{"abstract":"The adjugate is not divided by the determinant.","category":"Linear algebra","checks":4,"contract":"Integer matrix/vector entries, rectangular rows, compatible multiplication dimensions, and square matrices for determinant, trace, inverse, symmetry and characteristic-polynomial operations. Rational matrix outputs are reduced Fraction strings. Input is 2 by 2; return None for a singular matrix and otherwise a 2 by 2 list of reduced Fraction strings. Exact operational definition: [[str(Fraction(x,m[0][0]*m[1][1]-m[0][1]*m[1][0])) for x in row] for row in [[m[1][1],-m[0][1]],[-m[1][0],m[0][0]]]] if m[0][0]*m[1][1]!=m[0][1]*m[1][0] else None","evaluation_group":"model-d92548210f5aa39c","failed_approach":"Elementwise reciprocals do not invert a linear map.","family":"xn-two-by-two-inverse-rational","id":"FA-6066","implementations":{"attempt":{"sha256":"86fb3df6a436b71b0e35f1a283b2598594d726f149bce28ad3cfe9d5007a82a6","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(m):\n    return [[str(Fraction(1,x)) if x else \"0\" for x in row] for row in m]\ndef check(label, actual, expected):\n    observations.append({\"check\": label, \"actual\": actual, \"expected\": expected, \"passed\": actual == expected})\ncheck('fixture 1: ([[2, 0], [0, 4]],)', solve(*([[2, 0], [0, 4]],)), [['1/2', '0'], ['0', '1/4']])\ncheck('fixture 2: ([[1, 2], [3, 4]],)', solve(*([[1, 2], [3, 4]],)), [['-2', '1'], ['3/2', '-1/2']])\ncheck('fixture 3: ([[1, 2], [2, 4]],)', solve(*([[1, 2], [2, 4]],)), None)\ncheck('fixture 4: ([[1, 0], [0, 1]],)', solve(*([[1, 0], [0, 1]],)), [['1', '0'], ['0', '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":"30f0264bd4e8d655e6bc3f21b9f75f6738e01bcc87a5055b7ac11319025d9b56","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(m):\n    return [[str(x) for x in row] for row in [[m[1][1],-m[0][1]],[-m[1][0],m[0][0]]]]\ndef check(label, actual, expected):\n    observations.append({\"check\": label, \"actual\": actual, \"expected\": expected, \"passed\": actual == expected})\ncheck('fixture 1: ([[2, 0], [0, 4]],)', solve(*([[2, 0], [0, 4]],)), [['1/2', '0'], ['0', '1/4']])\ncheck('fixture 2: ([[1, 2], [3, 4]],)', solve(*([[1, 2], [3, 4]],)), [['-2', '1'], ['3/2', '-1/2']])\ncheck('fixture 3: ([[1, 2], [2, 4]],)', solve(*([[1, 2], [2, 4]],)), None)\ncheck('fixture 4: ([[1, 0], [0, 1]],)', solve(*([[1, 0], [0, 1]],)), [['1', '0'], ['0', '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":"e624fd0139a84f1a0fb2e51105dc99150c52dc2c217dc04ddc33b762d170a51e","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(m):\n    return [[str(Fraction(x,m[0][0]*m[1][1]-m[0][1]*m[1][0])) for x in row] for row in [[m[1][1],-m[0][1]],[-m[1][0],m[0][0]]]] if m[0][0]*m[1][1]!=m[0][1]*m[1][0] else None\ndef check(label, actual, expected):\n    observations.append({\"check\": label, \"actual\": actual, \"expected\": expected, \"passed\": actual == expected})\ncheck('fixture 1: ([[2, 0], [0, 4]],)', solve(*([[2, 0], [0, 4]],)), [['1/2', '0'], ['0', '1/4']])\ncheck('fixture 2: ([[1, 2], [3, 4]],)', solve(*([[1, 2], [3, 4]],)), [['-2', '1'], ['3/2', '-1/2']])\ncheck('fixture 3: ([[1, 2], [2, 4]],)', solve(*([[1, 2], [2, 4]],)), None)\ncheck('fixture 4: ([[1, 0], [0, 1]],)', solve(*([[1, 0], [0, 1]],)), [['1', '0'], ['0', '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":" 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-two-by-two-inverse-rational","generated_at":"2026-09-29T14:37:56.999706+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. Linear algebra results depend on the stated convention.","repair":"Apply the specified mathematical contract directly, preserving all terms and boundary cases: return [[str(Fraction(x,m[0][0]*m[1][1]-m[0][1]*m[1][0])) for x in row] for row in [[m[1][1],-m[0][1]],[-m[1][0],m[0][0]]]] if m[0][0]*m[1][1]!