{"abstract":"The exact gf2 field multiplication result violates the stated contract at field result projection.","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 out>>1 still violates the field result projection invariant.","family":"s3-numerics-gf2-field-multiplication-field-result-projection","id":"FA-15611","implementations":{"attempt":{"sha256":"08bf24022b0845ac4077742d9677a0aa06032c400d194f96da1d130e3a871f03","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>>1\ndef check(label, actual, expected):\n    observations.append({\"check\": label, \"actual\": actual, \"expected\": expected, \"passed\": actual == expected})\nfixtures = [[([1, 2, 7], 2), ([1, 1, 7], 1), ([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, 3, 7], 3), ([2, 1, 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)], [([2, 1, 7], 2), ([3, 1, 7], 3), ([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, 2, 7], 3), ([1, 1, 11], 1), ([0, 0, 7], 0), ([15, 15, 31], 8), ([4, 3, 11], 7), ([4, 4, 11], 6), ([4, 5, 11], 2), ([4, 6, 11], 5)], [([3, 1, 7], 3), ([1, 4, 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":"663036e3d7c8b490068472f6766c4bc8439c49231cd0c27eb32ba8dd209c842c","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&1\ndef check(label, actual, expected):\n    observations.append({\"check\": label, \"actual\": actual, \"expected\": expected, \"passed\": actual == expected})\nfixtures = [[([1, 2, 7], 2), ([1, 1, 7], 1), ([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, 3, 7], 3), ([2, 1, 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)], [([2, 1, 7], 2), ([3, 1, 7], 3), ([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, 2, 7], 3), ([1, 1, 11], 1), ([0, 0, 7], 0), ([15, 15, 31], 8), ([4, 3, 11], 7), ([4, 4, 11], 6), ([4, 5, 11], 2), ([4, 6, 11], 5)], [([3, 1, 7], 3), ([1, 4, 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":"2bbff4cca67de73dcaf48c942cb2cbc095464790923a5fe944ae5ee54af61749","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, 2, 7], 2), ([1, 1, 7], 1), ([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, 3, 7], 3), ([2, 1, 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)], [([2, 1, 7], 2), ([3, 1, 7], 3), ([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, 2, 7], 3), ([1, 1, 11], 1), ([0, 0, 7], 0), ([15, 15, 31], 8), ([4, 3, 11], 7), ([4, 4, 11], 6), ([4, 5, 11], 2), ([4, 6, 11], 5)], [([3, 1, 7], 3), ([1, 4, 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-field-result-projection","generated_at":"2026-09-29T14:39:28.595016+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 out at the field result projection step.","root_cause":"The field result projection step uses out&1 instead of out.","sha256":"f72e795314052d578dc873aca5e0865d3835e734cdcd737747e5cc67427feef7","title":"Gf2 field multiplication: field result projection · case 01","variant":1,"variant_policy":"Five numbered records share a model and may reuse boundary fixtures.","verification":{"attempt":{"elapsed_ms":42.248,"exit_code":1,"observations":[{"actual":1,"check":"explicit oracle 0","expected":2,"passed":false},{"actual":0,"check":"explicit oracle 1","expected":1,"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\": 2, \"passed\": false}, {\"check\": \"explicit oracle 1\", \"actual\": 0, \"expected\": 1, \"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":43.974,"exit_code":1,"observations":[{"actual":0,"check":"explicit oracle 0","expected":2,"passed":false},{"actual":1,"check":"explicit oracle 1","expected":1,"passed":true},{"actual":0,"check":"explicit oracle 2","expected":0,"passed":true},{"actual":0,"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\": 2, \"passed\": false}, {\"check\": \"explicit oracle 1\", \"actual\": 1, \"expected\": 1, \"passed\": true}, {\"check\": \"explicit oracle 2\", \"actual\": 0, \"expected\": 0, \"passed\": true}, {\"check\": \"explicit oracle 3\", \"actual\": 0, \"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":40.38,"exit_code":0,"observations":[{"actual":2,"check":"explicit oracle 0","expected":2,"passed":true},{"actual":1,"check":"explicit oracle 1","expected":1,"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\": 2, \"expected\": 2, \"passed\": true}, {\"check\": \"explicit oracle 1\", \"actual\": 1, \"expected\": 1, \"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"}