{"abstract":"The bounded binomial forest certificate reports an incorrect minimum.","category":"Heap invariants","checks":7,"contract":"For a binomial-forest certificate, roots is a list of [rank,key,node_count,child_ranks]. Report canonical rank order, duplicate ranks, root minimum, size, tree-cardinality failures, and descending child-rank failures. No implicit tree nodes are present.","evaluation_group":"s3-heap-model-binomial-forest","failed_approach":"The local patch uses r[0][1] if r else None and still violates the stated relation.","family":"s3-heap-binomial-forest-minimum","id":"FA-39851","implementations":{"attempt":{"sha256":"d6402b12eec78add8d8a5b6402656e4f27c4f3d61a0b457bf88df4caf47b05a4","source":"\"\"\"Failure Map reference implementation. Python standard library only.\"\"\"\nimport json\n\nN = 1\nobservations = []\ndef solve(d):\n    r = d['roots']\n    return {'canonical': sorted([x[0] for x in r]),\n    'collisions': sorted({x[0] for x in r if sum(y[0]==x[0] for y in r)>1}),\n    'minimum': r[0][1] if r else None,\n    'population': sum(x[2] for x in r),\n    'cardinality': [i for i,x in enumerate(r) if x[2] != 2**x[0]],\n    'children': [i for i,x in enumerate(r) if x[3] != list(range(x[0]-1,-1,-1))]}\ndef check(label, actual, expected):\n    observations.append({\"check\": label, \"actual\": actual, \"expected\": expected, \"passed\": actual == expected})\ncases = [[({'roots': []}, {'canonical': [], 'collisions': [], 'minimum': None, 'population': 0, 'cardinality': [], 'children': []}), ({'roots': [[0, 8, 1, []]]}, {'canonical': [0], 'collisions': [], 'minimum': 8, 'population': 1, 'cardinality': [], 'children': []}), ({'roots': [[2, 9, 4, [1, 0]], [0, 3, 1, []], [2, 2, 4, [1, 0]]]}, {'canonical': [0, 2, 2], 'collisions': [2], 'minimum': 2, 'population': 9, 'cardinality': [], 'children': []}), ({'roots': [[3, 1, 8, [2, 1, 0]], [1, 5, 2, [0]]]}, {'canonical': [1, 3], 'collisions': [], 'minimum': 1, 'population': 10, 'cardinality': [], 'children': []}), ({'roots': [[2, 4, 5, [0, 1]], [1, 4, 1, [0]]]}, {'canonical': [1, 2], 'collisions': [], 'minimum': 4, 'population': 6, 'cardinality': [0, 1], 'children': [0]}), ({'roots': [[1, 7, 2, []], [3, 0, 7, [2, 0, 1]], [1, 2, 2, [0]]]}, {'canonical': [1, 1, 3], 'collisions': [1], 'minimum': 0, 'population': 11, 'cardinality': [1], 'children': [0, 1]}), ({'roots': [[1, 8, 2, []], [3, 0, 7, [2, 0, 1]], [1, 2, 2, [0]]]}, {'canonical': [1, 1, 3], 'collisions': [1], 'minimum': 0, 'population': 11, 'cardinality': [1], 'children': [0, 1]})], [({'roots': []}, {'canonical': [], 'collisions': [], 'minimum': None, 'population': 0, 'cardinality': [], 'children': []}), ({'roots': [[0, 8, 1, []]]}, {'canonical': [0], 'collisions': [], 'minimum': 8, 'population': 1, 'cardinality': [], 'children': []}), ({'roots': [[2, 9, 4, [1, 0]], [0, 3, 1, []], [2, 2, 4, [1, 0]]]}, {'canonical': [0, 2, 2], 'collisions': [2], 'minimum': 2, 'population': 9, 'cardinality': [], 'children': []}), ({'roots': [[3, 1, 8, [2, 1, 0]], [1, 5, 2, [0]]]}, {'canonical': [1, 3], 'collisions': [], 'minimum': 1, 'population': 10, 'cardinality': [], 'children': []}), ({'roots': [[2, 4, 5, [0, 1]], [1, 4, 1, [0]]]}, {'canonical': [1, 2], 'collisions': [], 'minimum': 4, 'population': 6, 'cardinality': [0, 1], 'children': [0]}), ({'roots': [[1, 7, 2, []], [3, 0, 7, [2, 0, 1]], [1, 2, 2, [0]]]}, {'canonical': [1, 1, 3], 'collisions': [1], 'minimum': 0, 'population': 11, 'cardinality': [1], 'children': [0, 1]}), ({'roots': [[1, 9, 2, []], [3, 0, 7, [2, 0, 1]], [1, 2, 2, [0]]]}, {'canonical': [1, 1, 3], 'collisions': [1], 'minimum': 0, 'population': 11, 'cardinality': [1], 'children': [0, 1]})], [({'roots': []}, {'canonical': [], 'collisions': [], 'minimum': None, 'population': 0, 'cardinality': [], 'children': []}), ({'roots': [[0, 8, 1, []]]}, {'canonical': [0], 'collisions': [], 'minimum': 8, 'population': 1, 'cardinality': [], 