{"abstract":"A second derivative is divided by one step instead of squared step.","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. Central second derivative samples. Exact operational definition: str(Fraction(left-2*center+right,h*h))","evaluation_group":"model-9e4f5efd429ee93d","failed_approach":"The central sample coefficient is one instead of two.","family":"xn-central-second-derivative-samples","id":"FA-6391","implementations":{"attempt":{"sha256":"993dd4ec6811c6781948af4b3eff587024807936ff780b446a180edbfb7dd467","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(left, center, right, h):\n    return str(Fraction(left-center+right,h*h))\ndef check(label, actual, expected):\n    observations.append({\"check\": label, \"actual\": actual, \"expected\": expected, \"passed\": actual == expected})\ncheck('fixture 1: (0, 4, 16, 2)', solve(*(0, 4, 16, 2)), '2')\ncheck('fixture 2: (3, 3, 3, 1)', solve(*(3, 3, 3, 1)), '0')\ncheck('fixture 3: (1, 0, 1, 1)', solve(*(1, 0, 1, 1)), '2')\ncheck('fixture 4: (0, 1, 0, 2)', solve(*(0, 1, 0, 2)), '-1/2')\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":"e72a886cd8493285fca8cdc49ce579703a79d2145644adfc9ba7259f383efd2b","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(left, center, right, h):\n    return str(Fraction(left-2*center+right,h))\ndef check(label, actual, expected):\n    observations.append({\"check\": label, \"actual\": actual, \"expected\": expected, \"passed\": actual == expected})\ncheck('fixture 1: (0, 4, 16, 2)', solve(*(0, 4, 16, 2)), '2')\ncheck('fixture 2: (3, 3, 3, 1)', solve(*(3, 3, 3, 1)), '0')\ncheck('fixture 3: (1, 0, 1, 1)', solve(*(1, 0, 1, 1)), '2')\ncheck('fixture 4: (0, 1, 0, 2)', solve(*(0, 1, 0, 2)), '-1/2')\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":"11e9d342a492e31f4bc6512ccbf4dd914cddcbea4be5f350f4a5fc1d2089d96c","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(left, center, right, h):\n    return str(Fraction(left-2*center+right,h*h))\ndef check(label, actual, expected):\n    observations.append({\"check\": label, \"actual\": actual, \"expected\": expected, \"passed\": actual == expected})\ncheck('fixture 1: (0, 4, 16, 2)', solve(*(0, 4, 16, 2)), '2')\ncheck('fixture 2: (3, 3, 3, 1)', solve(*(3, 3, 3, 1)), '0')\ncheck('fixture 3: (1, 0, 1, 1)', solve(*(1, 0, 1, 1)), '2')\ncheck('fixture 4: (0, 1, 0, 2)', solve(*(0, 1, 0, 2)), '-1/2')\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-central-second-derivative-samples","generated_at":"2026-09-29T14:38:01.394936+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(left-2*center+right,h*h))","root_cause":"A second derivative is divided by one step instead of squared step.","sha256":"07d3ee1b294e5f0093df5be3358d98f73838fc98e7ea6b9212e63f3cd96cd738","title":"Central second derivative samples · case 01","variant":1,"variant_policy":"Five numbered records share a model and may reuse boundary fixtures.","verification":{"attempt":{"elapsed_ms":57.401,"exit_code":1,"observations":[{"actual":"3","check":"fixture 1: (0, 4, 16, 2)","expected":"2","passed":false},{"actual":"3","check":"fixture 2: (3, 3, 3, 1)","expected":"0","passed":false},{"actual":"2","check":"fixture 3: (1, 0, 1, 1)","expected":"2","passed":true},{"actual":"-1/4","check":"fixture 4: (0, 1, 0, 2)","expected":"-1/2","passed":false}],"passed":false,"stderr":"","stdout":"{\"observations\": [{\"check\": \"fixture 1: (0, 4, 16, 2)\", \"actual\": \"3\", \"expected\": \"2\", \"passed\": false}, {\"check\": \"fixture 2: (3, 3, 3, 1)\", \"actual\": \"3\", \"expected\": \"0\", \"passed\": false}, {\"check\": \"fixture 3: (1, 0, 1, 1)\", \"actual\": \"2\", \"expected\": \"2\", \"passed\": true}, {\"check\": \"fixture 4: (0, 1, 0, 2)\", \"actual\": \"-1/4\", \"expected\": \"-1/2\", \"passed\": false}], \"passed\": false}\n"},"broken":{"elapsed_ms":66.967,"exit_code":1,"observations":[{"actual":"4","check":"fixture 1: (0, 4, 16, 2)","expected":"2","passed":false},{"actual":"0","check":"fixture 2: (3, 3, 3, 1)","expected":"0","passed":true},{"actual":"2","check":"fixture 3: (1, 0, 1, 1)","expected":"2","passed":true},{"actual":"-1","check":"fixture 4: (0, 1, 0, 2)","expected":"-1/2","passed":false}],"passed":false,"stderr":"","stdout":"{\"observations\": [{\"check\": \"fixture 1: (0, 4, 16, 2)\", \"actual\": \"4\", \"expected\": \"2\", \"passed\": false}, {\"check\": \"fixture 2: (3, 3, 3, 1)\", \"actual\": \"0\", \"expected\": \"0\", \"passed\": true}, {\"check\": \"fixture 3: (1, 0, 1, 1)\", \"actual\": \"2\", \"expected\": \"2\", \"passed\": true}, {\"check\": \"fixture 4: (0, 1, 0, 2)\", \"actual\": \"-1\", \"expected\": \"-1/2\", \"passed\": false}], \"passed\": false}\n"},"fixed":{"elapsed_ms":57.447,"exit_code":0,"observations":[{"actual":"2","check":"fixture 1: (0, 4, 16, 2)","expected":"2","passed":true},{"actual":"0","check":"fixture 2: (3, 3, 3, 1)","expected":"0","passed":true},{"actual":"2","check":"fixture 3: (1, 0, 1, 1)","expected":"2","passed":true},{"actual":"-1/2","check":"fixture 4: (0, 1, 0, 2)","expected":"-1/2","passed":true}],"passed":true,"stderr":"","stdout":"{\"observations\": [{\"check\": \"fixture 1: (0, 4, 16, 2)\", \"actual\": \"2\", \"expected\": \"2\", \"passed\": true}, {\"check\": \"fixture 2: (3, 3, 3, 1)\", \"actual\": \"0\", \"expected\": \"0\", \"passed\": true}, {\"check\": \"fixture 3: (1, 0, 1, 1)\", \"actual\": \"2\", \"expected\": \"2\", \"passed\": true}, {\"check\": \"fixture 4: (0, 1, 0, 2)\", \"actual\": \"-1/2\", \"expected\": \"-1/2\", \"passed\": true}], \"passed\": true}\n"}},"verified":true,"visibility":"public"}