{"abstract":"The root-at-height-zero convention is shifted by one.","category":"Combinatorics","checks":4,"contract":"Nonnegative integer counts. Permutation and combination counts are zero when a requested selection exceeds the available objects. Shape sizes satisfy the preconditions stated below. Every level is full and the root has height zero; count all vertices through height. Exact operational definition: 2**(height+1)-1","contract_signature":"height","evaluation_group":"model-1f026ac51730e54f","failed_approach":"Only the last level is counted.","family":"xn-perfect-binary-tree-total-nodes","id":"FA-5721","implementations":{"attempt":{"sha256":"9be10b6e9e7730c3bc20dafdea68a0c30710e094e8a4e5c71f7dbf2ea3ab204f","source":"\"\"\"Failure Map reference implementation. Python standard library only.\"\"\"\nimport json\nimport math\nimport calendar\nimport statistics\nimport itertools\nfrom fractions import Fraction\nfrom datetime import date, datetime, timedelta, timezone\nfrom decimal import Decimal, ROUND_HALF_UP, ROUND_DOWN, ROUND_CEILING, ROUND_FLOOR\n\nN = 1\nobservations = []\ndef solve(height):\n    return 2**height\ndef check(label, actual, expected):\n    observations.append({\"check\": label, \"actual\": actual, \"expected\": expected, \"passed\": actual == expected})\ncheck('fixture 1: (0,)', solve(*(0,)), 1)\ncheck('fixture 2: (1,)', solve(*(1,)), 3)\ncheck('fixture 3: (3,)', solve(*(3,)), 15)\ncheck('fixture 4: (5,)', solve(*(5,)), 63)\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":"ebbdd1769cd1f9fad5d82cec9668fe874460fa04e9cfe8aab9446b1a9fd6fc2c","source":"\"\"\"Failure Map reference implementation. Python standard library only.\"\"\"\nimport json\nimport math\nimport calendar\nimport statistics\nimport itertools\nfrom fractions import Fraction\nfrom datetime import date, datetime, timedelta, timezone\nfrom decimal import Decimal, ROUND_HALF_UP, ROUND_DOWN, ROUND_CEILING, ROUND_FLOOR\n\nN = 1\nobservations = []\ndef solve(height):\n    return 2**height-1\ndef check(label, actual, expected):\n    observations.append({\"check\": label, \"actual\": actual, \"expected\": expected, \"passed\": actual == expected})\ncheck('fixture 1: (0,)', solve(*(0,)), 1)\ncheck('fixture 2: (1,)', solve(*(1,)), 3)\ncheck('fixture 3: (3,)', solve(*(3,)), 15)\ncheck('fixture 4: (5,)', solve(*(5,)), 63)\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":" 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":"xn-perfect-binary-tree-total-nodes","generated_at":"2026-09-29T14:37:52.825180+00:00","license":"CC0-1.0","python":"3.12.14","seed":1,"split":"open-access"},"relevance":"Small exact fixtures expose this error without platform timing, external services, or probabilistic observations. Combinatorics results depend on the stated convention.","root_cause":"The root-at-height-zero convention is shifted by one.","sha256":"254f019e9977e7cc96fed11a1da03b38050e77e646ca592e554d3769db8be164","title":"Perfect binary tree total nodes · case 01","variant":1,"variant_policy":"Five numbered records share a model and may reuse boundary fixtures.","verified":true,"visibility":"public","verification":{"attempt":{"elapsed_ms":44.557,"exit_code":1,"observations":[{"actual":1,"check":"fixture 1: (0,)","expected":1,"passed":true},{"actual":2,"check":"fixture 2: (1,)","expected":3,"passed":false},{"actual":8,"check":"fixture 3: (3,)","expected":15,"passed":false},{"actual":32,"check":"fixture 4: (5,)","expected":63,"passed":false}],"passed":false,"stderr":"","stdout":"{\"observations\": [{\"check\": \"fixture 1: (0,)\", \"actual\": 1, \"expected\": 1, \"passed\": true}, {\"check\": \"fixture 2: (1,)\", \"actual\": 2, \"expected\": 3, \"passed\": false}, {\"check\": \"fixture 3: (3,)\", \"actual\": 8, \"expected\": 15, \"passed\": false}, {\"check\": \"fixture 4: (5,)\", \"actual\": 32, \"expected\": 63, \"passed\": false}], \"passed\": false}\n"},"broken":{"elapsed_ms":48.475,"exit_code":1,"observations":[{"actual":0,"check":"fixture 1: (0,)","expected":1,"passed":false},{"actual":1,"check":"fixture 2: (1,)","expected":3,"passed":false},{"actual":7,"check":"fixture 3: (3,)","expected":15,"passed":false},{"actual":31,"check":"fixture 4: (5,)","expected":63,"passed":false}],"passed":false,"stderr":"","stdout":"{\"observations\": [{\"check\": \"fixture 1: (0,)\", \"actual\": 0, \"expected\": 1, \"passed\": false}, {\"check\": \"fixture 2: (1,)\", \"actual\": 1, \"expected\": 3, \"passed\": false}, {\"check\": \"fixture 3: (3,)\", \"actual\": 7, \"expected\": 15, \"passed\": false}, {\"check\": \"fixture 4: (5,)\", \"actual\": 31, \"expected\": 63, \"passed\": false}], \"passed\": false}\n"}},"member_only":{"stages":["fixed"],"fields":["implementations.fixed","verification.fixed","harness","repair"],"note":"The verified repair, its recorded checks, the repair description, and the scoring harness are available to members."}}