{"abstract":"The bounded pairing siblings certificate reports an incorrect degrees.","category":"Heap invariants","checks":7,"contract":"Pairing-tree certificate nodes are [key,parent,first_child,next_sibling,previous]. Missing links are -1. Previous denotes parent for first child, prior sibling otherwise. Report first-child backlinks, sibling-parent disagreements, previous-link disagreements, root sibling violations, direct heap-order violations, and child-chain degrees (stop on cycles).","evaluation_group":"s3-heap-model-pairing-siblings","failed_approach":"The local patch uses [max(0,len(c)-1) for c in chains] and still violates the stated relation.","family":"s3-heap-pairing-siblings-degrees","id":"FA-39956","implementations":{"attempt":{"sha256":"861f30ebb1cc974bc09cf2527d53d289bc508283d04523557c709281b0827359","source":"\"\"\"Failure Map reference implementation. Python standard library only.\"\"\"\nimport json\n\nN = 1\nobservations = []\ndef solve(d):\n    a=d['nodes']; root=d['root']; chains=[]\n    for i,x in enumerate(a):\n        chain=[]; j=x[2]\n        while j!=-1 and j not in chain:\n            chain.append(j); j=a[j][3]\n        chains.append(chain)\n    return {'first_previous': [i for i,x in enumerate(a) if x[2]!=-1 and a[x[2]][4]!=i],\n    'sibling_parent': [i for i,x in enumerate(a) if x[3]!=-1 and a[x[3]][1]!=x[1]],\n    'sibling_previous': [i for i,x in enumerate(a) if x[3]!=-1 and a[x[3]][4]!=i],\n    'root_links': [] if root==-1 else [k for k in (1,3,4) if a[root][k]!=-1],\n    'child_order': [[i,j] for i,c in enumerate(chains) for j in c if a[j][0]<a[i][0]],\n    'degrees': [max(0,len(c)-1) for c in chains]}\ndef check(label, actual, expected):\n    observations.append({\"check\": label, \"actual\": actual, \"expected\": expected, \"passed\": actual == expected})\ncases = [[({'nodes': [], 'root': -1}, {'first_previous': [], 'sibling_parent': [], 'sibling_previous': [], 'root_links': [], 'child_order': [], 'degrees': []}), ({'nodes': [[2, -1, -1, -1, -1]], 'root': 0}, {'first_previous': [], 'sibling_parent': [], 'sibling_previous': [], 'root_links': [], 'child_order': [], 'degrees': [0]}), ({'nodes': [[1, -1, 1, -1, -1], [7, 0, -1, 2, 0], [3, 0, -1, -1, 1]], 'root': 0}, {'first_previous': [], 'sibling_parent': [], 'sibling_previous': [], 'root_links': [], 'child_order': [], 'degrees': [2, 0, 0]}), ({'nodes': [[8, -1, 1, -1, -1], [9, 0, -1, 2, -1], [2, 1, -1, -1, 0]], 'root': 0}, {'first_previous': [0], 'sibling_parent': [1], 'sibling_previous': [1], 'root_links': [], 'child_order': [[0, 2]], 'degrees': [2, 0, 0]}), ({'nodes': [[1, -1, -1, -1, -1], [4, -1, 2, 0, 0], [4, 1, -1, -1, -1]], 'root': 1}, {'first_previous': [1], 'sibling_parent': [], 'sibling_previous': [1], 'root_links': [3, 4], 'child_order': [], 'degrees': [0, 1, 0]}), ({'nodes': [[7, -1, 1, -1, -1], [2, 0, -1, 2, 0], [8, -1, -1, 3, -1], [1, 0, -1, -1, 0]], 'root': 0}, {'first_previous': [], 'sibling_parent': [1, 2], 'sibling_previous': [1, 2], 'root_links': [], 'child_order': [[0, 1], [0, 3]], 'degrees': [3, 0, 0, 0]}), ({'nodes': [[8, -1, 1, -1, -1], [2, 0, -1, 2, 0], [8, -1, -1, 3, -1], [1, 0, -1, -1, 0]], 'root': 0}, {'first_previous': [], 'sibling_parent': [1, 2], 'sibling_previous': [1, 2], 'root_links': [], 'child_order': [[0, 1], [0, 3]], 'degrees': [3, 0, 0, 0]})], [({'nodes': [], 'root': -1}, {'first_previous': [], 'sibling_parent': [], 'sibling_previous': [], 'root_links': [], 'child_order': [], 'degrees': []}), ({'nodes': [[2, -1, -1, -1, -1]], 'root': 0}, {'first_previous': [], 'sibling_parent': [], 'sibling_previous': [], 'root_links': [], 'child_order': [], 'degrees': [0]}), ({'nodes': [[1, -1, 1, -1, -1], [7, 0, -1, 2, 0], [3, 0, -1, -1, 1]], 'root': 0}, {'first_previous': [], 'sibling_parent': [], 'sibling_previous': [], 'root_links': [], 'child_order': [], 'degrees': [2, 0, 0]}), ({'nodes': [[8, -1, 1, -1, -1], [9, 0, -1, 2, -1], [2, 1, -1, -1, 0]], 'root': 0}, {'first_previous': [0], 'sibling_parent': [1], 'sibling_previous': [1], 'root_links': [], 'child_order': [[0, 2]], 'degrees': [2, 0, 