'children': []}), ({'roots': [[2, 9, 4, [1, 0]], [0, 3, 1, []], [2, 2, 4, [1, 0]]]}, {'canonical': [0, 2, 2], 'collisions': [2], 'minimum': 2, 'population': 9, 'cardinality': [], 'children': []}), ({'roots': [[3, 1, 8, [2, 1, 0]], [1, 5, 2, [0]]]}, {'canonical': [1, 3], 'collisions': [], 'minimum': 1, 'population': 10, 'cardinality': [], 'children': []}), ({'roots': [[2, 4, 5, [0, 1]], [1, 4, 1, [0]]]}, {'canonical': [1, 2], 'collisions': [], 'minimum': 4, 'population': 6, 'cardinality': [0, 1], 'children': [0]}), ({'roots': [[1, 7, 2, []], [3, 0, 7, [2, 0, 1]], [1, 2, 2, [0]]]}, {'canonical': [1, 1, 3], 'collisions': [1], 'minimum': 0, 'population': 11, 'cardinality': [1], 'children': [0, 1]}), ({'roots': [[1, 10, 2, []], [3, 0, 7, [2, 0, 1]], [1, 2, 2, [0]]]}, {'canonical': [1, 1, 3], 'collisions': [1], 'minimum': 0, 'population': 11, 'cardinality': [1], 'children': [0, 1]})], [({'roots': []}, {'canonical': [], 'collisions': [], 'minimum': None, 'population': 0, 'cardinality': [], 'children': []}), ({'roots': [[0, 8, 1, []]]}, {'canonical': [0], 'collisions': [], 'minimum': 8, 'population': 1, 'cardinality': [], 'children': []}), ({'roots': [[2, 9, 4, [1, 0]], [0, 3, 1, []], [2, 2, 4, [1, 0]]]}, {'canonical': [0, 2, 2], 'collisions': [2], 'minimum': 2, 'population': 9, 'cardinality': [], 'children': []}), ({'roots': [[3, 1, 8, [2, 1, 0]], [1, 5, 2, [0]]]}, {'canonical': [1, 3], 'collisions': [], 'minimum': 1, 'population': 10, 'cardinality': [], 'children': []}), ({'roots': [[2, 4, 5, [0, 1]], [1, 4, 1, [0]]]}, {'canonical': [1, 2], 'collisions': [], 'minimum': 4, 'population': 6, 'cardinality': [0, 1], 'children': [0]}), ({'roots': [[1, 7, 2, []], [3, 0, 7, [2, 0, 1]], [1, 2, 2, [0]]]}, {'canonical': [1, 1, 3], 'collisions': [1], 'minimum': 0, 'population': 11, 'cardinality': [1], 'children': [0, 1]}), ({'roots': [[1, 11, 2, []], [3, 0, 7, [2, 0, 1]], [1, 2, 2, [0]]]}, {'canonical': [1, 1, 3], 'collisions': [1], 'minimum': 0, 'population': 11, 'cardinality': [1], 'children': [0, 1]})], [({'roots': []}, {'canonical': [], 'collisions': [], 'minimum': None, 'population': 0, 'cardinality': [], 'children': []}), ({'roots': [[0, 8, 1, []]]}, {'canonical': [0], 'collisions': [], 'minimum': 8, 'population': 1, 'cardinality': [], 'children': []}), ({'roots': [[2, 9, 4, [1, 0]], [0, 3, 1, []], [2, 2, 4, [1, 0]]]}, {'canonical': [0, 2, 2], 'collisions': [2], 'minimum': 2, 'population': 9, 'cardinality': [], 'children': []}), ({'roots': [[3, 1, 8, [2, 1, 0]], [1, 5, 2, [0]]]}, {'canonical': [1, 3], 'collisions': [], 'minimum': 1, 'population': 10, 'cardinality': [], 'children': []}), ({'roots': [[2, 4, 5, [0, 1]], [1, 4, 1, [0]]]}, {'canonical': [1, 2], 'collisions': [], 'minimum': 4, 'population': 6, 'cardinality': [0, 1], 'children': [0]}), ({'roots': [[1, 7, 2, []], [3, 0, 7, [2, 0, 1]], [1, 2, 2, [0]]]}, {'canonical': [1, 1, 3], 'collisions': [1], 'minimum': 0, 'population': 11, 'cardinality': [1], 'children': [0, 1]}), ({'roots': [[1, 12, 2, []], [3, 0, 7, [2, 0, 1]], [1, 2, 2, [0]]]}, {'canonical': [1, 1, 3], 'collisions': [1], 'minimum': 0, 'population': 11, 'cardinality': [1], 'children': [0, 1]})]][N-1]\ncheck('regression certificate 1', solve(cases[0][0]), cases[0][1])\ncheck('regression certificate 2', solve(cases[1][0]), cases[1][1])\ncheck('regression certificate 3', solve(cases[2][0]), cases[2][1])\ncheck('regression certificate 4', solve(cases[3][0]), cases[3][1])\ncheck('regression certificate 5', solve(cases[4][0]), cases[4][1])\ncheck('regression certificate 6', solve(cases[5][0]), cases[5][1])\ncheck('variant-dependent certificate', solve(cases[6][0]), cases[6][1])\nprint(json.dumps({\"observations\": observations, \"passed\": all(x[\"passed\"] for x in observations)}, ensure_ascii=False))\nraise SystemExit(0 if all(x[\"passed\"] for x in observations) else 1)\n"},"broken":{"sha256":"a262e44b83e454a3c71cdd2e8f6bc9d4021fa68ec65c682adf4d384a7b6d6366","source":"\"\"\"Failure Map reference implementation. Python standard library only.