{"abstract":"The exact gf2 field multiplication result violates the stated contract at multiplier coefficient test.","category":"Numerics","checks":8,"contract":"Input [a,b,irreducible polynomial bitmask], nonnegative field elements; return carryless product reduced by that monic polynomial.","evaluation_group":"s3-numerics-gf2-field-multiplication","failed_approach":"The partial repair b==1 still violates the multiplier coefficient test invariant.","family":"s3-numerics-gf2-field-multiplication-multiplier-coefficient-test","id":"FA-15591","implementations":{"attempt":{"sha256":"245acfa1938c75e96e9da76baba2cec5a5e3eb27cf166373fea1048332a0599f","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    a,b,poly=x;n=poly.bit_length()-1;out=0\n    for _ in range(n):\n     if b==1:out=out^a\n     b=b>>1\n     a<<=1\n     if a&(1<<n):a=a^poly\n    return out\ndef check(label, actual, expected):\n    observations.append({\"check\": label, \"actual\": actual, \"expected\": expected, \"passed\": actual == expected})\nfixtures = [[([1, 1, 7], 1), ([1, 3, 7], 3), ([0, 0, 7], 0), ([15, 15, 31], 8), ([0, 1, 7], 0), ([0, 2, 7], 0), ([0, 3, 7], 0), ([1, 0, 7], 0)], [([1, 2, 7], 2), ([1, 3, 11], 3), ([0, 0, 7], 0), ([15, 15, 31], 8), ([0, 1, 11], 0), ([0, 2, 11], 0), ([0, 3, 11], 0), ([0, 4, 11], 0)], [([1, 3, 7], 3), ([1, 7, 11], 7), ([0, 0, 7], 0), ([15, 15, 31], 8), ([2, 2, 11], 4), ([2, 3, 11], 6), ([2, 4, 11], 3), ([2, 5, 11], 1)], [([2, 1, 7], 2), ([2, 6, 11], 7), ([0, 0, 7], 0), ([15, 15, 31], 8), ([4, 3, 11], 7), ([4, 4, 11], 6), ([4, 5, 11], 2), ([4, 6, 11], 5)], [([2, 2, 7], 3), ([3, 5, 11], 4), ([0, 0, 7], 0), ([15, 15, 31], 8), ([6, 4, 11], 5), ([6, 5, 11], 3), ([6, 6, 11], 2), ([6, 7, 11], 4)]]\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":"c4205f7915cadba9e58344ff850f5ab5a7709ceaed93b0b69ed4f407188ab359","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    a,b,poly=x;n=poly.bit_length()-1;out=0\n    for _ in range(n):\n     if b&2:out=out^a\n     b=b>>1\n     a<<=1\n     if a&(1<<n):a=a^poly\n    return out\ndef check(label, actual, expected):\n    observations.append({\"check\": label, \"actual\": actual, \"expected\": expected, \"passed\": actual == expected})\nfixtures = [[([1, 1, 7], 1), ([1, 3, 7], 3), ([0, 0, 7], 0), ([15, 15, 31], 8), ([0, 1, 7], 0), ([0, 2, 7], 0), ([0, 3, 7], 0), ([1, 0, 7], 0)], [([1, 2, 7], 2), ([1, 3, 11], 3), ([0, 0, 7], 0), ([15, 15, 31], 8), ([0, 1, 11], 0), ([0, 2, 11], 0), ([0, 3, 11], 0), ([0, 4, 11], 0)], [([1, 3, 7], 3), ([1, 7, 11], 7), ([0, 0, 7], 0), ([15, 15, 31], 8), ([2, 2, 11], 4), ([2, 3, 11], 6), ([2, 4, 11], 3), ([2, 5, 11], 1)], [([2, 1, 7], 2), ([2, 6, 11], 7), ([0, 0, 7], 0), ([15, 15, 31], 8), ([4, 3, 11], 7), ([4, 4, 11], 6), ([4, 5, 11], 2), ([4, 6, 11], 5)], [([2, 2, 7], 3), ([3, 5, 11], 4), ([0, 0, 7], 0), ([15, 15, 31], 8), ([6, 4, 11], 5), ([6, 5, 11], 3), ([6, 6, 11], 2), ([6, 7, 11], 