0]}), ({'nodes': [[1, -1, -1, -1, -1], [4, -1, 2, 0, 0], [4, 1, -1, -1, -1]], 'root': 1}, {'first_previous': [1], 'sibling_parent': [], 'sibling_previous': [1], 'root_links': [3, 4], 'child_order': [], 'degrees': [0, 1, 0]}), ({'nodes': [[7, -1, 1, -1, -1], [2, 0, -1, 2, 0], [8, -1, -1, 3, -1], [1, 0, -1, -1, 0]], 'root': 0}, {'first_previous': [], 'sibling_parent': [1, 2], 'sibling_previous': [1, 2], 'root_links': [], 'child_order': [[0, 1], [0, 3]], 'degrees': [3, 0, 0, 0]}), ({'nodes': [[9, -1, 1, -1, -1], [2, 0, -1, 2, 0], [8, -1, -1, 3, -1], [1, 0, -1, -1, 0]], 'root': 0}, {'first_previous': [], 'sibling_parent': [1, 2], 'sibling_previous': [1, 2], 'root_links': [], 'child_order': [[0, 1], [0, 2], [0, 3]], 'degrees': [3, 0, 0, 0]})], [({'nodes': [], 'root': -1}, {'first_previous': [], 'sibling_parent': [], 'sibling_previous': [], 'root_links': [], 'child_order': [], 'degrees': []}), ({'nodes': [[2, -1, -1, -1, -1]], 'root': 0}, {'first_previous': [], 'sibling_parent': [], 'sibling_previous': [], 'root_links': [], 'child_order': [], 'degrees': [0]}), ({'nodes': [[1, -1, 1, -1, -1], [7, 0, -1, 2, 0], [3, 0, -1, -1, 1]], 'root': 0}, {'first_previous': [], 'sibling_parent': [], 'sibling_previous': [], 'root_links': [], 'child_order': [], 'degrees': [2, 0, 0]}), ({'nodes': [[8, -1, 1, -1, -1], [9, 0, -1, 2, -1], [2, 1, -1, -1, 0]], 'root': 0}, {'first_previous': [0], 'sibling_parent': [1], 'sibling_previous': [1], 'root_links': [], 'child_order': [[0, 2]], 'degrees': [2, 0, 0]}), ({'nodes': [[1, -1, -1, -1, -1], [4, -1, 2, 0, 0], [4, 1, -1, -1, -1]], 'root': 1}, {'first_previous': [1], 'sibling_parent': [], 'sibling_previous': [1], 'root_links': [3, 4], 'child_order': [], 'degrees': [0, 1, 0]}), ({'nodes': [[7, -1, 1, -1, -1], [2, 0, -1, 2, 0], [8, -1, -1, 3, -1], [1, 0, -1, -1, 0]], 'root': 0}, {'first_previous': [], 'sibling_parent': [1, 2], 'sibling_previous': [1, 2], 'root_links': [], 'child_order': [[0, 1], [0, 3]], 'degrees': [3, 0, 0, 0]}), ({'nodes': [[10, -1, 1, -1, -1], [2, 0, -1, 2, 0], [8, -1, -1, 3, -1], [1, 0, -1, -1, 0]], 'root': 0}, {'first_previous': [], 'sibling_parent': [1, 2], 'sibling_previous': [1, 2], 'root_links': [], 'child_order': [[0, 1], [0, 2], [0, 3]], 'degrees': [3, 0, 0, 0]})], [({'nodes': [], 'root': -1}, {'first_previous': [], 'sibling_parent': [], 'sibling_previous': [], 'root_links': [], 'child_order': [], 'degrees': []}), ({'nodes': [[2, -1, -1, -1, -1]], 'root': 0}, {'first_previous': [], 'sibling_parent': [], 'sibling_previous': [], 'root_links': [], 'child_order': [], 'degrees': [0]}), ({'nodes': [[1, -1, 1, -1, -1], [7, 0, -1, 2, 0], [3, 0, -1, -1, 1]], 'root': 0}, {'first_previous': [], 'sibling_parent': [], 'sibling_previous': [], 'root_links': [], 'child_order': [], 'degrees': [2, 0, 0]}), ({'nodes': [[8, -1, 1, -1, -1], [9, 0, -1, 2, -1], [2, 1, -1, -1, 0]], 'root': 0}, {'first_previous': [0], 'sibling_parent': [1], 'sibling_previous': [1], 'root_links': [], 'child_order': [[0, 2]], 'degrees': [2, 0, 0]}), ({'nodes': [[1, -1, -1, -1, -1], [4, -1, 2, 0, 0], [4, 1, -1, -1, -1]], 'root': 1}, {'first_previous': [1], 'sibling_parent': [], 'sibling_previous': [1], 'root_links': [3, 4], 'child_order': [], 'degrees': [0, 1, 0]}), ({'nodes': [[7, -1, 1, -1, -1], [2, 0, -1, 2, 0], [8, -1, -1, 3, -1], [1, 0, -1, -1, 0]], 'root': 0}, {'first_previous': [], 'sibling_parent': [1, 2], 'sibling_previous': [1, 2], 'root_links': [], 'child_order': [[0, 1], [0, 3]], 'degrees': [3, 0, 0, 0]}), ({'nodes': [[11, -1, 1, -1, -1], [2, 0, -1, 2, 0], [8, -1, -1, 3, -1], [1, 0, -1, -1, 0]], 'root': 0}, {'first_previous': [], 'sibling_parent': [1, 2], 'sibling_previous': [1, 2], 'root_links': [], 'child_order': [[0, 1], [0, 2], [0, 3]], 'degrees': [3, 0, 0, 0]})], [({'nodes': [], 'root': -1}, {'first_previous': [], 'sibling_parent': [], 'sibling_previous': [], 'root_links': [], 'child_order': [], 'degrees': []}), ({'nodes': [[2, -1, -1, -1, -1]], 'root': 0}, {'first_previous': [], 'sibling_parent': [], 'sibling_previous': [], 'root_links': [], 'child_order': [], 'degrees': [0]}), ({'nodes': [[1, -1, 