{"abstract":"Elementwise multiplication omits cross-degree products.","category":"Polynomial arithmetic","checks":4,"contract":"Integer coefficients in ascending power order and integer evaluation bounds; missing high-degree coefficients are zero. Rational outputs are reduced Fraction strings; the zero-polynomial degree is -1. Polynomial coefficient convolution. Exact operational definition: [sum(a[j]*b[i-j] for j in range(len(a)) if 0<=i-j<len(b)) for i in range(len(a)+len(b)-1)] if a and b else []","evaluation_group":"model-9befd813d946ce9a","failed_approach":"The result is truncated to an input length and loses high powers.","family":"xn-polynomial-coefficient-convolution","id":"FA-6116","implementations":{"attempt":{"sha256":"9e894569c6128b5fa0d0e2fae92d98ea9540cb29a00963e505004768bd07b419","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(a, b):\n    return [sum(a[j]*b[i-j] for j in range(len(a)) if 0<=i-j<len(b)) for i in range(max(len(a),len(b)))]\ndef check(label, actual, expected):\n    observations.append({\"check\": label, \"actual\": actual, \"expected\": expected, \"passed\": actual == expected})\ncheck('fixture 1: ([1, 2], [3, 4])', solve(*([1, 2], [3, 4])), [3, 10, 8])\ncheck('fixture 2: ([0, 1], [0, 1])', solve(*([0, 1], [0, 1])), [0, 0, 1])\ncheck('fixture 3: ([2], [3])', solve(*([2], [3])), [6])\ncheck('fixture 4: ([], [1])', solve(*([], [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":"cd13bd9fd264876f54aaa70efb6ac3b11756ee0539a96fa41c1d2e3482685c53","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(a, b):\n    return [x*y for x,y in zip(a,b)]\ndef check(label, actual, expected):\n    observations.append({\"check\": label, \"actual\": actual, \"expected\": expected, \"passed\": actual == expected})\ncheck('fixture 1: ([1, 2], [3, 4])', solve(*([1, 2], [3, 4])), [3, 10, 8])\ncheck('fixture 2: ([0, 1], [0, 1])', solve(*([0, 1], [0, 1])), [0, 0, 1])\ncheck('fixture 3: ([2], [3])', solve(*([2], [3])), [6])\ncheck('fixture 4: ([], [1])', solve(*([], [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":"0d07f9a3438f6a81ec37c342a02407ef5c8aaece6bab65e672b2888530df77f0","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(a, b):\n    return [sum(a[j]*b[i-j] for j in range(len(a)) if 0<=i-j<len(b)) for i in range(len(a)+len(b)-1)] if a and b else []\ndef check(label, actual, expected):\n    observations.append({\"check\": label, \"actual\": actual, \"expected\": expected, \"passed\": actual == expected})\ncheck('fixture 1: ([1, 2], [3, 4])', solve(*([1, 2], [3, 4])), [3, 10, 8])\ncheck('fixture 2: ([0, 1], [0, 1])', solve(*([0, 1], [0, 1])), [0, 0, 1])\ncheck('fixture 3: ([2], [3])', solve(*([2], [3])), [6])\ncheck('fixture 4: ([], [1])', solve(*([], [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-polynomial-coefficient-convolution","generated_at":"2026-09-29T14:37:57.801253+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. Polynomial arithmetic results depend on the stated convention.","repair":"Apply the specified mathematical contract directly, preserving all terms and boundary cases: return [sum(a[j]*b[i-j] for j in range(len(a)) if 0<=i-j<len(b)) for i in range(len(a)+len(b)-1)] if a and b else []","root_cause":"Elementwise multiplication omits cross-degree products.","sha256":"db0fbd6621346b2e226aa18dffbb0e19525da30f42f82704c2854565bc9a5de7","title":"Polynomial coefficient convolution · case 01","variant":1,"variant_policy":"Five numbered records share a model and may reuse boundary fixtures.","verification":{"attempt":{"elapsed_ms":44.021,"exit_code":1,"observations":[{"actual":[3,10],"check":"fixture 1: ([1, 2], [3, 4])","expected":[3,10,8],"passed":false},{"actual":[0,0],"check":"fixture 2: ([0, 1], [0, 1])","expected":[0,0,1],"passed":false},{"actual":[6],"check":"fixture 3: ([2], [3])","expected":[6],"passed":true},{"actual":[0],"check":"fixture 4: ([], [1])","expected":[],"passed":false}],"passed":false,"stderr":"","stdout":"{\"observations\": [{\"check\": \"fixture 1: ([1, 2], [3, 4])\", \"actual\": [3, 10], \"expected\": [3, 10, 8], \"passed\": false}, {\"check\": \"fixture 2: ([0, 1], [0, 1])\", \"actual\": [0, 0], \"expected\": [0, 0, 1], \"passed\": false}, {\"check\": \"fixture 3: ([2], [3])\", \"actual\": [6], \"expected\": [6], \"passed\": true}, {\"check\": \"fixture 4: ([], [1])\", \"actual\": [0], \"expected\": [], \"passed\": false}], \"passed\": false}\n"},"broken":{"elapsed_ms":42.925,"exit_code":1,"observations":[{"actual":[3,8],"check":"fixture 1: ([1, 2], [3, 4])","expected":[3,10,8],"passed":false},{"actual":[0,1],"check":"fixture 2: ([0, 1], [0, 1])","expected":[0,0,1],"passed":false},{"actual":[6],"check":"fixture 3: ([2], [3])","expected":[6],"passed":true},{"actual":[],"check":"fixture 4: ([], [1])","expected":[],"passed":true}],"passed":false,"stderr":"","stdout":"{\"observations\": [{\"check\": \"fixture 1: ([1, 2], [3, 4])\", \"actual\": [3, 8], \"expected\": [3, 10, 8], \"passed\": false}, {\"check\": \"fixture 2: ([0, 1], [0, 1])\", \"actual\": [0, 1], \"expected\": [0, 0, 1], \"passed\": false}, {\"check\": \"fixture 3: ([2], [3])\", \"actual\": [6], \"expected\": [6], \"passed\": true}, {\"check\": \"fixture 4: ([], [1])\", \"actual\": [], \"expected\": [], \"passed\": true}], \"passed\": false}\n"},"fixed":{"elapsed_ms":41.023,"exit_code":0,"observations":[{"actual":[3,10,8],"check":"fixture 1: ([1, 2], [3, 4])","expected":[3,10,8],"passed":true},{"actual":[0,0,1],"check":"fixture 2: ([0, 1], [0, 1])","expected":[0,0,1],"passed":true},{"actual":[6],"check":"fixture 3: ([2], [3])","expected":[6],"passed":true},{"actual":[],"check":"fixture 4: ([], [1])","expected":[],"passed":true}],"passed":true,"stderr":"","stdout":"{\"observations\": [{\"check\": \"fixture 1: ([1, 2], [3, 4])\", \"actual\": [3, 10, 8], \"expected\": [3, 10, 8], \"passed\": true}, {\"check\": \"fixture 2: ([0, 1], [0, 1])\", \"actual\": [0, 0, 1], \"expected\": [0, 0, 1], \"passed\": true}, {\"check\": \"fixture 3: ([2], [3])\", \"actual\": [6], \"expected\": [6], \"passed\": true}, {\"check\": \"fixture 4: ([], [1])\", \"actual\": [], \"expected\": [], \"passed\": true}], \"passed\": true}\n"}},"verified":true,"visibility":"public"}