4)]]\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"},"fixed":{"sha256":"a2ebb15678f53090f53db68a8a85335820757086b0713b74976a0178457767ff","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    a,b,poly=x;n=poly.bit_length()-1;out=0\n    for _ in range(n):\n     if b&1:out=out^a\n     b=b>>1\n     a<<=1\n     if a&(1<<n):a=a^poly\n    return out\ndef check(label, actual, expected):\n    observations.append({\"check\": label, \"actual\": actual, \"expected\": expected, \"passed\": actual == expected})\nfixtures = [[([1, 1, 7], 1), ([1, 3, 7], 3), ([0, 0, 7], 0), ([15, 15, 31], 8), ([0, 1, 7], 0), ([0, 2, 7], 0), ([0, 3, 7], 0), ([1, 0, 7], 0)], [([1, 2, 7], 2), ([1, 3, 11], 3), ([0, 0, 7], 0), ([15, 15, 31], 8), ([0, 1, 11], 0), ([0, 2, 11], 0), ([0, 3, 11], 0), ([0, 4, 11], 0)], [([1, 3, 7], 3), ([1, 7, 11], 7), ([0, 0, 7], 0), ([15, 15, 31], 8), ([2, 2, 11], 4), ([2, 3, 11], 6), ([2, 4, 11], 3), ([2, 5, 11], 1)], [([2, 1, 7], 2), ([2, 6, 11], 7), ([0, 0, 7], 0), ([15, 15, 31], 8), ([4, 3, 11], 7), ([4, 4, 11], 6), ([4, 5, 11], 2), ([4, 6, 11], 5)], [([2, 2, 7], 3), ([3, 5, 11], 4), ([0, 0, 7], 0), ([15, 15, 31], 8), ([6, 4, 11], 5), ([6, 5, 11], 3), ([6, 6, 11], 2), ([6, 7, 11], 4)]]\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-gf2-field-multiplication-multiplier-coefficient-test","generated_at":"2026-09-29T14:39:28.191996+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.","repair":"Use b&1 at the multiplier coefficient test step.","root_cause":"The multiplier coefficient test step uses b&2 instead of b&1.","sha256":"c332b12a5fabad160887e9af085cf906a39c4bd67420703634c94458e5f038d8","title":"Gf2 field multiplication: multiplier coefficient test · case 01","variant":1,"variant_policy":"Five numbered records share a model and may reuse boundary fixtures.","verification":{"attempt":{"elapsed_ms":45.115,"exit_code":1,"observations":[{"actual":1,"check":"explicit oracle 0","expected":1,"passed":true},{"actual":2,"check":"explicit oracle 1","expected":3,"passed":false},{"actual":0,"check":"explicit oracle 2","expected":0,"passed":true},{"actual":4,"check":"explicit oracle 3","expected":8,"passed":false},{"actual":0,"check":"explicit oracle 4","expected":0,"passed":true},{"actual":0,"check":"explicit oracle 5","expected":0,"passed":true},{"actual":0,"check":"explicit oracle 6","expected":0,"passed":true},{"actual":0,"check":"explicit oracle 7","expected":0,"passed":true}],"passed":false,"stderr":"","stdout":"{\"observations\": [{\"check\": \"explicit oracle 0\", \"actual\": 1, \"expected\": 1, \"passed\": true}, {\"check\": \"explicit oracle 1\", \"actual\": 2, \"expected\": 3, \"passed\": false}, {\"check\": \"explicit oracle 2\", \"actual\": 0, \"expected\": 0, \"passed\": true}, {\"check\": \"explicit oracle 3\", \"actual\": 4, \"expected\": 8, \"passed\": false}, {\"check\": \"explicit oracle 4\", \"actual\": 0, \"expected\": 0, \"passed\": true}, {\"check\": \"explicit oracle 5\", \"actual\": 0, \"expected\": 0, \"passed\": true}, {\"check\": \"explicit