 2], [746, 4], [747, 4], [748, 0], [749, 2], [750, 11], [751, 11], [752, 0], [753, 3], [754, 1], [755, 3], [756, 0], [757, 11], [758, 1], [759, 11], [760, 0], [761, 4], [762, 4], [763, 4], [764, 0], [765, 11], [766, 11], [767, 11], [768, 0], [769, 2], [770, 1], [771, 3], [772, 0], [773, 2], [774, 1], [775, 5], [776, 0], [777, 2], [778, 4], [779, 4], [780, 0], [781, 2], [782, 5], [783, 5], [784, 0], [785, 3], [786, 1], [787, 3], [788, 0], [789, 6], [790, 1], [791, 6], [792, 0], [793, 4], [794, 4], [795, 4], [796, 0], [797, 6], [798, 6], [799, 6], [800, 0], [801, 2], [802, 1], [803, 3], [804, 0], [805, 2], [806, 1], [807, 7], [808, 0], [809, 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v3, v4, v5, v6, v7, v9], [v0, v1, v10, v11, v2, v3, v4, v5, v6, v7, v9], [v0, v10, v2, v4, v6, v8, v9], [v0, v1, v10, v2, v4, v6, v8, v9], [v0, v11, v2, v4, v6, v8, v9], [v1, v11, v2, v4, v6, v8, v9], [v0, v1, v11, v2, v4, v6, v8, v9], [v0, v10, v11, v2, v4, v6, v8, v9], [v1, v10, v11, v2, v4, v6, v8, v9], [v0, v1, v10, v11, v2, v4, v6, v8, v9], [v0, v1, v10, v3, v4, v6, v8, v9], [v1, v11, v3, v4, v6, v8, v9], [v0, v1, v11, v3, v4, v6, v8, v9], [v1, v10, v11, v3, v4, v6, v8, v9], [v0, v1, v10, v11, v3, v4, v6, v8, v9], [v0, v10, v2, v3, v4, v6, v8, v9], [v0, v1, v10, v2, v3, v4, v6, v8, v9], [v0, v11, v2, v3, v4, v6, v8, v9], [v1, v11, v2, v3, v4, v6, v8, v9], [v0, v1, v11, v2, v3, v4, v6, v8, v9], [v0, v10, v11, v2, v3, v4, v6, v8, v9], [v1, v10, v11, v2, v3, v4, v6, v8, v9], [v0, v1, v10, v11, v2, v3, v4, v6, v8, v9], [v0, v1, v10, v3, v5, v6, v8, v9], [v1, v11, v3, v5, v6, v8, v9], [v0, v1, v11, v3, v5, v6, v8, v9], [v1, v10, v11, v3, v5, v6, v8, v9], [v0, v1, v10, v11, v3, v5, v6, v8, v9], [v0, v10, v2, v3, v5, v6, v8, v9], [v0, v1, v10, v2, v3, v5, v6, v8, v9], [v0, v11, v2, v3, v5, v6, v8, v9], [v1, v11, v2, v3, v5, v6, v8, v9], [v0, v1, v11, v2, v3, v5, v6, v8, v9], [v0, v10, v11, v2, v3, v5, v6, v8, v9], [v1, v10, v11, v2, v3, v5, v6, v8, v9], [v0, v1, v10, v11, v2, v3, v5, v6, v8, v9], [v0, v10, v2, v4, v5, v6, v8, v9], [v0, v1, v10, v2, v4, v5, v6, v8, v9], [v0, v11, v2, v4, v5, v6, v8, v9], [v1, v11, v2, v4, v5, v6, v8, v9], [v0, v1, v11, v2, v4, v5, v6, v8, v9], [v0, v10, v11, v2, v4, v5, v6, v8, v9], [v1, v10, v11, v2, v4, v5, v6, v8, v9], [v0, v1, v10, v11, v2, v4, v5, v6, v8, v9], [v0, v1, v10, v3, v4, v5, v6, v8, v9], [v1, v11, v3, v4, v5, v6, v8, v9], [v0, v1, v11, v3, v4, v5, v6, v8, v9], [v1, v10, v11, v3, v4, v5, v6, v8, v9], [v0, v1, v10, v11, v3, v4, v5, v6, v8, v9], [v