\"\"\"\nimport json\n\nN = 1\nobservations = []\ndef solve(d):\n    r = d['roots']\n    return {'canonical': sorted([x[0] for x in r]),\n    'collisions': sorted({x[0] for x in r if sum(y[0]==x[0] for y in r)>1}),\n    'minimum': min(r,default=[0,None],key=lambda x:x[0])[1],\n    'population': sum(x[2] for x in r),\n    'cardinality': [i for i,x in enumerate(r) if x[2] != 2**x[0]],\n    'children': [i for i,x in enumerate(r) if x[3] != list(range(x[0]-1,-1,-1))]}\ndef check(label, actual, expected):\n    observations.append({\"check\": label, \"actual\": actual, \"expected\": expected, \"passed\": actual == expected})\ncases = [[({'roots': []}, {'canonical': [], 'collisions': [], 'minimum': None, 'population': 0, 'cardinality': [], 'children': []}), ({'roots': [[0, 8, 1, []]]}, {'canonical': [0], 'collisions': [], 'minimum': 8, 'population': 1, 'cardinality': [], 'children': []}), ({'roots': [[2, 9, 4, [1, 0]], [0, 3, 1, []], [2, 2, 4, [1, 0]]]}, {'canonical': [0, 2, 2], 'collisions': [2], 'minimum': 2, 'population': 9, 'cardinality': [], 'children': []}), ({'roots': [[3, 1, 8, [2, 1, 0]], [1, 5, 2, [0]]]}, {'canonical': [1, 3], 'collisions': [], 'minimum': 1, 'population': 10, 'cardinality': [], 'children': []}), ({'roots': [[2, 4, 5, [0, 1]], [1, 4, 1, [0]]]}, {'canonical': [1, 2], 'collisions': [], 'minimum': 4, 'population': 6, 'cardinality': [0, 1], 'children': [0]}), ({'roots': [[1, 7, 2, []], [3, 0, 7, [2, 0, 1]], [1, 2, 2, [0]]]}, {'canonical': [1, 1, 3], 'collisions': [1], 'minimum': 0, 'population': 11, 'cardinality': [1], 'children': [0, 1]}), ({'roots': [[1, 8, 2, []], [3, 0, 7, [2, 0, 1]], [1, 2, 2, [0]]]}, {'canonical': [1, 1, 3], 'collisions': [1], 'minimum': 0, 'population': 11, 'cardinality': [1], 'children': [0, 1]})], [({'roots': []}, {'canonical': [], 'collisions': [], 'minimum': None, 'population': 0, 'cardinality': [], 'children': []}), ({'roots': [[0, 8, 1, []]]}, {'canonical': [0], 'collisions': [], 'minimum': 8, 'population': 1, 'cardinality': [], 'children': []}), ({'roots': [[2, 9, 4, [1, 0]], [0, 3, 1, []], [2, 2, 4, [1, 0]]]}, {'canonical': [0, 2, 2], 'collisions': [2], 'minimum': 2, 'population': 9, 'cardinality': [], 'children': []}), ({'roots': [[3, 1, 8, [2, 1, 0]], [1, 5, 2, [0]]]}, {'canonical': [1, 3], 'collisions': [], 'minimum': 1, 'population': 10, 'cardinality': [], 'children': []}), ({'roots': [[2, 4, 5, [0, 1]], [1, 4, 1, [0]]]}, {'canonical': [1, 2], 'collisions': [], 'minimum': 4, 'population': 6, 'cardinality': [0, 1], 'children': [0]}), ({'roots': [[1, 7, 2, []], [3, 0, 7, [2, 0, 1]], [1, 2, 2, [0]]]}, {'canonical': [1, 1, 3], 'collisions': [1], 'minimum': 0, 'population': 11, 'cardinality': [1], 'children': [0, 1]}), ({'roots': [[1, 9, 2, []], [3, 0, 7, [2, 0, 1]], [1, 2, 2, [0]]]}, {'canonical': [1, 1, 3], 'collisions': [1], 'minimum': 0, 'population': 11, 'cardinality': [1], 'children': [0, 1]})], [({'roots': []}, {'canonical': [], 'collisions': [], 'minimum': None, 'population': 0, 'cardinality': [], 'children': []}), ({'roots': [[0, 8, 1, []]]}, {'canonical': [0], 'collisions': [], 'minimum': 8, 'population': 1, 'cardinality': [], 'children': []}), ({'roots': [[2, 9, 4, [1, 0]], [0, 3, 1, []], [2, 2, 4, [1, 0]]]}, {'canonical': [0, 2, 2], 'collisions': [2], 'minimum': 2, 'population': 9, 'cardinality': [], 'children': []}), ({'roots': [[3, 1, 8, [2, 1, 0]], [1, 5, 2, [0]]]}, {'canonical': [1, 3], 'collisions': [], 'minimum': 1, 'population': 10, 'cardinality': [], 'children': []}), ({'roots': [[2, 4, 5, [0, 1]], [1, 4, 1, [0]]]}, {'canonical': [1, 2], 'collisions': [], 'minimum': 4, 'population': 6, 'cardinality': [0, 1], 'children': [0]}), ({'roots': [[1, 7, 2, []], [3, 0, 7, [2, 0, 1]], [1, 2, 2, [0]]]}, {'canonical': [1, 1, 3], 'collisions': [1], 'minimum': 0, 'population': 11, 'cardinality': [1], 