{"abstract":"The exact gf2 field multiplication result violates the stated contract at field addition operation.","category":"Numerics","checks":8,"contract":"Input [a,b,irreducible polynomial bitmask], nonnegative field elements; return carryless product reduced by that monic polynomial.","contract_signature":"x","evaluation_group":"s3-numerics-gf2-field-multiplication","failed_approach":"The partial repair out|a still violates the field addition operation invariant.","family":"s3-numerics-gf2-field-multiplication-field-addition-operation","id":"FA-15596","implementations":{"attempt":{"sha256":"234867347f45c9b0eab45cc4f30ed47ab767e5031f599b4a73e2d550b95d65fa","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, 3, 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, 1, 7], 1)], [([3, 3, 7], 2), ([2, 7, 11], 5), ([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, 5, 11], 1), ([3, 6, 11], 1), ([0, 0, 7], 0), ([15, 15, 31], 8), ([2, 2, 11], 4), ([2, 3, 11], 6), ([2, 4, 11], 3), ([2, 6, 11], 7)], [([2, 7, 11], 5), ([4, 6, 11], 5), ([0, 0, 7], 0), ([15, 15, 31], 8), ([4, 3, 11], 7), ([4, 4, 11], 6), ([4, 5, 11], 2), ([4, 7, 11], 1)], [([3, 3, 11], 5), ([5, 7, 11], 6), ([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":"5a01b8479d2719edc5e1145f691cc98f16c4d05b3456a00d012e6bb5967bec89","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, 3, 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, 1, 7], 1)], [([3, 3, 7], 2), ([2, 7, 11], 5), ([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, 5, 11], 1), ([3, 6, 11], 1), ([0, 0, 7], 0), ([15, 15, 31], 8), ([2, 2, 11], 4), ([2, 3, 11], 6), ([2, 4, 11], 3), ([2, 6, 11], 7)], [([2, 7, 11], 5), ([4, 6, 11], 5), ([0, 0, 7], 0), ([15, 15, 31], 8), ([4, 3, 11], 7), ([4, 4, 11], 6), ([4, 5, 11], 2), ([4, 7, 11], 1)], [([3, 3, 11], 5), ([5, 7, 11], 6), ([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-addition-operation","generated_at":"2026-09-29T14:39:28.367423+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.","root_cause":"The field addition operation step uses out+a instead of out^a.","sha256":"a91016dcab2e5c6186bbb1323f8ef2cd9b407274172dd6a892c8ea67338f58b1","title":"Gf2 field multiplication: field addition operation · case 01","variant":1,"variant_policy":"Five numbered records share a model and may reuse boundary fixtures.","verified":true,"visibility":"public","verification":{"attempt":{"elapsed_ms":40.006,"exit_code":1,"observations":[{"actual":3,"check":"explicit oracle 0","expected":1,"passed":false},{"actual":0,"check":"explicit oracle 1","expected":0,"passed":true},{"actual":15,"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\": 1, \"passed\": false}, {\"check\": \"explicit oracle 1\", \"actual\": 0, \"expected\": 0, \"passed\": true}, {\"check\": \"explicit oracle 2\", \"actual\": 15, \"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":43.37,"exit_code":1,"observations":[{"actual":5,"check":"explicit oracle 0","expected":1,"passed":false},{"actual":0,"check":"explicit oracle 1","expected":0,"passed":true},{"actual":22,"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\": 5, \"expected\": 1, \"passed\": false}, {\"check\": \"explicit oracle 1\", \"actual\": 0, \"expected\": 0, \"passed\": true}, {\"check\": \"explicit oracle 2\", \"actual\": 22, \"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"}},"member_only":{"stages":["fixed"],"fields":["implementations.fixed","verification.fixed","harness","repair"],"note":"The verified repair, its recorded checks, the repair description, and the scoring harness are available to members."}}