{"abstract":"The exact lagrange interpolation value result violates the stated contract at basis denominator orientation.","category":"Numerics","checks":8,"contract":"Input [points,t] distinct integer abscissae and integer ordinates; exact rational interpolation value [p,q].","contract_signature":"x","evaluation_group":"s3-numerics-lagrange-interpolation-value","failed_approach":"The partial repair abs(xi-xj) still violates the basis denominator orientation invariant.","family":"s3-numerics-lagrange-interpolation-value-basis-denominator-orientation","id":"FA-14676","implementations":{"attempt":{"sha256":"eea85b9275c86b82518ef5ee2f8e24f5cfd5d304ca528ed958a5d819db21d0d2","source":"\"\"\"Failure Map reference implementation. Python standard library only.\"\"\"\nimport json\nimport math\nimport itertools\nfrom fractions import Fraction\nN = 1\nobservations = []\ndef solve(x):\n    points,t=x\n    ans=Fraction(0)\n    for i,(xi,yi) in enumerate(points):\n     term=Fraction(yi)\n     for j,(xj,yj) in enumerate(points):\n      if i==j:continue\n      den=abs(xi-xj)\n      if den==0:return None\n      term*=Fraction(t-xj,den)\n     ans=ans+term\n    return [ans.numerator,ans.denominator]\ndef check(label, actual, expected):\n    observations.append({\"check\": label, \"actual\": actual, \"expected\": expected, \"passed\": actual == expected})\nfixtures = [[([[[0, -2], [2, -2]], -2], [-2, 1]), ([[[-2, -2], [-1, -2], [0, -2]], -3], [-2, 1]), ([[[0, 2], [2, 2]], 3], [2, 1]), ([[[-2, -2], [-1, -2], [0, -2]], -2], [-2, 1]), ([[[-2, -2], [-1, -2], [0, -2]], -1], [-2, 1]), ([[[-2, -2], [-1, -2], [0, -2]], 0], [-2, 1]), ([[[-2, -2], [-1, -2], [0, -2]], 1], [-2, 1]), ([[[-2, -2], [-1, -2], [0, -2]], 2], [-2, 1])], [([[[0, -2], [2, -2]], -1], [-2, 1]), ([[[-2, -2], [-1, -2], [0, -2]], 2], [-2, 1]), ([[[-2, -2], [-1, -2], [0, -2]], -3], [-2, 1]), ([[[0, 2], [2, 2]], 3], [2, 1]), ([[[-2, -2], [-1, -2], [0, 0]], 0], [0, 1]), ([[[-2, -2], [-1, -2], [0, 0]], 1], [4, 1]), ([[[-2, -2], [-1, -2], [0, 0]], 2], [10, 1]), ([[[-2, -2], [-1, -2], [0, 0]], 3], [18, 1])], [([[[0, -2], [2, -2]], 0], [-2, 1]), ([[[-2, -2], [-1, -2], [0, -1]], -1], [-2, 1]), ([[[-2, -2], [-1, -2], [0, -2]], -3], [-2, 1]), ([[[0, 2], [2, 2]], 3], [2, 1]), ([[[-2, -2], [-1, -2], [0, 2]], 3], [38, 1]), ([[[-2, -2], [-1, -1], [0, -2]], -3], [-5, 1]), ([[[-2, -2], [-1, -1], [0, -2]], -2], [-2, 1]), ([[[-2, -2], [-1, -1], [0, -2]], -1], [-1, 1])], [([[[0, -2], [2, -2]], 1], [-2, 1]), ([[[-2, -2], [-1, -2], [0, -1]], 3], [8, 1]), ([[[-2, -2], [-1, -2], [0, -2]], -3], [-2, 1]), ([[[0, 2], [2, 2]], 3], [2, 1]), ([[[-2, -2], [-1, -1], [0, 0]], -1], [-1, 1]), ([[[-2, -2], [-1, -1], [0, 0]], 0], [0, 1]), ([[[-2, -2], [-1, -1], [0, 0]], 1], [1, 1]), ([[[-2, -2], [-1, -1], [0, 0]], 2], [2, 1])], [([[[0, -2], [2, -2]], 2], [-2, 1]), ([[[-2, -2], [-1, -2], [0, 0]], 1], [4, 1]), ([[[-2, -2], [-1, -2], [0, -2]], -3], [-2, 1]), ([[[0, 2], [2, 2]], 3], [2, 1]), ([[[-2, -2], [-1, -1], [0, 2]], 2], [14, 1]), ([[[-2, -2], [-1, -1], [0, 2]], 3], [23, 1]), ([[[-2, -2], [-1, 0], [0, -2]], -3], [-8, 1]), ([[[-2, -2], [-1, 0], [0, -2]], -2], [-2, 1])]]\nfor i, (args, expected) in enumerate(fixtures[N-1]):\n    check(\"explicit oracle %d\" % i, solve(args), expected)\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":"7e2f498dc5fdc67fc028e41a4e798e2c550da0b6783023938695ec2ef7806626","source":"\"\"\"Failure Map reference implementation. Python standard library only.