'children': [0, 1]}), ({'roots': [[1, 10, 2, []], [3, 0, 7, [2, 0, 1]], [1, 2, 2, [0]]]}, {'canonical': [1, 1, 3], 'collisions': [1], 'minimum': 0, 'population': 11, 'cardinality': [1], 'children': [0, 1]})], [({'roots': []}, {'canonical': [], 'collisions': [], 'minimum': None, 'population': 0, 'cardinality': [], 'children': []}), ({'roots': [[0, 8, 1, []]]}, {'canonical': [0], 'collisions': [], 'minimum': 8, 'population': 1, 'cardinality': [], 'children': []}), ({'roots': [[2, 9, 4, [1, 0]], [0, 3, 1, []], [2, 2, 4, [1, 0]]]}, {'canonical': [0, 2, 2], 'collisions': [2], 'minimum': 2, 'population': 9, 'cardinality': [], 'children': []}), ({'roots': [[3, 1, 8, [2, 1, 0]], [1, 5, 2, [0]]]}, {'canonical': [1, 3], 'collisions': [], 'minimum': 1, 'population': 10, 'cardinality': [], 'children': []}), ({'roots': [[2, 4, 5, [0, 1]], [1, 4, 1, [0]]]}, {'canonical': [1, 2], 'collisions': [], 'minimum': 4, 'population': 6, 'cardinality': [0, 1], 'children': [0]}), ({'roots': [[1, 7, 2, []], [3, 0, 7, [2, 0, 1]], [1, 2, 2, [0]]]}, {'canonical': [1, 1, 3], 'collisions': [1], 'minimum': 0, 'population': 11, 'cardinality': [1], 'children': [0, 1]}), ({'roots': [[1, 11, 2, []], [3, 0, 7, [2, 0, 1]], [1, 2, 2, [0]]]}, {'canonical': [1, 1, 3], 'collisions': [1], 'minimum': 0, 'population': 11, 'cardinality': [1], 'children': [0, 1]})], [({'roots': []}, {'canonical': [], 'collisions': [], 'minimum': None, 'population': 0, 'cardinality': [], 'children': []}), ({'roots': [[0, 8, 1, []]]}, {'canonical': [0], 'collisions': [], 'minimum': 8, 'population': 1, 'cardinality': [], 'children': []}), ({'roots': [[2, 9, 4, [1, 0]], [0, 3, 1, []], [2, 2, 4, [1, 0]]]}, {'canonical': [0, 2, 2], 'collisions': [2], 'minimum': 2, 'population': 9, 'cardinality': [], 'children': []}), ({'roots': [[3, 1, 8, [2, 1, 0]], [1, 5, 2, [0]]]}, {'canonical': [1, 3], 'collisions': [], 'minimum': 1, 'population': 10, 'cardinality': [], 'children': []}), ({'roots': [[2, 4, 5, [0, 1]], [1, 4, 1, [0]]]}, {'canonical': [1, 2], 'collisions': [], 'minimum': 4, 'population': 6, 'cardinality': [0, 1], 'children': [0]}), ({'roots': [[1, 7, 2, []], [3, 0, 7, [2, 0, 1]], [1, 2, 2, [0]]]}, {'canonical': [1, 1, 3], 'collisions': [1], 'minimum': 0, 'population': 11, 'cardinality': [1], 'children': [0, 1]}), ({'roots': [[1, 12, 2, []], [3, 0, 7, [2, 0, 1]], [1, 2, 2, [0]]]}, {'canonical': [1, 1, 3], 'collisions': [1], 'minimum': 0, 'population': 11, 'cardinality': [1], 'children': [0, 1]})]][N-1]\ncheck('regression certificate 1', solve(cases[0][0]), cases[0][1])\ncheck('regression certificate 2', solve(cases[1][0]), cases[1][1])\ncheck('regression certificate 3', solve(cases[2][0]), cases[2][1])\ncheck('regression certificate 4', solve(cases[3][0]), cases[3][1])\ncheck('regression certificate 5', solve(cases[4][0]), cases[4][1])\ncheck('regression certificate 6', solve(cases[5][0]), cases[5][1])\ncheck('variant-dependent certificate', solve(cases[6][0]), cases[6][1])\nprint(json.dumps({\"observations\": observations, \"passed\": all(x[\"passed\"] for x in observations)}, ensure_ascii=False))\nraise SystemExit(0 if all(x[\"passed\"] for x in observations) else 1)\n"},"fixed":{"sha256":"7801cf55b67d2ae1bafcf85034e3d253b8c6a27a75ba40d6413404288499ad49","source":"\"\"\"Failure Map reference implementation. Python standard library only.