1, -1, -1], [7, 0, -1, 2, 0], [3, 0, -1, -1, 1]], 'root': 0}, {'first_previous': [], 'sibling_parent': [], 'sibling_previous': [], 'root_links': [], 'child_order': [], 'degrees': [2, 0, 0]}), ({'nodes': [[8, -1, 1, -1, -1], [9, 0, -1, 2, -1], [2, 1, -1, -1, 0]], 'root': 0}, {'first_previous': [0], 'sibling_parent': [1], 'sibling_previous': [1], 'root_links': [], 'child_order': [[0, 2]], 'degrees': [2, 0, 0]}), ({'nodes': [[1, -1, -1, -1, -1], [4, -1, 2, 0, 0], [4, 1, -1, -1, -1]], 'root': 1}, {'first_previous': [1], 'sibling_parent': [], 'sibling_previous': [1], 'root_links': [3, 4], 'child_order': [], 'degrees': [0, 1, 0]}), ({'nodes': [[7, -1, 1, -1, -1], [2, 0, -1, 2, 0], [8, -1, -1, 3, -1], [1, 0, -1, -1, 0]], 'root': 0}, {'first_previous': [], 'sibling_parent': [1, 2], 'sibling_previous': [1, 2], 'root_links': [], 'child_order': [[0, 1], [0, 3]], 'degrees': [3, 0, 0, 0]}), ({'nodes': [[12, -1, 1, -1, -1], [2, 0, -1, 2, 0], [8, -1, -1, 3, -1], [1, 0, -1, -1, 0]], 'root': 0}, {'first_previous': [], 'sibling_parent': [1, 2], 'sibling_previous': [1, 2], 'root_links': [], 'child_order': [[0, 1], [0, 2], [0, 3]], 'degrees': [3, 0, 0, 0]})]][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":"c386ad2663c64d43a2897bb55906a34071c6e800e8644842d6bb744704429ed4","source":"\"\"\"Failure Map reference implementation. Python standard library only.\"\"\"\nimport json\n\nN = 1\nobservations = []\ndef solve(d):\n    a=d['nodes']; root=d['root']; chains=[]\n    for i,x in enumerate(a):\n        chain=[]; j=x[2]\n        while j!=-1 and j not in chain:\n            chain.append(j); j=a[j][3]\n        chains.append(chain)\n    return {'first_previous': [i for i,x in enumerate(a) if x[2]!=-1 and a[x[2]][4]!=i],\n    'sibling_parent': [i for i,x in enumerate(a) if x[3]!=-1 and a[x[3]][1]!=x[1]],\n    'sibling_previous': [i for i,x in enumerate(a) if x[3]!=-1 and a[x[3]][4]!=i],\n    'root_links': [] if root==-1 else [k for k in (1,3,4) if a[root][k]!=-1],\n    'child_order': [[i,j] for i,c in enumerate(chains) for j in c if a[j][0]<a[i][0]],\n    'degrees': [int(bool(c)) for c in chains]}\ndef check(label, actual, expected):\n    observations.append({\"check\": label, \"actual\": actual, \"expected\": expected, \"passed\": actual == expected})\ncases = [[({'nodes': [], 'root': -1}, {'first_previous': [], 'sibling_parent': [], 'sibling_previous': [], 'root_links': [], 'child_order': [], 'degrees': []}), ({'nodes': [[2, -1, -1, -1, -1]], 'root': 0}, {'first_previous': [], 'sibling_parent': [], 'sibling_previous': [], 'root_links': [], 'child_order': [], 'degrees': [0]}), ({'nodes': [[1, -1, 1, -1, -1], [7, 0, -1, 2, 0], [3, 0, -1, -1, 1]], 'root': 0}, {'first_previous': [], 'sibling_parent': [], 'sibling_previous': [], 'root_links': [], 'child_order': [], 'degrees': [2, 0, 0]}), ({'nodes': [[8, -1, 1, -1, -1], [9, 0, -1, 2, -1], [2, 1, -1, -1, 0]], 'root': 0}, {'first_previous': [0], 'sibling_parent': [1], 'sibling_previous': [1], 'root_links': [], 'child_order': [[0, 2]], 'degrees': [2, 0, 0]}), ({'nodes': [[1, -1, -1, -1, -1], [4, -1, 2, 0, 0], [4, 1, -1, -1, -1]], 'root': 1}, {'first_previous': [1], 'sibling_parent': [], 'sibling_previous': [1], 'root_links': [3, 4], 'child_order': [], 'degrees': [0, 1, 0]}), ({'nodes': [[7, -1, 1, -1, -1], [2, 0, -1, 2, 0], [8, -1, -1, 3, -1], [1, 0, -1, -1, 0]], 'root': 0}, {'first_previous': [], 'sibling_parent': [1, 2], 'sibling_previous': [1, 2], 'root_links': [], 'child_order': [[0, 1], [0, 3]], 'degrees': [3, 0, 0, 0]}), ({'nodes': [[8, -1, 1, -1, -1], [2, 0, -1, 2, 0], [8, -1, -1, 3, -1], [1, 0, -1, -1, 0]], 'root': 0}, {'first_previous': [], 'sibling_parent': [1, 2], 'sibling_previous': [1, 2], 'root_links': [], 'child_order': [[0, 1], [0, 3]], 'degrees': [3, 0, 0, 0]})], [({'nodes': [], 'root': -1}, {'first_previous': [], 'sibling_parent': [], 'sibling_previous': [], 'root_links': [], 'child_order': [], 'degrees': []}), ({'nodes': [[2, -1, -1, -1, -1]], 'root': 0}, {'first_previous': [], 'sibling_parent': [], 'sibling_previous': [], 'root_links': [], 'child_order': [], 