\"\"\"\nimport json\nimport math\nimport itertools\nfrom fractions import Fraction\nN = 1\nobservations = []\ndef solve(x):\n    points,t=x\n    ans=Fraction(0)\n    for i,(xi,yi) in enumerate(points):\n     term=Fraction(yi)\n     for j,(xj,yj) in enumerate(points):\n      if i==j:continue\n      den=xj-xi\n      if den==0:return None\n      term*=Fraction(t-xj,den)\n     ans=ans+term\n    return [ans.numerator,ans.denominator]\ndef check(label, actual, expected):\n    observations.append({\"check\": label, \"actual\": actual, \"expected\": expected, \"passed\": actual == expected})\nfixtures = [[([[[0, -2], [2, -2]], -2], [-2, 1]), ([[[-2, -2], [-1, -2], [0, -2]], -3], [-2, 1]), ([[[0, 2], [2, 2]], 3], [2, 1]), ([[[-2, -2], [-1, -2], [0, -2]], -2], [-2, 1]), ([[[-2, -2], [-1, -2], [0, -2]], -1], [-2, 1]), ([[[-2, -2], [-1, -2], [0, -2]], 0], [-2, 1]), ([[[-2, -2], [-1, -2], [0, -2]], 1], [-2, 1]), ([[[-2, -2], [-1, -2], [0, -2]], 2], [-2, 1])], [([[[0, -2], [2, -2]], -1], [-2, 1]), ([[[-2, -2], [-1, -2], [0, -2]], 2], [-2, 1]), ([[[-2, -2], [-1, -2], [0, -2]], -3], [-2, 1]), ([[[0, 2], [2, 2]], 3], [2, 1]), ([[[-2, -2], [-1, -2], [0, 0]], 0], [0, 1]), ([[[-2, -2], [-1, -2], [0, 0]], 1], [4, 1]), ([[[-2, -2], [-1, -2], [0, 0]], 2], [10, 1]), ([[[-2, -2], [-1, -2], [0, 0]], 3], [18, 1])], [([[[0, -2], [2, -2]], 0], [-2, 1]), ([[[-2, -2], [-1, -2], [0, -1]], -1], [-2, 1]), ([[[-2, -2], [-1, -2], [0, -2]], -3], [-2, 1]), ([[[0, 2], [2, 2]], 3], [2, 1]), ([[[-2, -2], [-1, -2], [0, 2]], 3], [38, 1]), ([[[-2, -2], [-1, -1], [0, -2]], -3], [-5, 1]), ([[[-2, -2], [-1, -1], [0, -2]], -2], [-2, 1]), ([[[-2, -2], [-1, -1], [0, -2]], -1], [-1, 1])], [([[[0, -2], [2, -2]], 1], [-2, 1]), ([[[-2, -2], [-1, -2], [0, -1]], 3], [8, 1]), ([[[-2, -2], [-1, -2], [0, -2]], -3], [-2, 1]), ([[[0, 2], [2, 2]], 3], [2, 1]), ([[[-2, -2], [-1, -1], [0, 0]], -1], [-1, 1]), ([[[-2, -2], [-1, -1], [0, 0]], 0], [0, 1]), ([[[-2, -2], [-1, -1], [0, 0]], 1], [1, 1]), ([[[-2, -2], [-1, -1], [0, 0]], 2], [2, 1])], [([[[0, -2], [2, -2]], 2], [-2, 1]), ([[[-2, -2], [-1, -2], [0, 0]], 1], [4, 1]), ([[[-2, -2], [-1, -2], [0, -2]], -3], [-2, 1]), ([[[0, 2], [2, 2]], 3], [2, 1]), ([[[-2, -2], [-1, -1], [0, 2]], 2], [14, 1]), ([[[-2, -2], [-1, -1], [0, 2]], 3], [23, 1]), ([[[-2, -2], [-1, 0], [0, -2]], -3], [-8, 1]), ([[[-2, -2], [-1, 0], [0, -2]], -2], [-2, 1])]]\nfor i, (args, expected) in enumerate(fixtures[N-1]):\n    check(\"explicit oracle %d\" % i, solve(args), expected)\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 deterministic bounded teaching model. Inputs are restricted to the explicit contract; this is not a production algebra 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-numerics-lagrange-interpolation-value-basis-denominator-orientation","generated_at":"2026-09-29T14:39:19.197651+00:00","license":"CC0-1.0","python":"3.12.14","seed":1,"split":"open-access"},"relevance":"Exact discrete arithmetic with observable algorithmic state; no floating point approximation is used.","root_cause":"The basis denominator orientation step uses xj-xi instead of