\"\"\"\nimport json\n\nN = 1\nobservations = []\ndef solve(d):\n    r = d['roots']\n    return {'canonical': sorted([x[0] for x in r]),\n    'collisions': sorted({x[0] for x in r if sum(y[0]==x[0] for y in r)>1}),\n    'minimum': min([x[1] for x in r],default=None),\n    'population': sum(x[2] for x in r),\n    'cardinality': [i for i,x in enumerate(r) if x[2] != 2**x[0]],\n    'children': [i for i,x in enumerate(r) if x[3] != list(range(x[0]-1,-1,-1))]}\ndef check(label, actual, expected):\n    observations.append({\"check\": label, \"actual\": actual, \"expected\": expected, \"passed\": actual == expected})\ncases = [[({'roots': []}, {'canonical': [], 'collisions': [], 'minimum': None, 'population': 0, 'cardinality': [], 'children': []}), ({'roots': [[0, 8, 1, []]]}, {'canonical': [0], 'collisions': [], 'minimum': 8, 'population': 1, 'cardinality': [], 'children': []}), ({'roots': [[2, 9, 4, [1, 0]], [0, 3, 1, []], [2, 2, 4, [1, 0]]]}, {'canonical': [0, 2, 2], 'collisions': [2], 'minimum': 2, 'population': 9, 'cardinality': [], 'children': []}), ({'roots': [[3, 1, 8, [2, 1, 0]], [1, 5, 2, [0]]]}, {'canonical': [1, 3], 'collisions': [], 'minimum': 1, 'population': 10, 'cardinality': [], 'children': []}), ({'roots': [[2, 4, 5, [0, 1]], [1, 4, 1, [0]]]}, {'canonical': [1, 2], 'collisions': [], 'minimum': 4, 'population': 6, 'cardinality': [0, 1], 'children': [0]}), ({'roots': [[1, 7, 2, []], [3, 0, 7, [2, 0, 1]], [1, 2, 2, [0]]]}, {'canonical': [1, 1, 3], 'collisions': [1], 'minimum': 0, 'population': 11, 'cardinality': [1], 'children': [0, 1]}), ({'roots': [[1, 8, 2, []], [3, 0, 7, [2, 0, 1]], [1, 2, 2, [0]]]}, {'canonical': [1, 1, 3], 'collisions': [1], 'minimum': 0, 'population': 11, 'cardinality': [1], 'children': [0, 1]})], [({'roots': []}, {'canonical': [], 'collisions': [], 'minimum': None, 'population': 0, 'cardinality': [], 'children': []}), ({'roots': [[0, 8, 1, []]]}, {'canonical': [0], 'collisions': [], 'minimum': 8, 'population': 1, 'cardinality': [], 'children': []}), ({'roots': [[2, 9, 4, [1, 0]], [0, 3, 1, []], [2, 2, 4, [1, 0]]]}, {'canonical': [0, 2, 2], 'collisions': [2], 'minimum': 2, 'population': 9, 'cardinality': [], 'children': []}), ({'roots': [[3, 1, 8, [2, 1, 0]], [1, 5, 2, [0]]]}, {'canonical': [1, 3], 'collisions': [], 'minimum': 1, 'population': 10, 'cardinality': [], 'children': []}), ({'roots': [[2, 4, 5, [0, 1]], [1, 4, 1, [0]]]}, {'canonical': [1, 2], 'collisions': [], 'minimum': 4, 'population': 6, 'cardinality': [0, 1], 'children': [0]}), ({'roots': [[1, 7, 2, []], [3, 0, 7, [2, 0, 1]], [1, 2, 2, [0]]]}, {'canonical': [1, 1, 3], 'collisions': [1], 'minimum': 0, 'population': 11, 'cardinality': [1], 'children': [0, 1]}), ({'roots': [[1, 9, 2, []], [3, 0, 7, [2, 0, 1]], [1, 2, 2, [0]]]}, {'canonical': [1, 1, 3], 'collisions': [1], 'minimum': 0, 'population': 11, 'cardinality': [1], 'children': [0, 1]})], [({'roots': []}, {'canonical': [], 'collisions': [], 'minimum': None, 'population': 0, 'cardinality': [], 'children': []}), ({'roots': [[0, 8, 1, []]]}, {'canonical': [0], 'collisions': [], 'minimum': 8, 'population': 1, 'cardinality': [], 'children': []}), ({'roots': [[2, 9, 4, [1, 0]], [0, 3, 1, []], [2, 2, 4, [1, 0]]]}, {'canonical': [0, 2, 2], 'collisions': [2], 'minimum': 2, 'population': 9, 'cardinality': [], 'children': []}), ({'roots': [[3, 1, 8, [2, 1, 0]], [1, 5, 2, [0]]]}, {'canonical': [1, 3], 'collisions': [], 'minimum': 1, 'population': 10, 'cardinality': [], 'children': []}), ({'roots': [[2, 4, 5, [0, 1]], [1, 4, 1, [0]]]}, {'canonical': [1, 2], 'collisions': [], 'minimum': 4, 'population': 6, 'cardinality': [0, 1], 'children': [0]}), ({'roots': [[1, 7, 2, []], [3, 0, 7, [2, 0, 1]], [1, 2, 2, [0]]]}, {'canonical': [1, 1, 3], 'collisions': [1], 'minimum': 0, 'population': 11, 'cardinality': [1], 'children': [0, 1]}), ({'roots': [[1, 10, 2, []], [3, 0, 7, [2, 0, 1]], [1, 2, 2, [0]]]}, {'canonical': [1, 1, 3], 'collisions': [1], 'minimum': 0, 'population': 11, 'cardinality': [1], 'children': [0, 1]})], [({'roots': []}, {'canonical': [], 'collisions': [], 'minimum': None, 'population': 0, 'cardinality': [], 'children': []}), ({'roots': [[0, 8, 1, []]]}, {'canonical': [0], 'collisions': [], 'minimum': 8, 'population': 1, 'cardinality': [], 'children': []}), ({'roots': [[2, 9, 4, [1, 0]], [0, 3, 1, []], [2, 2, 4, [1, 0]]]}, {'canonical': [0, 2, 2], 'collisions': [2], 'minimum': 2, 'population': 9, 'cardinality': [], 'children': []}), ({'roots': [[3, 1, 8, [2, 1, 0]], [1, 5, 