{"abstract":"Uniform-grid second differences are applied to nonuniform nodes.","category":"Discrete calculus","checks":4,"contract":"Integer sample values and compatible sample-array lengths. Rational results are reduced Fraction strings; spacing is positive unless explicitly stated otherwise. x0,x1,x2 are pairwise distinct integer nodes with integer observations; return the second divided difference. Exact operational definition: str((Fraction(y2-y1,x2-x1)-Fraction(y1-y0,x1-x0))/Fraction(x2-x0))","evaluation_group":"model-d2f4846dd3a5d1bd","failed_approach":"The difference of first slopes is not divided by the full node span.","family":"xn-newton-divided-difference-two","id":"FA-6456","implementations":{"attempt":{"sha256":"eb60c83f240bd1bc025591de8f76913a063f81806a7cb39f0684985c877ac1cf","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(x0, y0, x1, y1, x2, y2):\n    return str(Fraction(y2-y1,x2-x1)-Fraction(y1-y0,x1-x0))\ndef check(label, actual, expected):\n    observations.append({\"check\": label, \"actual\": actual, \"expected\": expected, \"passed\": actual == expected})\ncheck('fixture 1: (0, 0, 1, 1, 3, 9)', solve(*(0, 0, 1, 1, 3, 9)), '1')\ncheck('fixture 2: (0, 1, 2, 5, 4, 17)', solve(*(0, 1, 2, 5, 4, 17)), '1')\ncheck('fixture 3: (0, 3, 1, 5, 2, 7)', solve(*(0, 3, 1, 5, 2, 7)), '0')\ncheck('fixture 4: (1, 1, 2, 4, 4, 16)', solve(*(1, 1, 2, 4, 4, 16)), '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":"157714a418001485499e8a52fa975a3c4970d249b8a1ddd4ed9357a13ae081a5","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(x0, y0, x1, y1, x2, y2):\n    return str(Fraction(y2-2*y1+y0,x2-x0))\ndef check(label, actual, expected):\n    observations.append({\"check\": label, \"actual\": actual, \"expected\": expected, \"passed\": actual == expected})\ncheck('fixture 1: (0, 0, 1, 1, 3, 9)', solve(*(0, 0, 1, 1, 3, 9)), '1')\ncheck('fixture 2: (0, 1, 2, 5, 4, 17)', solve(*(0, 1, 2, 5, 4, 17)), '1')\ncheck('fixture 3: (0, 3, 1, 5, 2, 7)', solve(*(0, 3, 1, 5, 2, 7)), '0')\ncheck('fixture 4: (1, 1, 2, 4, 4, 16)', solve(*(1, 1, 2, 4, 4, 16)), '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":"8629132a3ab9cadca6d3b1d08cbb583fbd061cb10f15c29ad40724469c87b65c","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(x0, y0, x1, y1, x2, y2):\n    return str((Fraction(y2-y1,x2-x1)-Fraction(y1-y0,x1-x0))/Fraction(x2-x0))\ndef check(label, actual, expected):\n    observations.append({\"check\": label, \"actual\": actual, \"expected\": expected, \"passed\": actual == expected})\ncheck('fixture 1: (0, 0, 1, 1, 3, 9)', solve(*(0, 0, 1, 1, 3, 9)), '1')\ncheck('fixture 2: (0, 1, 2, 5, 4, 17)', solve(*(0, 1, 2, 5, 4, 17)), '1')\ncheck('fixture 3: (0, 3, 1, 5, 2, 7)', solve(*(0, 3, 1, 5, 2, 7)), '0')\ncheck('fixture 4: (1, 1, 2, 4, 4, 16)', solve(*(1, 1, 2, 4, 4, 16)), '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-newton-divided-difference-two","generated_at":"2026-09-29T14:38:02.205490+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. Discrete calculus results depend on the stated convention.","repair":"Apply the specified mathematical contract directly, preserving all terms and boundary cases: return str((Fraction(y2-y1,x2-x1)-Fraction(y1-y0,x1-x0))/Fraction(x2-x0))","root_cause":"Uniform-grid second differences are applied to nonuniform nodes.","sha256":"35c138b2bcb97b2e91ca9f43227a451e1f472165ce9ccbe3dcbcc039d4e7e53d","title":"Newton divided difference two · case 01","variant":1,"variant_policy":"Five numbered records share a model and may reuse boundary fixtures.","verification":{"attempt":{"elapsed_ms":49.392,"exit_code":1,"observations":[{"actual":"3","check":"fixture 1: (0, 0, 1, 1, 3, 9)","expected":"1","passed":false},{"actual":"4","check":"fixture 2: (0, 1, 2, 5, 4, 17)","expected":"1","passed":false},{"actual":"0","check":"fixture 3: (0, 3, 1, 5, 2, 7)","expected":"0","passed":true},{"actual":"3","check":"fixture 4: (1, 1, 2, 4, 4, 16)","expected":"1","passed":false}],"passed":false,"stderr":"","stdout":"{\"observations\": [{\"check\": \"fixture 1: (0, 0, 1, 1, 3, 9)\", \"actual\": \"3\", \"expected\": \"1\", \"passed\": false}, {\"check\": \"fixture 2: (0, 1, 2, 5, 4, 17)\", \"actual\": \"4\", \"expected\": \"1\", \"passed\": false}, {\"check\": \"fixture 3: (0, 3, 1, 5, 2, 7)\", \"actual\": \"0\", \"expected\": \"0\", \"passed\": true}, {\"check\": \"fixture 4: (1, 1, 2, 4, 4, 16)\", \"actual\": \"3\", \"expected\": \"1\", \"passed\": false}], \"passed\": false}\n"},"broken":{"elapsed_ms":42.728,"exit_code":1,"observations":[{"actual":"7/3","check":"fixture 1: (0, 0, 1, 1, 3, 9)","expected":"1","passed":false},{"actual":"2","check":"fixture 2: (0, 1, 2, 5, 4, 17)","expected":"1","passed":false},{"actual":"0","check":"fixture 3: (0, 3, 1, 5, 2, 7)","expected":"0","passed":true},{"actual":"3","check":"fixture 4: (1, 1, 2, 4, 4, 16)","expected":"1","passed":false}],"passed":false,"stderr":"","stdout":"{\"observations\": [{\"check\": \"fixture 1: (0, 0, 1, 1, 3, 9)\", \"actual\": \"7/3\", \"expected\": \"1\", \"passed\": false}, {\"check\": \"fixture 2: (0, 1, 2, 5, 4, 17)\", \"actual\": \"2\", \"expected\": \"1\", \"passed\": false}, {\"check\": \"fixture 3: (0, 3, 1, 5, 2, 7)\", \"actual\": \"0\", \"expected\": \"0\", \"passed\": true}, {\"check\": \"fixture 4: (1, 1, 2, 4, 4, 16)\", \"actual\": \"3\", \"expected\": \"1\", \"passed\": false}], \"passed\": false}\n"},"fixed":{"elapsed_ms":47.043,"exit_code":0,"observations":[{"actual":"1","check":"fixture 1: (0, 0, 1, 1, 3, 9)","expected":"1","passed":true},{"actual":"1","check":"fixture 2: (0, 1, 2, 5, 4, 17)","expected":"1","passed":true},{"actual":"0","check":"fixture 3: (0, 3, 1, 5, 2, 7)","expected":"0","passed":true},{"actual":"1","check":"fixture 4: (1, 1, 2, 4, 4, 16)","expected":"1","passed":true}],"passed":true,"stderr":"","stdout":"{\"observations\": [{\"check\": \"fixture 1: (0, 0, 1, 1, 3, 9)\", \"actual\": \"1\", \"expected\": \"1\", \"passed\": true}, {\"check\": \"fixture 2: (0, 1, 2, 5, 4, 17)\", \"actual\": \"1\", \"expected\": \"1\", \"passed\": true}, {\"check\": \"fixture 3: (0, 3, 1, 5, 2, 7)\", \"actual\": \"0\", \"expected\": \"0\", \"passed\": true}, {\"check\": \"fixture 4: (1, 1, 2, 4, 4, 16)\", \"actual\": \"1\", \"expected\": \"1\", \"passed\": true}], \"passed\": true}\n"}},"verified":true,"visibility":"public"}