0, v10, v2, v3, v4, v5, v6, v8, v9], [v0, v1, v10, v2, v3, v4, v5, v6, v8, v9], [v0, v11, v2, v3, v4, v5, v6, v8, v9], [v1, v11, v2, v3, v4, v5, v6, v8, v9], [v0, v1, v11, v2, v3, v4, v5, v6, v8, v9], [v0, v10, v11, v2, v3, v4, v5, v6, v8, v9], [v1, v10, v11, v2, v3, v4, v5, v6, v8, v9], [v0, v1, v10, v11, v2, v3, v4, v5, v6, v8, v9], [v0, v1, v10, v3, v5, v7, v8, v9], [v1, v11, v3, v5, v7, v8, v9], [v0, v1, v11, v3, v5, v7, v8, v9], [v1, v10, v11, v3, v5, v7, v8, v9], [v0, v1, v10, v11, v3, v5, v7, v8, v9], [v0, v10, v2, v3, v5, v7, v8, v9], [v0, v1, v10, v2, v3, v5, v7, v8, v9], [v0, v11, v2, v3, v5, v7, v8, v9], [v1, v11, v2, v3, v5, v7, v8, v9], [v0, v1, v11, v2, v3, v5, v7, v8, v9], [v0, v10, v11, v2, v3, v5, v7, v8, v9], [v1, v10, v11, v2, v3, v5, v7, v8, v9], [v0, v1, v10, v11, v2, v3, v5, v7, v8, v9], [v0, v10, v2, v4, v5, v7, v8, v9], [v0, v1, v10, v2, v4, v5, v7, v8, v9], [v0, v11, v2, v4, v5, v7, v8, v9], [v1, v11, v2, v4, v5, v7, v8, v9], [v0, v1, v11, v2, v4, v5, v7, v8, v9], [v0, v10, v11, v2, v4, v5, v7, v8, v9], [v1, v10, v11, v2, v4, v5, v7, v8, v9], [v0, v1, v10, v11, v2, v4, v5, v7, v8, v9], [v0, v1, v10, v3, v4, v5, v7, v8, v9], [v1, v11, v3, v4, v5, v7, v8, v9], [v0, v1, v11, v3, v4, v5, v7, v8, v9], [v1, v10, v11, v3, v4, v5, v7, v8, v9], [v0, v1, v10, v11, v3, v4, v5, v7, v8, v9], [v0, v10, v2, v3, v4, v5, v7, v8, v9], [v0, v1, v10, v2, v3, v4, v5, v7, v8, v9], [v0, v11, v2, v3, v4, v5, v7, v8, v9], [v1, v11, v2, v3, v4, v5, v7, v8, v9], [v0, v1, v11, v2, v3, v4, v5, v7, v8, v9], [v0, v10, v11, v2, v3, v4, v5, v7, v8, v9], [v1, v10, v11, v2, v3, v4, v5, v7, v8, v9], [v0, v1, v10, v11, v2, v3, v4, v5, v7, v8, v9], [v0, v10, v2, v4, v6, v7, v8, v9], [v0, v1, v10, v2, v4, v6, v7, v8, v9], [v0, v11, v2, v4, v6, v7, v8, v9], [v1, v11, v2, v4, v6, v7, v8, v9], [v0, v1, v11, v2, v4, v6, v7, v8, v9], [v0, v10, v11, v2, v4, v6, v7, v8, v9], [v1, v10, v11, v2, v4, v6, v7, v8, v9], [v0, v1, v10, v11, v2, v4, v6, v7, v8, v9], [v0, v1, v10, v3, v4, v6, v7, v8, v9], [v1, v11, v3, v4, v6, v7, v8, v9], [v0, v1, v11, v3, v4, v6, v7, v8, v9], [v1, v10, v11, v3, v4, v6, v7, v8, v9], [v0, v1, v10, v11, v3, v4, v6, v7, v8, v9], [v0, v10, v2, v3, v4, v6, v7, v8, v9], [v0, v1, v10, v2, v3, v4, v6, v7, v8, v9], [v0, v11, v2, v3, v4, v6, v7, v8, v9], [v1, v11, v2, v3, v4, v6, v7, v8, v9], [v0, v1, v11, v2, v3, v4, v6, v7, v8, v9], [v0, v10, v11, v2, v3, v4, v6, v7, v8, v9], [v1, v10, v11, v2, v3, v4, v6, v7, v8, v9], [v0, v1, v10, v11, v2, v3, v4, v6, v7, v8, v9], [v0, v1, v10, v3, v5, v6, v7, v8, v9], [v1, v11, v3, v5, v6, v7, v8, v9], [v0, v1, v11, v3, v5, v6, v7, v8, v9], [v1, v10, v11, v3, v5, v6, v7, v8, v9], [v0, v1, v10, v11, v3, v5, v6, v7, v8, v9], [v0, v10, v2, v3, v5, v6, v7, v8, v9], [v0, v1, v10, v2, v3, v5, v6, v7, v8, v9], [v0, v11, v2, v3, v5, v6, v7, v8, v9], [v1, v11, v2, v3, v5, v6, v7, v8, v9], [v0, v1, v11, v2, v3, v5, v6, v7, v8, v9], [v0, v10, v11, v2, v3, v5, v6, v7, v8, v9], [v1, v10, v11, v2, v3, v5, v6, v7, v8, v9], [v0, v1, v10, v11, v2, v3, v5, v6, v7, v8, v9], [v0, v10, v2, v4, v5, v6, v7, v8, v9], [v0, v1, v10, v2, v4, v5, v6, v7, v8, v9], [v0, v11, v2, v4, v5, v6, v7, v8, v9], [v1, v11, v2, v4, v5, v6, v7, v8, v9], [v0, v1, v11, v2, v4, v5, v6, v7, v8, v9], [v0, v10, v11, v2, v4, v5, v6, v7, v8, v9], [v1, v10, v11, v2, v4, v5, v6, v7, v8, v9], [v0, v1, v10, v11, v2, v4, v5, v6, v7, v8, v9], [v0, v1, v10, v3, v4, v5, v6, v7, v8, v9], [v1, v11, v3, v4, v5, v6, v7, v8, v9], [v0, v1, v11, v3, v4, v5, v6, v7, v8, v9], [v1, v10, v11, v3, v4, v5, v6, v7, v8, v9], [v0, v1, v10, v11, v3, v4, v5, v6, v7, v8, v9], [v0, v10, v2, v3, v4, v5, v6, v7, v8, v9], [v0, v1, v10, v2, v3, v4, v5, v6, v7, v8, v9], [v0, v11, v2, v3, v4, v5, v6, v7, v8, v9], [v1, v11, v2, v3, v4, v5, v6, v7, v8, v9], [v0, v1, v11, v2, v3, v4, v5, v6, v7, v8, v9], [v0, v10, v11, v2, v3, v4, v5, v6, v7, v8, v9], [v1, v10, v11, v2, v3, v4, v5, v6, v7, v8, v9], [v0, v1, v10, v11, v2, v3, v4, v5, v6, v7, v8, v9]], [0, 0, 0, 0, 0, 0, 2, 36, 105, 112, 54, 12, 1], 6, [[v0, v10, v2, v4, v6, v8], [v1, v11, v3, v5, v7, v9]]]
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, v10, v11, v2, v3, v4, v5, v6, v7, v8, v9], [[v0, v1], [v0, v11], [v1, v2], [v10, v11], [v10, v9], [v2, v3], [v3, v4], [v4, v5], [v5, v6], [v6, v7], [v7, v8], [v8, v9]]]
After
derived = [6, [[v0, v10, v2, v4, v6, v8], [v1, v11, v3, v5, v7, v9]], 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, v11], [v1, v2], [v10, v11], [v10, v9], [v2, v3], [v3, v4], [v4, v5], [v5, v6], [v6, v7], [v7, v8], [v8, v9]], 6, [[v0, v10, v2, v4, v6, v8], [v1, v11, v3, v5, v7, v9]], 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: 6560b485d6a7a029ce6b018dd58be28f14d408514c5eb2c723d112d57170d4cccandidate record SHA-256: 774e89c1b4060c6eb592ac6affc6438be8dcb0c93d146e02447088cdc9bb1b84semantic review receipt SHA-256: 90df1e889242e9afdf01e068788ee8500ef366ffcfc0496e30a6800de32631c3question-first witness SHA-256: ae1e849608590ca856c9f3b6911ba58e3bf53c608eb7369606192a770f48bd5cLean: NOT_RUN · human review: NOT_PERFORMED