2, [0]]]}, {'canonical': [1, 3], 'collisions': [], 'minimum': 1, 'population': 10, 'cardinality': [], 'children': []}), ({'roots': [[2, 4, 5, [0, 1]], [1, 4, 1, [0]]]}, {'canonical': [1, 2], 'collisions': [], 'minimum': 4, 'population': 6, 'cardinality': [0, 1], 'children': [0]}), ({'roots': [[1, 7, 2, []], [3, 0, 7, [2, 0, 1]], [1, 2, 2, [0]]]}, {'canonical': [1, 1, 3], 'collisions': [1], 'minimum': 0, 'population': 11, 'cardinality': [1], 'children': [0, 1]}), ({'roots': [[1, 11, 2, []], [3, 0, 7, [2, 0, 1]], [1, 2, 2, [0]]]}, {'canonical': [1, 1, 3], 'collisions': [1], 'minimum': 0, 'population': 11, 'cardinality': [1], 'children': [0, 1]})], [({'roots': []}, {'canonical': [], 'collisions': [], 'minimum': None, 'population': 0, 'cardinality': [], 'children': []}), ({'roots': [[0, 8, 1, []]]}, {'canonical': [0], 'collisions': [], 'minimum': 8, 'population': 1, 'cardinality': [], 'children': []}), ({'roots': [[2, 9, 4, [1, 0]], [0, 3, 1, []], [2, 2, 4, [1, 0]]]}, {'canonical': [0, 2, 2], 'collisions': [2], 'minimum': 2, 'population': 9, 'cardinality': [], 'children': []}), ({'roots': [[3, 1, 8, [2, 1, 0]], [1, 5, 2, [0]]]}, {'canonical': [1, 3], 'collisions': [], 'minimum': 1, 'population': 10, 'cardinality': [], 'children': []}), ({'roots': [[2, 4, 5, [0, 1]], [1, 4, 1, [0]]]}, {'canonical': [1, 2], 'collisions': [], 'minimum': 4, 'population': 6, 'cardinality': [0, 1], 'children': [0]}), ({'roots': [[1, 7, 2, []], [3, 0, 7, [2, 0, 1]], [1, 2, 2, [0]]]}, {'canonical': [1, 1, 3], 'collisions': [1], 'minimum': 0, 'population': 11, 'cardinality': [1], 'children': [0, 1]}), ({'roots': [[1, 12, 2, []], [3, 0, 7, [2, 0, 1]], [1, 2, 2, [0]]]}, {'canonical': [1, 1, 3], 'collisions': [1], 'minimum': 0, 'population': 11, 'cardinality': [1], 'children': [0, 1]})]][N-1]\ncheck('regression certificate 1', solve(cases[0][0]), cases[0][1])\ncheck('regression certificate 2', solve(cases[1][0]), cases[1][1])\ncheck('regression certificate 3', solve(cases[2][0]), cases[2][1])\ncheck('regression certificate 4', solve(cases[3][0]), cases[3][1])\ncheck('regression certificate 5', solve(cases[4][0]), cases[4][1])\ncheck('regression certificate 6', solve(cases[5][0]), cases[5][1])\ncheck('variant-dependent certificate', solve(cases[6][0]), cases[6][1])\nprint(json.dumps({\"observations\": observations, \"passed\": all(x[\"passed\"] for x in observations)}, ensure_ascii=False))\nraise SystemExit(0 if all(x[\"passed\"] for x in observations) else 1)\n"}},"limitations":"A stipulated offline diagnostic model; it does not implement a production allocator, concurrency protocol, or complete heap library. This reproducer isolates one failure mechanism. Results cover the supplied fixtures. Variants within a family share a test contract and should remain grouped when constructing evaluation splits. Related mechanisms with a shared evaluation_group must also remain together; these controlled models are not independent production incidents.","method":"Deterministic executable model with adversarial boundary fixtures.","provenance":{"created_by":"Failure Map","dependencies":"Python standard library","family":"s3-heap-binomial-forest-minimum","generated_at":"2026-09-29T14:43:25.352505+00:00","license":"CC0-1.0","python":"3.12.14","seed":1,"split":"open-access"},"relevance":"This isolates an internal heap representation or priority-structure invariant using deterministic finite records.","repair":"Derive minimum using min([x[1] for x in r],default=None) under the stated bounded certificate contract.","root_cause":"The smallest-rank root need not carry the smallest priority.","sha256":"fa6015450bfbb4a93aa4bfe16c3d7ade35e50befb15ff68d76660f4b45e16808","title":"Binomial minimum pointer is chosen by tree rank · case 01","variant":1,"variant_policy":"Six explicit regression certificates are retained; a seventh changes structural size, position, priority, or bounds with