{"abstract":"The exact gf2 field multiplication result violates the stated contract at irreducible polynomial reduction.","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 a&((1<<n)-1) still violates the irreducible polynomial reduction invariant.","family":"s3-numerics-gf2-field-multiplication-irreducible-polynomial-reduction","id":"FA-15606","implementations":{"attempt":{"sha256":"0a1d429d3e9019b5fa40b9de3ad1a49e1cdf0d01bcfa60bae2cd95d7fa203375","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&((1<<n)-1)\n    return out\ndef check(label, actual, expected):\n    observations.append({\"check\": label, \"actual\": actual, \"expected\": expected, \"passed\": actual == expected})\nfixtures = [[([2, 2, 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, 1, 7], 1)], [([2, 3, 7], 1), ([3, 3, 7], 2), ([0, 0, 7], 0), ([15, 15, 31], 8), ([0, 1, 11], 0), ([0, 2, 11], 0), ([0, 3, 11], 0), ([0, 4, 11], 0)], [([3, 2, 7], 1), ([2, 6, 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)], [([3, 3, 7], 2), ([3, 5, 11], 4), ([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, 4, 11], 3), ([4, 2, 11], 3), ([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":"c720a1d31782f41ff7ccfc4bfab55cd4db8ec749b97ebfdfd0e7a3134c5d86a9","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 = [[([2, 2, 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, 1, 7], 1)], [([2, 3, 7], 1), ([3, 3, 7], 2), ([0, 0, 7], 0), ([15, 15, 31], 8), ([0, 1, 11], 0), ([0, 2, 11], 0), ([0, 3, 11], 0), ([0, 4, 11], 0)], [([3, 2, 7], 1), ([2, 6, 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)], [([3, 3, 7], 2), ([3, 5, 11], 4), ([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, 4, 11], 3), ([4, 2, 11], 3), ([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":"7c3278051d0cba3555edfdabda1e772a4fe87d78c7cd41ea53589bc8c46ed88c","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 = [[([2, 2, 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, 1, 7], 1)], [([2, 3, 7], 1), ([3, 3, 7], 2), ([0, 0, 7], 0), ([15, 15, 31], 8), ([0, 1, 11], 0), ([0, 2, 11], 0), ([0, 3, 11], 0), ([0, 4, 11], 0)], [([3, 2, 7], 1), ([2, 6, 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)], [([3, 3, 7], 2), ([3, 5, 11], 4), ([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, 4, 11], 3), ([4, 2, 11], 3), ([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-irreducible-polynomial-reduction","generated_at":"2026-09-29T14:39:28.416556+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 a^poly at the irreducible polynomial reduction step.","root_cause":"The irreducible polynomial reduction step uses a-poly instead of a^poly.","sha256":"2e03a69aaad0c9b096adbc2db6e834d1e415679748509d3e9cce53650c4cf4d7","title":"Gf2 field multiplication: irreducible polynomial reduction · case 01","variant":1,"variant_policy":"Five numbered records share a model and may reuse boundary fixtures.","verification":{"attempt":{"elapsed_ms":47.522,"exit_code":1,"observations":[{"actual":0,"check":"explicit oracle 0","expected":3,"passed":false},{"actual":0,"check":"explicit oracle 1","expected":0,"passed":true},{"actual":5,"check":"explicit oracle 2","expected":8,"passed":false},{"actual":0,"check":"explicit oracle 3","expected":0,"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":1,"check":"explicit oracle 7","expected":1,"passed":true}],"passed":false,"stderr":"","stdout":"{\"observations\": [{\"check\": \"explicit oracle 0\", \"actual\": 0, \"expected\": 3, \"passed\": false}, {\"check\": \"explicit oracle 1\", \"actual\": 0, \"expected\": 0, \"passed\": true}, {\"check\": \"explicit oracle 2\", \"actual\": 5, \"expected\": 8, \"passed\": false}, {\"check\": \"explicit oracle 3\", \"actual\": 0, \"expected\": 0, \"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\": 1, \"expected\": 1, \"passed\": true}], \"passed\": false}\n"},"broken":{"elapsed_ms":43.284,"exit_code":1,"observations":[{"actual":-3,"check":"explicit oracle 0","expected":3,"passed":false},{"actual":0,"check":"explicit oracle 1","expected":0,"passed":true},{"actual":-80,"check":"explicit oracle 2","expected":8,"passed":false},{"actual":0,"check":"explicit oracle 3","expected":0,"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":1,"check":"explicit oracle 7","expected":1,"passed":true}],"passed":false,"stderr":"","stdout":"{\"observations\": [{\"check\": \"explicit oracle 0\", \"actual\": -3, \"expected\": 3, \"passed\": false}, {\"check\": \"explicit oracle 1\", \"actual\": 0, \"expected\": 0, \"passed\": true}, {\"check\": \"explicit oracle 2\", \"actual\": -80, \"expected\": 8, \"passed\": false}, {\"check\": \"explicit oracle 3\", \"actual\": 0, \"expected\": 0, \"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\": 1, \"expected\": 1, \"passed\": true}], \"passed\": false}\n"},"fixed":{"elapsed_ms":43.223,"exit_code":0,"observations":[{"actual":3,"check":"explicit oracle 0","expected":3,"passed":true},{"actual":0,"check":"explicit oracle 1","expected":0,"passed":true},{"actual":8,"check":"explicit oracle 2","expected":8,"passed":true},{"actual":0,"check":"explicit oracle 3","expected":0,"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":1,"check":"explicit oracle 7","expected":1,"passed":true}],"passed":true,"stderr":"","stdout":"{\"observations\": [{\"check\": \"explicit oracle 0\", \"actual\": 3, \"expected\": 3, \"passed\": true}, {\"check\": \"explicit oracle 1\", \"actual\": 0, \"expected\": 0, \"passed\": true}, {\"check\": \"explicit oracle 2\", \"actual\": 8, \"expected\": 8, \"passed\": true}, {\"check\": \"explicit oracle 3\", \"actual\": 0, \"expected\": 0, \"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\": 1, \"expected\": 1, \"passed\": true}], \"passed\": true}\n"}},"verified":true,"visibility":"public"}