'degrees': [0]}), ({'nodes': [[1, -1, 1, -1, -1], [7, 0, -1, 2, 0], [3, 0, -1, -1, 1]], 'root': 0}, {'first_previous': [], 'sibling_parent': [], 'sibling_previous': [], 'root_links': [], 'child_order': [], 'degrees': [2, 0, 0]}), ({'nodes': [[8, -1, 1, -1, -1], [9, 0, -1, 2, -1], [2, 1, -1, -1, 0]], 'root': 0}, {'first_previous': [0], 'sibling_parent': [1], 'sibling_previous': [1], 'root_links': [], 'child_order': [[0, 2]], 'degrees': [2, 0, 0]}), ({'nodes': [[1, -1, -1, -1, -1], [4, -1, 2, 0, 0], [4, 1, -1, -1, -1]], 'root': 1}, {'first_previous': [1], 'sibling_parent': [], 'sibling_previous': [1], 'root_links': [3, 4], 'child_order': [], 'degrees': [0, 1, 0]}), ({'nodes': [[7, -1, 1, -1, -1], [2, 0, -1, 2, 0], [8, -1, -1, 3, -1], [1, 0, -1, -1, 0]], 'root': 0}, {'first_previous': [], 'sibling_parent': [1, 2], 'sibling_previous': [1, 2], 'root_links': [], 'child_order': [[0, 1], [0, 3]], 'degrees': [3, 0, 0, 0]}), ({'nodes': [[9, -1, 1, -1, -1], [2, 0, -1, 2, 0], [8, -1, -1, 3, -1], [1, 0, -1, -1, 0]], 'root': 0}, {'first_previous': [], 'sibling_parent': [1, 2], 'sibling_previous': [1, 2], 'root_links': [], 'child_order': [[0, 1], [0, 2], [0, 3]], 'degrees': [3, 0, 0, 0]})], [({'nodes': [], 'root': -1}, {'first_previous': [], 'sibling_parent': [], 'sibling_previous': [], 'root_links': [], 'child_order': [], 'degrees': []}), ({'nodes': [[2, -1, -1, -1, -1]], 'root': 0}, {'first_previous': [], 'sibling_parent': [], 'sibling_previous': [], 'root_links': [], 'child_order': [], 'degrees': [0]}), ({'nodes': [[1, -1, 1, -1, -1], [7, 0, -1, 2, 0], [3, 0, -1, -1, 1]], 'root': 0}, {'first_previous': [], 'sibling_parent': [], 'sibling_previous': [], 'root_links': [], 'child_order': [], 'degrees': [2, 0, 0]}), ({'nodes': [[8, -1, 1, -1, -1], [9, 0, -1, 2, -1], [2, 1, -1, -1, 0]], 'root': 0}, {'first_previous': [0], 'sibling_parent': [1], 'sibling_previous': [1], 'root_links': [], 'child_order': [[0, 2]], 'degrees': [2, 0, 0]}), ({'nodes': [[1, -1, -1, -1, -1], [4, -1, 2, 0, 0], [4, 1, -1, -1, -1]], 'root': 1}, {'first_previous': [1], 'sibling_parent': [], 'sibling_previous': [1], 'root_links': [3, 4], 'child_order': [], 'degrees': [0, 1, 0]}), ({'nodes': [[7, -1, 1, -1, -1], [2, 0, -1, 2, 0], [8, -1, -1, 3, -1], [1, 0, -1, -1, 0]], 'root': 0}, {'first_previous': [], 'sibling_parent': [1, 2], 'sibling_previous': [1, 2], 'root_links': [], 'child_order': [[0, 1], [0, 3]], 'degrees': [3, 0, 0, 0]}), ({'nodes': [[10, -1, 1, -1, -1], [2, 0, -1, 2, 0], [8, -1, -1, 3, -1], [1, 0, -1, -1, 0]], 'root': 0}, {'first_previous': [], 'sibling_parent': [1, 2], 'sibling_previous': [1, 2], 'root_links': [], 'child_order': [[0, 1], [0, 2], [0, 3]], 'degrees': [3, 0, 0, 0]})], [({'nodes': [], 'root': -1}, {'first_previous': [], 'sibling_parent': [], 'sibling_previous': [], 'root_links': [], 'child_order': [], 'degrees': []}), ({'nodes': [[2, -1, -1, -1, -1]], 'root': 0}, {'first_previous': [], 'sibling_parent': [], 'sibling_previous': [], 'root_links': [], 'child_order': [], 'degrees': [0]}), ({'nodes': [[1, -1, 1, -1, -1], [7, 0, -1, 2, 0], [3, 0, -1, -1, 1]], 'root': 0}, {'first_previous': [], 'sibling_parent': [], 'sibling_previous': [], 'root_links': [], 'child_order': [], 'degrees': [2, 0, 0]}), ({'nodes': [[8, -1, 1, -1, -1], [9, 0, -1, 2, -1], [2, 1, -1, -1, 0]], 'root': 0}, {'first_previous': [0], 'sibling_parent': [1], 'sibling_previous': [1], 'root_links': [], 'child_order': [[0, 2]], 'degrees': [2, 0, 0]}), ({'nodes': [[1, -1, -1, -1, -1], [4, -1, 2, 0, 0], [4, 1, -1, -1, -1]], 'root': 1}, {'first_previous': [1], 'sibling_parent': [], 'sibling_previous': [1], 'root_links': [3, 4], 'child_order': [], 'degrees': [0, 1, 0]}), ({'nodes': [[7, -1, 1, -1, -1], [2, 0, -1, 2, 0], [8, -1, -1, 3, -1], [1, 0, -1, -1, 0]], 'root': 0}, {'first_previous': [], 'sibling_parent': [1, 2], 'sibling_previous': [1, 2], 'root_links': [], 'child_order': [[0, 1], [0, 3]], 'degrees': [3, 0, 0, 0]}), ({'nodes': [[11, -1, 1, -1, -1], [2, 0, -1, 2, 0], [8, -1, -1, 3, -1], [1, 0, -1, -1, 0]], 'root': 0}, {'first_previous': [], 'sibling_parent': [1, 2], 'sibling_previous': [1, 