N.","verification":{"attempt":{"elapsed_ms":42.223,"exit_code":1,"observations":[{"actual":{"canonical":[],"cardinality":[],"children":[],"collisions":[],"minimum":null,"population":0},"check":"regression certificate 1","expected":{"canonical":[],"cardinality":[],"children":[],"collisions":[],"minimum":null,"population":0},"passed":true},{"actual":{"canonical":[0],"cardinality":[],"children":[],"collisions":[],"minimum":8,"population":1},"check":"regression certificate 2","expected":{"canonical":[0],"cardinality":[],"children":[],"collisions":[],"minimum":8,"population":1},"passed":true},{"actual":{"canonical":[0,2,2],"cardinality":[],"children":[],"collisions":[2],"minimum":9,"population":9},"check":"regression certificate 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\"collisions\": [], \"minimum\": 8, \"population\": 1, \"cardinality\": [], \"children\": []}, \"passed\": true}, {\"check\": \"regression certificate 3\", \"actual\": {\"canonical\": [0, 2, 2], \"collisions\": [2], \"minimum\": 9, \"population\": 9, \"cardinality\": [], \"children\": []}, \"expected\": {\"canonical\": [0, 2, 2], \"collisions\": [2], \"minimum\": 2, \"population\": 9, \"cardinality\": [], \"children\": []}, \"passed\": false}, {\"check\": \"regression certificate 4\", \"actual\": {\"canonical\": [1, 3], \"collisions\": [], \"minimum\": 1, \"population\": 10, \"cardinality\": [], \"children\": []}, \"expected\": {\"canonical\": [1, 3], \"collisions\": [], \"minimum\": 1, \"population\": 10, \"cardinality\": [], \"children\": []}, \"passed\": true}, {\"check\": \"regression certificate 5\", \"actual\": {\"canonical\": [1, 2], \"collisions\": [], \"minimum\": 4, \"population\": 6, \"cardinality\": [0, 1], \"children\": [0]}, \"expected\": {\"canonical\": [1, 2], \"collisions\": [], \"minimum\": 4, \"population\": 6, \"cardinality\": [0, 1], \"children\": [0]}, \"passed\": true}, {\"check\": \"regression certificate 6\", \"actual\": {\"canonical\": [1, 1, 3], \"collisions\": [1], \"minimum\": 7, \"population\": 11, \"cardinality\": [1], \"children\": [0, 1]}, \"expected\": {\"canonical\": [1, 1, 3], \"collisions\": [1], \"minimum\": 0, \"population\": 11, \"cardinality\": [1], \"children\": [0, 1]}, \"passed\": false}, {\"check\": \"variant-dependent certificate\", \"actual\": {\"canonical\": [1, 1, 3], \"collisions\": [1], \"minimum\": 8, \"population\": 11, \"cardinality\": [1], \"children\": [0, 1]}, \"expected\": {\"canonical\": [1, 1, 3], \"collisions\": [1], \"minimum\": 0, \"population\": 11, \"cardinality\": [1], \"children\": [0, 1]}, \"passed\": false}], \"passed\": 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certificate","expected":{"canonical":[1,1,3],"cardinality":[1],"children":[0,1],"collisions":[1],"minimum":0,"population":11},"passed":false}],"passed":false,"stderr":"","stdout":"{\"observations\": [{\"check\": \"regression certificate 1\", \"actual\": {\"canonical\": [], \"collisions\": [], \"minimum\": null, \"population\": 0, \"cardinality\": [], \"children\": []}, \"expected\": {\"canonical\": [], \"collisions\": [], \"minimum\": null, \"population\": 0, \"cardinality\": [], \"children\": []}, \"passed\": true}, {\"check\": \"regression certificate 2\", \"actual\": {\"canonical\": [0], \"collisions\": [], \"minimum\": 8, \"population\": 1, \"cardinality\": [], \"children\": []}, \"expected\": {\"canonical\": [0], \"collisions\": [], \"minimum\": 8, \"population\": 1, \"cardinality\": [], \"children\": []}, \"passed\": true}, {\"check\": \"regression certificate 3\", \"actual\": {\"canonical\": [0, 2, 2], \"collisions\": [2], \"minimum\": 3, \"population\": 9, \"cardinality\": [], \"children\": []}, \"expected\": {\"canonical\": [0, 2, 2], \"collisions\": [2], \"minimum\": 2, \"population\": 9, \"cardinality\": [], \"children\": []}, \"passed\": false}, {\"check\": \"regression certificate 4\", \"actual\": {\"canonical\": [1, 3], \"collisions\": [], \"minimum\": 