oracle 6\", \"actual\": 0, \"expected\": 0, \"passed\": true}, {\"check\": \"explicit oracle 7\", \"actual\": 0, \"expected\": 0, \"passed\": true}], \"passed\": false}\n"},"broken":{"elapsed_ms":44.038,"exit_code":1,"observations":[{"actual":0,"check":"explicit oracle 0","expected":1,"passed":false},{"actual":1,"check":"explicit oracle 1","expected":3,"passed":false},{"actual":0,"check":"explicit oracle 2","expected":0,"passed":true},{"actual":12,"check":"explicit oracle 3","expected":8,"passed":false},{"actual":0,"check":"explicit oracle 4","expected":0,"passed":true},{"actual":0,"check":"explicit oracle 5","expected":0,"passed":true},{"actual":0,"check":"explicit oracle 6","expected":0,"passed":true},{"actual":0,"check":"explicit oracle 7","expected":0,"passed":true}],"passed":false,"stderr":"","stdout":"{\"observations\": [{\"check\": \"explicit oracle 0\", \"actual\": 0, \"expected\": 1, \"passed\": false}, {\"check\": \"explicit oracle 1\", \"actual\": 1, \"expected\": 3, \"passed\": false}, {\"check\": \"explicit oracle 2\", \"actual\": 0, \"expected\": 0, \"passed\": true}, {\"check\": \"explicit oracle 3\", \"actual\": 12, \"expected\": 8, \"passed\": false}, {\"check\": \"explicit oracle 4\", \"actual\": 0, \"expected\": 0, \"passed\": true}, {\"check\": \"explicit oracle 5\", \"actual\": 0, \"expected\": 0, \"passed\": true}, {\"check\": \"explicit oracle 6\", \"actual\": 0, \"expected\": 0, \"passed\": true}, {\"check\": \"explicit oracle 7\", \"actual\": 0, \"expected\": 0, \"passed\": true}], \"passed\": false}\n"},"fixed":{"elapsed_ms":43.935,"exit_code":0,"observations":[{"actual":1,"check":"explicit oracle 0","expected":1,"passed":true},{"actual":3,"check":"explicit oracle 1","expected":3,"passed":true},{"actual":0,"check":"explicit oracle 2","expected":0,"passed":true},{"actual":8,"check":"explicit oracle 3","expected":8,"passed":true},{"actual":0,"check":"explicit oracle 4","expected":0,"passed":true},{"actual":0,"check":"explicit oracle 5","expected":0,"passed":true},{"actual":0,"check":"explicit oracle 6","expected":0,"passed":true},{"actual":0,"check":"explicit oracle 7","expected":0,"passed":true}],"passed":true,"stderr":"","stdout":"{\"observations\": [{\"check\": \"explicit oracle 0\", \"actual\": 1, \"expected\": 1, \"passed\": true}, {\"check\": \"explicit oracle 1\", \"actual\": 3, \"expected\": 3, \"passed\": true}, {\"check\": \"explicit oracle 2\", \"actual\": 0, \"expected\": 0, \"passed\": true}, {\"check\": \"explicit oracle 3\", \"actual\": 8, \"expected\": 8, \"passed\": true}, {\"check\": \"explicit oracle 4\", \"actual\": 0, \"expected\": 0, \"passed\": true}, {\"check\": \"explicit oracle 5\", \"actual\": 0, \"expected\": 0, \"passed\": true}, {\"check\": \"explicit oracle 6\", \"actual\": 0, \"expected\": 0, \"passed\": true}, {\"check\": \"explicit oracle 7\", \"actual\": 0, \"expected\": 0, \"passed\": true}], \"passed\": true}\n"}},"verified":true,"visibility":"public"}