2], 'root_links': [], 'child_order': [[0, 1], [0, 2], [0, 3]], 'degrees': [3, 0, 0, 0]})], [({'nodes': [], 'root': -1}, {'first_previous': [], 'sibling_parent': [], 'sibling_previous': [], 'root_links': [], 'child_order': [], 'degrees': []}), ({'nodes': [[2, -1, -1, -1, -1]], 'root': 0}, {'first_previous': [], 'sibling_parent': [], 'sibling_previous': [], 'root_links': [], 'child_order': [], 'degrees': [0]}), ({'nodes': [[1, -1, 1, -1, -1], [7, 0, -1, 2, 0], [3, 0, -1, -1, 1]], 'root': 0}, {'first_previous': [], 'sibling_parent': [], 'sibling_previous': [], 'root_links': [], 'child_order': [], 'degrees': [2, 0, 0]}), ({'nodes': [[8, -1, 1, -1, -1], [9, 0, -1, 2, -1], [2, 1, -1, -1, 0]], 'root': 0}, {'first_previous': [0], 'sibling_parent': [1], 'sibling_previous': [1], 'root_links': [], 'child_order': [[0, 2]], 'degrees': [2, 0, 0]}), ({'nodes': [[1, -1, -1, -1, -1], [4, -1, 2, 0, 0], [4, 1, -1, -1, -1]], 'root': 1}, {'first_previous': [1], 'sibling_parent': [], 'sibling_previous': [1], 'root_links': [3, 4], 'child_order': [], 'degrees': [0, 1, 0]}), ({'nodes': [[7, -1, 1, -1, -1], [2, 0, -1, 2, 0], [8, -1, -1, 3, -1], [1, 0, -1, -1, 0]], 'root': 0}, {'first_previous': [], 'sibling_parent': [1, 2], 'sibling_previous': [1, 2], 'root_links': [], 'child_order': [[0, 1], [0, 3]], 'degrees': [3, 0, 0, 0]}), ({'nodes': [[12, -1, 1, -1, -1], [2, 0, -1, 2, 0], [8, -1, -1, 3, -1], [1, 0, -1, -1, 0]], 'root': 0}, {'first_previous': [], 'sibling_parent': [1, 2], 'sibling_previous': [1, 2], 'root_links': [], 'child_order': [[0, 1], [0, 2], [0, 3]], 'degrees': [3, 0, 0, 0]})]][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":"c1da3dd990cdebd93e1c96c1808c57fca8226f272076f0500db19507ec380abd","source":"\"\"\"Failure Map reference implementation. Python standard library only.\"\"\"\nimport json\n\nN = 1\nobservations = []\ndef solve(d):\n    a=d['nodes']; root=d['root']; chains=[]\n    for i,x in enumerate(a):\n        chain=[]; j=x[2]\n        while j!=-1 and j not in chain:\n            chain.append(j); j=a[j][3]\n        chains.append(chain)\n    return {'first_previous': [i for i,x in enumerate(a) if x[2]!=-1 and a[x[2]][4]!=i],\n    'sibling_parent': [i for i,x in enumerate(a) if x[3]!=-1 and a[x[3]][1]!=x[1]],\n    'sibling_previous': [i for i,x in enumerate(a) if x[3]!=-1 and a[x[3]][4]!=i],\n    'root_links': [] if root==-1 else [k for k in (1,3,4) if a[root][k]!=-1],\n    'child_order': [[i,j] for i,c in enumerate(chains) for j in c if a[j][0]<a[i][0]],\n    'degrees': [len(c) for c in chains]}\ndef check(label, actual, expected):\n    observations.append({\"check\": label, \"actual\": actual, \"expected\": expected, \"passed\": actual == expected})\ncases = [[({'nodes': [], 'root': -1}, {'first_previous': [], 'sibling_parent': [], 'sibling_previous': [], 'root_links': [], 'child_order': [], 'degrees': []}), ({'nodes': [[2, -1, -1, -1, -1]], 'root': 0}, {'first_previous': [], 'sibling_parent': [], 'sibling_previous': [], 'root_links': [], 'child_order': [], 'degrees': [0]}), ({'nodes': [[1, -1, 1, -1, -1], [7, 0, -1, 2, 0], [3, 0, -1, -1, 1]], 'root': 0}, {'first_previous': [], 'sibling_parent': [], 'sibling_previous': [], 'root_links': [], 'child_order': [], 'degrees': [2, 0, 0]}), ({'nodes': [[8, -1, 1, -1, -1], [9, 0, -1, 2, -1], [2, 1, -1, -1, 0]], 'root': 0}, {'first_previous': [0], 'sibling_parent': [1], 'sibling_previous': [1], 'root_links': [], 'child_order': [[0, 2]], 'degrees': [2, 0, 0]}), ({'nodes': [[1, -1, -1, -1, -1], [4, -1, 2, 0, 0], [4, 1, -1, -1, -1]], 'root': 1}, {'first_previous': [1], 'sibling_parent': [], 'sibling_previous': [1], 'root_links': [3, 4], 'child_order': [], 'degrees': [0, 1, 0]}), ({'nodes': [[7, -1, 1, -1, -1], [2, 0, -1, 2, 0], [8, -1, -1, 3, -1], [1, 0, -1, -1, 0]], 'root': 0}, {'first_previous': [], 'sibling_parent': [1, 2], 'sibling_previous': [1, 2], 'root_links': [], 'child_order': [[0, 1], [0, 3]], 'degrees': [3, 0, 0, 0]}), ({'nodes': [[8, -1, 1, -1, -1], [2, 0, -1, 2, 0], [8, -1, -1, 3, -1], [1, 0, -1, -1, 0]], 'root': 0}, {'first_previous': [], 