xi-xj.","sha256":"5c03f1385ef14202a0921e9e015ac9f503bcf12bd1e7179d4a232ca802acb257","title":"Lagrange interpolation value: basis denominator orientation · 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":42.555,"exit_code":1,"observations":[{"actual":[6,1],"check":"explicit oracle 0","expected":[-2,1],"passed":false},{"actual":[-14,1],"check":"explicit oracle 1","expected":[-2,1],"passed":false},{"actual":[4,1],"check":"explicit oracle 2","expected":[2,1],"passed":false},{"actual":[-2,1],"check":"explicit oracle 3","expected":[-2,1],"passed":true},{"actual":[2,1],"check":"explicit oracle 4","expected":[-2,1],"passed":false},{"actual":[-2,1],"check":"explicit oracle 5","expected":[-2,1],"passed":true},{"actual":[-14,1],"check":"explicit oracle 6","expected":[-2,1],"passed":false},{"actual":[-34,1],"check":"explicit oracle 7","expected":[-2,1],"passed":false}],"passed":false,"stderr":"","stdout":"{\"observations\": [{\"check\": \"explicit oracle 0\", \"actual\": [6, 1], \"expected\": [-2, 1], \"passed\": false}, {\"check\": \"explicit oracle 1\", \"actual\": [-14, 1], \"expected\": [-2, 1], \"passed\": false}, {\"check\": \"explicit oracle 2\", \"actual\": [4, 1], \"expected\": [2, 1], \"passed\": false}, {\"check\": \"explicit oracle 3\", \"actual\": [-2, 1], \"expected\": [-2, 1], \"passed\": true}, {\"check\": \"explicit oracle 4\", \"actual\": [2, 1], \"expected\": [-2, 1], \"passed\": false}, {\"check\": \"explicit oracle 5\", \"actual\": [-2, 1], \"expected\": [-2, 1], \"passed\": true}, {\"check\": \"explicit oracle 6\", \"actual\": [-14, 1], \"expected\": [-2, 1], \"passed\": false}, {\"check\": \"explicit oracle 7\", \"actual\": [-34, 1], \"expected\": [-2, 1], \"passed\": false}], \"passed\": false}\n"},"broken":{"elapsed_ms":40.635,"exit_code":1,"observations":[{"actual":[2,1],"check":"explicit oracle 0","expected":[-2,1],"passed":false},{"actual":[-2,1],"check":"explicit oracle 1","expected":[-2,1],"passed":true},{"actual":[-2,1],"check":"explicit oracle 2","expected":[2,1],"passed":false},{"actual":[-2,1],"check":"explicit oracle 3","expected":[-2,1],"passed":true},{"actual":[-2,1],"check":"explicit oracle 4","expected":[-2,1],"passed":true},{"actual":[-2,1],"check":"explicit oracle 5","expected":[-2,1],"passed":true},{"actual":[-2,1],"check":"explicit oracle 6","expected":[-2,1],"passed":true},{"actual":[-2,1],"check":"explicit oracle 7","expected":[-2,1],"passed":true}],"passed":false,"stderr":"","stdout":"{\"observations\": [{\"check\": \"explicit oracle 0\", \"actual\": [2, 1], \"expected\": [-2, 1], \"passed\": false}, {\"check\": \"explicit oracle 1\", \"actual\": [-2, 1], \"expected\": [-2, 1], \"passed\": true}, {\"check\": \"explicit oracle 2\", \"actual\": [-2, 1], \"expected\": [2, 1], \"passed\": false}, {\"check\": \"explicit oracle 3\", \"actual\": [-2, 1], \"expected\": [-2, 1], \"passed\": true}, {\"check\": \"explicit oracle 4\", \"actual\": [-2, 1], \"expected\": [-2, 1], \"passed\": true}, {\"check\": \"explicit oracle 5\", \"actual\": [-2, 1], \"expected\": [-2, 1], \"passed\": true}, {\"check\": \"explicit oracle 6\", \"actual\": [-2, 1], \"expected\": [-2, 1], \"passed\": true}, {\"check\": \"explicit oracle 7\", \"actual\": [-2, 1], \"expected\": [-2, 1], \"passed\": true}], \"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."}}