=m[0][1]*m[1][0] else None","root_cause":"The adjugate is not divided by the determinant.","sha256":"b0bf0ebce921f4f6b8f4b694b581a1acad3b944ef05ca6f69395fc48f906e557","title":"Two by two inverse rational · case 01","variant":1,"variant_policy":"Five numbered records share a model and may reuse boundary fixtures.","verification":{"attempt":{"elapsed_ms":46.148,"exit_code":1,"observations":[{"actual":[["1/2","0"],["0","1/4"]],"check":"fixture 1: ([[2, 0], [0, 4]],)","expected":[["1/2","0"],["0","1/4"]],"passed":true},{"actual":[["1","1/2"],["1/3","1/4"]],"check":"fixture 2: ([[1, 2], [3, 4]],)","expected":[["-2","1"],["3/2","-1/2"]],"passed":false},{"actual":[["1","1/2"],["1/2","1/4"]],"check":"fixture 3: ([[1, 2], [2, 4]],)","expected":null,"passed":false},{"actual":[["1","0"],["0","1"]],"check":"fixture 4: ([[1, 0], [0, 1]],)","expected":[["1","0"],["0","1"]],"passed":true}],"passed":false,"stderr":"","stdout":"{\"observations\": [{\"check\": \"fixture 1: ([[2, 0], [0, 4]],)\", \"actual\": [[\"1/2\", \"0\"], [\"0\", \"1/4\"]], \"expected\": [[\"1/2\", \"0\"], [\"0\", \"1/4\"]], \"passed\": true}, {\"check\": \"fixture 2: ([[1, 2], [3, 4]],)\", \"actual\": [[\"1\", \"1/2\"], [\"1/3\", \"1/4\"]], \"expected\": [[\"-2\", \"1\"], [\"3/2\", \"-1/2\"]], \"passed\": false}, {\"check\": \"fixture 3: ([[1, 2], [2, 4]],)\", \"actual\": [[\"1\", \"1/2\"], [\"1/2\", \"1/4\"]], \"expected\": null, \"passed\": false}, {\"check\": \"fixture 4: ([[1, 0], [0, 1]],)\", \"actual\": [[\"1\", \"0\"], [\"0\", \"1\"]], \"expected\": [[\"1\", \"0\"], [\"0\", \"1\"]], \"passed\": true}], \"passed\": false}\n"},"broken":{"elapsed_ms":45.744,"exit_code":1,"observations":[{"actual":[["4","0"],["0","2"]],"check":"fixture 1: ([[2, 0], [0, 4]],)","expected":[["1/2","0"],["0","1/4"]],"passed":false},{"actual":[["4","-2"],["-3","1"]],"check":"fixture 2: ([[1, 2], [3, 4]],)","expected":[["-2","1"],["3/2","-1/2"]],"passed":false},{"actual":[["4","-2"],["-2","1"]],"check":"fixture 3: ([[1, 2], [2, 4]],)","expected":null,"passed":false},{"actual":[["1","0"],["0","1"]],"check":"fixture 4: ([[1, 0], [0, 1]],)","expected":[["1","0"],["0","1"]],"passed":true}],"passed":false,"stderr":"","stdout":"{\"observations\": [{\"check\": \"fixture 1: ([[2, 0], [0, 4]],)\", \"actual\": [[\"4\", \"0\"], [\"0\", \"2\"]], \"expected\": [[\"1/2\", \"0\"], [\"0\", \"1/4\"]], \"passed\": false}, {\"check\": \"fixture 2: ([[1, 2], [3, 4]],)\", \"actual\": [[\"4\", \"-2\"], [\"-3\", \"1\"]], \"expected\": [[\"-2\", \"1\"], [\"3/2\", \"-1/2\"]], \"passed\": false}, {\"check\": \"fixture 3: ([[1, 2], [2, 4]],)\", \"actual\": [[\"4\", \"-2\"], [\"-2\", \"1\"]], \"expected\": null, \"passed\": false}, {\"check\": \"fixture 4: ([[1, 0], [0, 1]],)\", \"actual\": [[\"1\", \"0\"], [\"0\", \"1\"]], \"expected\": [[\"1\", \"0\"], [\"0\", \"1\"]], \"passed\": true}], \"passed\": false}\n"},"fixed":{"elapsed_ms":47.748,"exit_code":0,"observations":[{"actual":[["1/2","0"],["0","1/4"]],"check":"fixture 1: ([[2, 0], [0, 4]],)","expected":[["1/2","0"],["0","1/4"]],"passed":true},{"actual":[["-2","1"],["3/2","-1/2"]],"check":"fixture 2: ([[1, 2], [3, 4]],)","expected":[["-2","1"],["3/2","-1/2"]],"passed":true},{"actual":null,"check":"fixture 3: ([[1, 2], [2, 4]],)","expected":null,"passed":true},{"actual":[["1","0"],["0","1"]],"check":"fixture 4: ([[1, 0], [0, 1]],)","expected":[["1","0"],["0","1"]],"passed":true}],"passed":true,"stderr":"","stdout":"{\"observations\": [{\"check\": \"fixture 1: ([[2, 0], [0, 4]],)\", \"actual\": [[\"1/2\", \"0\"], [\"0\", \"1/4\"]], \"expected\": [[\"1/2\", \"0\"], [\"0\", \"1/4\"]], \"passed\": true}, {\"check\": \"fixture 2: ([[1, 2], [3, 4]],)\", \"actual\": [[\"-2\", \"1\"], [\"3/2\", \"-1/2\"]], \"expected\": [[\"-2\", \"1\"], [\"3/2\", \"-1/2\"]], \"passed\": true}, {\"check\": \"fixture 3: ([[1, 2], [2, 4]],)\", \"actual\": null, \"expected\": null, \"passed\": true}, {\"check\": \"fixture 4: ([[1, 0], [0, 1]],)\", \"actual\": [[\"1\", \"0\"], [\"0\", \"1\"]], \"expected\": [[\"1\", \"0\"], [\"0\", \"1\"]], \"passed\": true}], \"passed\": true}\n"}},"verified":true,"visibility":"public"}