'sibling_parent': [1, 2], 'sibling_previous': [1, 2], 'root_links': [], 'child_order': [[0, 1], [0, 3]], 'degrees': [3, 0, 0, 0]})], [({'nodes': [], 'root': -1}, {'first_previous': [], 'sibling_parent': [], 'sibling_previous': [], 'root_links': [], 'child_order': [], 'degrees': []}), ({'nodes': [[2, -1, -1, -1, -1]], 'root': 0}, {'first_previous': [], 'sibling_parent': [], 'sibling_previous': [], 'root_links': [], 'child_order': [], 'degrees': [0]}), ({'nodes': [[1, -1, 1, -1, -1], [7, 0, -1, 2, 0], [3, 0, -1, -1, 1]], 'root': 0}, {'first_previous': [], 'sibling_parent': [], 'sibling_previous': [], 'root_links': [], 'child_order': [], 'degrees': [2, 0, 0]}), ({'nodes': [[8, -1, 1, -1, -1], [9, 0, -1, 2, -1], [2, 1, -1, -1, 0]], 'root': 0}, {'first_previous': [0], 'sibling_parent': [1], 'sibling_previous': [1], 'root_links': [], 'child_order': [[0, 2]], 'degrees': [2, 0, 0]}), ({'nodes': [[1, -1, -1, -1, -1], [4, -1, 2, 0, 0], [4, 1, -1, -1, -1]], 'root': 1}, {'first_previous': [1], 'sibling_parent': [], 'sibling_previous': [1], 'root_links': [3, 4], 'child_order': [], 'degrees': [0, 1, 0]}), ({'nodes': [[7, -1, 1, -1, -1], [2, 0, -1, 2, 0], [8, -1, -1, 3, -1], [1, 0, -1, -1, 0]], 'root': 0}, {'first_previous': [], 'sibling_parent': [1, 2], 'sibling_previous': [1, 2], 'root_links': [], 'child_order': [[0, 1], [0, 3]], 'degrees': [3, 0, 0, 0]}), ({'nodes': [[9, -1, 1, -1, -1], [2, 0, -1, 2, 0], [8, -1, -1, 3, -1], [1, 0, -1, -1, 0]], 'root': 0}, {'first_previous': [], 'sibling_parent': [1, 2], 'sibling_previous': [1, 2], 'root_links': [], 'child_order': [[0, 1], [0, 2], [0, 3]], 'degrees': [3, 0, 0, 0]})], [({'nodes': [], 'root': -1}, {'first_previous': [], 'sibling_parent': [], 'sibling_previous': [], 'root_links': [], 'child_order': [], 'degrees': []}), ({'nodes': [[2, -1, -1, -1, -1]], 'root': 0}, {'first_previous': [], 'sibling_parent': [], 'sibling_previous': [], 'root_links': [], 'child_order': [], 'degrees': [0]}), ({'nodes': [[1, -1, 1, -1, -1], [7, 0, -1, 2, 0], [3, 0, -1, -1, 1]], 'root': 0}, {'first_previous': [], 'sibling_parent': [], 'sibling_previous': [], 'root_links': [], 'child_order': [], 'degrees': [2, 0, 0]}), ({'nodes': [[8, -1, 1, -1, -1], [9, 0, -1, 2, -1], [2, 1, -1, -1, 0]], 'root': 0}, {'first_previous': [0], 'sibling_parent': [1], 'sibling_previous': [1], 'root_links': [], 'child_order': [[0, 2]], 'degrees': [2, 0, 0]}), ({'nodes': [[1, -1, -1, -1, -1], [4, -1, 2, 0, 0], [4, 1, -1, -1, -1]], 'root': 1}, {'first_previous': [1], 'sibling_parent': [], 'sibling_previous': [1], 'root_links': [3, 4], 'child_order': [], 'degrees': [0, 1, 0]}), ({'nodes': [[7, -1, 1, -1, -1], [2, 0, -1, 2, 0], [8, -1, -1, 3, -1], [1, 0, -1, -1, 0]], 'root': 0}, {'first_previous': [], 'sibling_parent': [1, 2], 'sibling_previous': [1, 2], 'root_links': [], 'child_order': [[0, 1], [0, 3]], 'degrees': [3, 0, 0, 0]}), ({'nodes': [[10, -1, 1, -1, -1], [2, 0, -1, 2, 0], [8, -1, -1, 3, -1], [1, 0, -1, -1, 0]], 'root': 0}, {'first_previous': [], 'sibling_parent': [1, 2], 'sibling_previous': [1, 2], 'root_links': [], 'child_order': [[0, 1], [0, 2], [0, 3]], 'degrees': [3, 0, 0, 0]})], [({'nodes': [], 'root': -1}, {'first_previous': [], 'sibling_parent': [], 'sibling_previous': [], 'root_links': [], 'child_order': [], 'degrees': []}), ({'nodes': [[2, -1, -1, -1, -1]], 'root': 0}, {'first_previous': [], 'sibling_parent': [], 'sibling_previous': [], 'root_links': [], 'child_order': [], 'degrees': [0]}), ({'nodes': [[1, -1, 1, -1, -1], [7, 0, -1, 2, 0], [3, 0, -1, -1, 1]], 'root': 0}, {'first_previous': [], 'sibling_parent': [], 'sibling_previous': [], 'root_links': [], 'child_order': [], 'degrees': [2, 0, 0]}), ({'nodes': [[8, -1, 1, -1, -1], [9, 0, -1, 2, -1], [2, 1, -1, -1, 0]], 'root': 0}, {'first_previous': [0], 'sibling_parent': [1], 'sibling_previous': [1], 'root_links': [], 'child_order': [[0, 2]], 'degrees': [2, 0, 0]}), ({'nodes': [[1, -1, -1, -1, -1], [4, -1, 2, 0, 0], [4, 1, -1, -1, -1]], 'root': 1}, {'first_previous': [1], 'sibling_parent': [], 'sibling_previous': [1], 'root_links': [3, 4], 'child_order': [], 