5, \"population\": 10, \"cardinality\": [], \"children\": []}, \"expected\": {\"canonical\": [1, 3], \"collisions\": [], \"minimum\": 1, \"population\": 10, \"cardinality\": [], \"children\": []}, \"passed\": false}, {\"check\": \"regression certificate 5\", \"actual\": {\"canonical\": [1, 2], \"collisions\": [], \"minimum\": 4, \"population\": 6, \"cardinality\": [0, 1], \"children\": [0]}, \"expected\": {\"canonical\": [1, 2], \"collisions\": [], \"minimum\": 4, \"population\": 6, \"cardinality\": [0, 1], \"children\": [0]}, \"passed\": true}, {\"check\": \"regression certificate 6\", \"actual\": {\"canonical\": [1, 1, 3], \"collisions\": [1], \"minimum\": 7, \"population\": 11, \"cardinality\": [1], \"children\": [0, 1]}, \"expected\": {\"canonical\": [1, 1, 3], \"collisions\": [1], \"minimum\": 0, \"population\": 11, \"cardinality\": [1], \"children\": [0, 1]}, \"passed\": false}, {\"check\": \"variant-dependent certificate\", \"actual\": {\"canonical\": [1, 1, 3], \"collisions\": [1], \"minimum\": 8, \"population\": 11, \"cardinality\": [1], \"children\": [0, 1]}, \"expected\": {\"canonical\": [1, 1, 3], \"collisions\": [1], \"minimum\": 0, \"population\": 11, \"cardinality\": [1], \"children\": [0, 1]}, \"passed\": false}], \"passed\": false}\n"},"fixed":{"elapsed_ms":42.165,"exit_code":0,"observations":[{"actual":{"canonical":[],"cardinality":[],"children":[],"collisions":[],"minimum":null,"population":0},"check":"regression certificate 1","expected":{"canonical":[],"cardinality":[],"children":[],"collisions":[],"minimum":null,"population":0},"passed":true},{"actual":{"canonical":[0],"cardinality":[],"children":[],"collisions":[],"minimum":8,"population":1},"check":"regression 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0, \"cardinality\": [], \"children\": []}, \"passed\": true}, {\"check\": \"regression certificate 2\", \"actual\": {\"canonical\": [0], \"collisions\": [], \"minimum\": 8, \"population\": 1, \"cardinality\": [], \"children\": []}, \"expected\": {\"canonical\": [0], \"collisions\": [], \"minimum\": 8, \"population\": 1, \"cardinality\": [], \"children\": []}, \"passed\": true}, {\"check\": \"regression certificate 3\", \"actual\": {\"canonical\": [0, 2, 2], \"collisions\": [2], \"minimum\": 2, \"population\": 9, \"cardinality\": [], \"children\": []}, \"expected\": {\"canonical\": [0, 2, 2], \"collisions\": [2], \"minimum\": 2, \"population\": 9, \"cardinality\": [], \"children\": []}, \"passed\": true}, {\"check\": \"regression certificate 4\", \"actual\": {\"canonical\": [1, 3], \"collisions\": [], \"minimum\": 1, \"population\": 10, \"cardinality\": [], \"children\": []}, \"expected\": {\"canonical\": [1, 3], \"collisions\": [], \"minimum\": 1, \"population\": 10, \"cardinality\": [], \"children\": []}, \"passed\": true}, {\"check\": \"regression certificate 5\", \"actual\": {\"canonical\": [1, 2], \"collisions\": [], \"minimum\": 4, \"population\": 6, \"cardinality\": [0, 1], \"children\": [0]}, \"expected\": {\"canonical\": [1, 2], \"collisions\": [], \"minimum\": 4, \"population\": 6, \"cardinality\": [0, 1], \"children\": [0]}, \"passed\": true}, {\"check\": \"regression certificate 6\", \"actual\": {\"canonical\": [1, 1, 3], \"collisions\": [1], \"minimum\": 0, \"population\": 11, \"cardinality\": [1], \"children\": [0, 1]}, \"expected\": {\"canonical\": [1, 1, 3], \"collisions\": [1], \"minimum\": 0, \"population\": 11, \"cardinality\": [1], \"children\": [0, 1]}, \"passed\": true}, {\"check\": \"variant-dependent certificate\", \"actual\": {\"canonical\": [1, 1, 3], \"collisions\": [1], \"minimum\": 0, \"population\": 11, \"cardinality\": [1], \"children\": [0, 1]}, \"expected\": {\"canonical\": [1, 1, 3], \"collisions\": [1], \"minimum\": 0, \"population\": 11, \"cardinality\": [1], \"children\": [0, 1]}, \"passed\": true}], \"passed\": true}\n"}},"verified":true,"visibility":"public"}