'degrees': [0, 1, 0]}), ({'nodes': [[7, -1, 1, -1, -1], [2, 0, -1, 2, 0], [8, -1, -1, 3, -1], [1, 0, -1, -1, 0]], 'root': 0}, {'first_previous': [], 'sibling_parent': [1, 2], 'sibling_previous': [1, 2], 'root_links': [], 'child_order': [[0, 1], [0, 3]], 'degrees': [3, 0, 0, 0]}), ({'nodes': [[11, -1, 1, -1, -1], [2, 0, -1, 2, 0], [8, -1, -1, 3, -1], [1, 0, -1, -1, 0]], 'root': 0}, {'first_previous': [], 'sibling_parent': [1, 2], 'sibling_previous': [1, 2], 'root_links': [], 'child_order': [[0, 1], [0, 2], [0, 3]], 'degrees': [3, 0, 0, 0]})], [({'nodes': [], 'root': -1}, {'first_previous': [], 'sibling_parent': [], 'sibling_previous': [], 'root_links': [], 'child_order': [], 'degrees': []}), ({'nodes': [[2, -1, -1, -1, -1]], 'root': 0}, {'first_previous': [], 'sibling_parent': [], 'sibling_previous': [], 'root_links': [], 'child_order': [], 'degrees': [0]}), ({'nodes': [[1, -1, 1, -1, -1], [7, 0, -1, 2, 0], [3, 0, -1, -1, 1]], 'root': 0}, {'first_previous': [], 'sibling_parent': [], 'sibling_previous': [], 'root_links': [], 'child_order': [], 'degrees': [2, 0, 0]}), ({'nodes': [[8, -1, 1, -1, -1], [9, 0, -1, 2, -1], [2, 1, -1, -1, 0]], 'root': 0}, {'first_previous': [0], 'sibling_parent': [1], 'sibling_previous': [1], 'root_links': [], 'child_order': [[0, 2]], 'degrees': [2, 0, 0]}), ({'nodes': [[1, -1, -1, -1, -1], [4, -1, 2, 0, 0], [4, 1, -1, -1, -1]], 'root': 1}, {'first_previous': [1], 'sibling_parent': [], 'sibling_previous': [1], 'root_links': [3, 4], 'child_order': [], 'degrees': [0, 1, 0]}), ({'nodes': [[7, -1, 1, -1, -1], [2, 0, -1, 2, 0], [8, -1, -1, 3, -1], [1, 0, -1, -1, 0]], 'root': 0}, {'first_previous': [], 'sibling_parent': [1, 2], 'sibling_previous': [1, 2], 'root_links': [], 'child_order': [[0, 1], [0, 3]], 'degrees': [3, 0, 0, 0]}), ({'nodes': [[12, -1, 1, -1, -1], [2, 0, -1, 2, 0], [8, -1, -1, 3, -1], [1, 0, -1, -1, 0]], 'root': 0}, {'first_previous': [], 'sibling_parent': [1, 2], 'sibling_previous': [1, 2], 'root_links': [], 'child_order': [[0, 1], [0, 2], [0, 3]], 'degrees': [3, 0, 0, 0]})]][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-pairing-siblings-degrees","generated_at":"2026-09-29T14:43:26.390185+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 degrees using [len(c) for c in chains] under the stated bounded certificate contract.","root_cause":"The full sibling chain contains direct children, including its first member.","sha256":"f2bba8967fd502e67d5abda5882ff9ee8de08e5f4359929af906735283f07ae2","title":"Pairing degree counts only the first child pointer · 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":39.588,"exit_code":1,"observations":[{"actual":{"child_order":[],"degrees":[],"first_previous":[],"root_links":[],"sibling_parent":[],"sibling_previous":[]},"check":"regression certificate 1","expected":{"child_order":[],"degrees":[],"first_previous":[],"root_links":[],"sibling_parent":[],"sibling_previous":[]},"passed":true},{"actual":{"child_order":[],"degrees":[0],"first_previous":[],"root_links":[],"sibling_parent":[],"sibling_previous":[]},"check":"regression certificate 2","expected":{"child_order":[],"degrees":[0],"first_previous":[],"root_links":[],"sibling_parent":[],"sibling_previous":[]},"passed":true},{"actual":{"child_order":[],"degrees":[1,0,0],"first_previous":[],"root_links":[],"sibling_parent":[],"sibling_previous":[]},"check":"regression certificate 3","expected":{"child_order":[],"degrees":[2,0,0],"first_previous":[],"root_links":[],"sibling_parent":[],"sibling_previous":[]},"passed":false},{"actual":{"child_order":[[0,2]],"degrees":[1,0,0],"first_previous":[0],"root_links":[],"sibling_parent":[1],"sibling_previous":[1]},"check":"regression certificate 4","expected":{"child_order":[[0,2]],"degrees":[2,0,0],"first_previous":[0],"root_links":[],"sibling_parent":[1],"sibling_previous":[1]},"passed":false},{"actual":{"child_order":[],"degrees":[0,0,0],"first_previous":[1],"root_links":[3,4],"sibling_parent":[],"sibling_previous":[1]},"check":"regression certificate 5","expected":{"child_order":[],"degrees":[0,1,0],"first_previous":[1],"root_links":[3,4],"sibling_parent":[],"sibling_previous":[1]},"passed":false},{"actual":{"child_order":[[0,1],[0,3]],"degrees":[2,0,0,0],"first_previous":[],"root_links":[],"sibling_parent":[1,2],"sibling_previous":[1,2]},"check":"regression certificate 6","expected":{"child_order":[[0,1],[0,3]],"degrees":[3,0,0,0],"first_previous":[],"root_links":[],"sibling_parent":[1,2],"sibling_previous":[1,2]},"passed":false},{"actual":{"child_order":[[0,1],[0,3]],"degrees":[2,0,0,0],"first_previous":[],"root_links":[],"sibling_parent":[1,2],"sibling_previous":[1,2]},"check":"variant-dependent certificate","expected":{"child_order":[[0,1],[0,3]],"degrees":[3,0,0,0],"first_previous":[],"root_links":[],"sibling_parent":[1,2],"sibling_previous":[1,2]},"passed":false}],"passed":false,"stderr":"","stdout":"{\"observations\": [{\"check\": \"regression certificate 1\", \"actual\": {\"first_previous\": [], \"sibling_parent\": [], \"sibling_previous\": [], \"root_links\": [], \"child_order\": [], \"degrees\": []}, \"expected\": {\"first_previous\": [], \"sibling_parent\": [], \"sibling_previous\": [], \"root_links\": [], \"child_order\": [], \"degrees\": []}, \"passed\": true}, {\"check\": \"regression certificate 2\", \"actual\": {\"first_previous\": [], \"sibling_parent\": [], \"sibling_previous\": [], \"root_links\": [], \"child_order\": [], \"degrees\": [0]}, \"expected\": {\"first_previous\": [], \"sibling_parent\": [], \"sibling_previous\": [], \"root_links\": [], \"child_order\": [], \"degrees\": [0]}, \"passed\": true}, {\"check\": \"regression certificate 3\", \"actual\": {\"first_previous\": [], \"sibling_parent\": [], \"sibling_previous\": [], \"root_links\": [], \"child_order\": [], \"degrees\": [1, 0, 0]}, \"expected\": {\"first_previous\": [], \"sibling_parent\": [], \"sibling_previous\": [], \"root_links\": [], \"child_order\": [], \"degrees\": [2, 0, 0]}, \"passed\": false}, {\"check\": \"regression certificate 4\", \"actual\": {\"first_previous\": [0], \"sibling_parent\": [1], \"sibling_previous\": [1], \"root_links\": [], \"child_order\": [[0, 2]], \"degrees\": [1, 0, 0]}, \"expected\": {\"first_previous\": [0], \"sibling_parent\": [1], \"sibling_previous\": [1], \"root_links\": [], \"child_order\": [[0, 2]], \"degrees\": [2, 0, 0]}, \"passed\": false}, {\"check\": \"regression certificate 5\", \"actual\": {\"first_previous\": [1], \"sibling_parent\": [], \"sibling_previous\": [1], \"root_links\": [3, 4], \"child_order\": [], \"degrees\": [0, 0, 0]}, \"expected\": {\"first_previous\": [1], \"sibling_parent\": [], \"sibling_previous\": [1], \"root_links\": [3, 4], \"child_order\": [], \"degrees\": [0, 1, 0]}, \"passed\": false}, {\"check\": \"regression certificate 6\", \"actual\": {\"first_previous\": [], \"sibling_parent\": [1, 2], \"sibling_previous\": [1, 2], \"root_links\": [], \"child_order\": [[0, 1], [0, 3]], \"degrees\": [2, 0, 0, 0]}, \"expected\": {\"first_previous\": [], \"sibling_parent\": [1, 2], \"sibling_previous\": [1, 2], \"root_links\": [], \"child_order\": [[0, 1], [0, 3]], \"degrees\": [3, 0, 0, 0]}, \"passed\": false}, {\"check\": \"variant-dependent certificate\", \"actual\": {\"first_previous\": [], \"sibling_parent\": [1, 2], \"sibling_previous\": [1, 2], \"root_links\": [], \"child_order\": [[0, 1], [0, 3]], \"degrees\": [2, 0, 0, 0]}, \"expected\": {\"first_previous\": [], \"sibling_parent\": [1, 2], \"sibling_previous\": [1, 2], \"root_links\": [], \"child_order\": [[0, 1], [0, 3]], \"degrees\": [3, 0, 0, 0]}, \"passed\": false}], \"passed\": false}\n"},"broken":{"elapsed_ms":39.669,"exit_code":1,"observations":[{"actual":{"child_order":[],"degrees":[],"first_previous":[],"root_links":[],"sibling_parent":[],"sibling_previous":[]},"check":"regression certificate 1","expected":{"child_order":[],"degrees":[],"first_previous":[],"root_links":[],"sibling_parent":[],"sibling_previous":[]},"passed":true},{"actual":{"child_order":[],"degrees":[0],"first_previous":[],"root_links":[],"sibling_parent":[],"sibling_previous":[]},"check":"regression certificate 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