{"abstract":"The exact binary field trace result violates the stated contract at trace field addition.","category":"Numerics","checks":8,"contract":"Input [a,irreducible bitmask]; return absolute field trace a+a^2+...+a^(2^(n-1)), an F2 element represented integer 0 or1.","contract_signature":"x","evaluation_group":"s3-numerics-binary-field-trace","failed_approach":"The partial repair out|v still violates the trace field addition invariant.","family":"s3-numerics-binary-field-trace-trace-field-addition","id":"FA-15676","implementations":{"attempt":{"sha256":"e525c7dc417e4fd7c16e561ba539a3315aa4c257588eed40fc3244b8b9feb13f","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,poly=x;n=poly.bit_length()-1\n    def mul(a,b):\n     z=0\n     while b:\n      if b&1:z^=a\n      b>>=1;a<<=1\n      if a&(1<<n):a^=poly\n     return z\n    v=a;out=0\n    for _ in range(n):\n     out=out|v\n     v=mul(v,v)\n    return out\ndef check(label, actual, expected):\n    observations.append({\"check\": label, \"actual\": actual, \"expected\": expected, \"passed\": actual == expected})\nfixtures = [[([1, 7], 0), ([0, 7], 0), ([63, 67], 1), ([2, 7], 1), ([3, 7], 1), ([0, 11], 0), ([1, 11], 1), ([2, 11], 0)], [([2, 7], 1), ([2, 11], 0), ([0, 7], 0), ([63, 67], 1), ([5, 19], 0), ([6, 19], 0), ([7, 19], 0), ([8, 19], 1)], [([3, 7], 1), ([5, 11], 1), ([0, 7], 0), ([63, 67], 1), ([6, 25], 0), ([7, 25], 0), ([8, 25], 1), ([9, 25], 1)], [([1, 11], 1), ([1, 19], 0), ([0, 7], 0), ([63, 67], 1), ([7, 31], 0), ([8, 31], 1), ([9, 31], 1), ([10, 31], 0)], [([2, 11], 0), ([4, 19], 0), ([0, 7], 0), ([63, 67], 1), ([8, 37], 1), ([9, 37], 0), ([10, 37], 1), ([11, 37], 0)]]\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":"5b20731f41e87fd5d28295be9b1ae1219bb177505d01d0872e7aeeff6d977c23","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,poly=x;n=poly.bit_length()-1\n    def mul(a,b):\n     z=0\n     while b:\n      if b&1:z^=a\n      b>>=1;a<<=1\n      if a&(1<<n):a^=poly\n     return z\n    v=a;out=0\n    for _ in range(n):\n     out=out+v\n     v=mul(v,v)\n    return out\ndef check(label, actual, expected):\n    observations.append({\"check\": label, \"actual\": actual, \"expected\": expected, \"passed\": actual == expected})\nfixtures = [[([1, 7], 0), ([0, 7], 0), ([63, 67], 1), ([2, 7], 1), ([3, 7], 1), ([0, 11], 0), ([1, 11], 1), ([2, 11], 0)], [([2, 7], 1), ([2, 11], 0), ([0, 7], 0), ([63, 67], 1), ([5, 19], 0), ([6, 19], 0), ([7, 19], 0), ([8, 19], 1)], [([3, 7], 1), ([5, 11], 1), ([0, 7], 0), ([63, 67], 1), ([6, 25], 0), ([7, 25], 0), ([8, 25], 1), ([9, 25], 1)], [([1, 11], 1), ([1, 19], 0), ([0, 7], 0), ([63, 67], 1), ([7, 31], 0), ([8, 31], 1), ([9, 31], 1), ([10, 31], 0)], [([2, 11], 0), ([4, 19], 0), ([0, 7], 0), ([63, 67], 1), ([8, 37], 1), ([9, 37], 0), ([10, 37], 1), ([11, 37], 0)]]\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-binary-field-trace-trace-field-addition","generated_at":"2026-09-29T14:39:29.104773+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 trace field addition step uses out+v instead of out^v.","sha256":"354465a4b27e428c5973472747021e918e6033116fa3f852580433c420b5126d","title":"Binary field trace: trace field addition · 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":45.696,"exit_code":1,"observations":[{"actual":1,"check":"explicit oracle 0","expected":0,"passed":false},{"actual":0,"check":"explicit oracle 1","expected":0,"passed":true},{"actual":63,"check":"explicit oracle 2","expected":1,"passed":false},{"actual":3,"check":"explicit oracle 3","expected":1,"passed":false},{"actual":3,"check":"explicit oracle 4","expected":1,"passed":false},{"actual":0,"check":"explicit oracle 5","expected":0,"passed":true},{"actual":1,"check":"explicit oracle 6","expected":1,"passed":true},{"actual":6,"check":"explicit oracle 7","expected":0,"passed":false}],"passed":false,"stderr":"","stdout":"{\"observations\": [{\"check\": \"explicit oracle 0\", \"actual\": 1, \"expected\": 0, \"passed\": false}, {\"check\": \"explicit oracle 1\", \"actual\": 0, \"expected\": 0, \"passed\": true}, {\"check\": \"explicit oracle 2\", \"actual\": 63, \"expected\": 1, \"passed\": false}, {\"check\": \"explicit oracle 3\", \"actual\": 3, \"expected\": 1, \"passed\": false}, {\"check\": \"explicit oracle 4\", \"actual\": 3, \"expected\": 1, \"passed\": false}, {\"check\": \"explicit oracle 5\", \"actual\": 0, \"expected\": 0, \"passed\": true}, {\"check\": \"explicit oracle 6\", \"actual\": 1, \"expected\": 1, \"passed\": true}, {\"check\": \"explicit oracle 7\", \"actual\": 6, \"expected\": 0, \"passed\": false}], \"passed\": false}\n"},"broken":{"elapsed_ms":43.41,"exit_code":1,"observations":[{"actual":2,"check":"explicit oracle 0","expected":0,"passed":false},{"actual":0,"check":"explicit oracle 1","expected":0,"passed":true},{"actual":307,"check":"explicit oracle 2","expected":1,"passed":false},{"actual":5,"check":"explicit oracle 3","expected":1,"passed":false},{"actual":5,"check":"explicit oracle 4","expected":1,"passed":false},{"actual":0,"check":"explicit oracle 5","expected":0,"passed":true},{"actual":3,"check":"explicit oracle 6","expected":1,"passed":false},{"actual":12,"check":"explicit oracle 7","expected":0,"passed":false}],"passed":false,"stderr":"","stdout":"{\"observations\": [{\"check\": \"explicit oracle 0\", \"actual\": 2, \"expected\": 0, \"passed\": false}, {\"check\": \"explicit oracle 1\", \"actual\": 0, \"expected\": 0, \"passed\": true}, {\"check\": \"explicit oracle 2\", \"actual\": 307, \"expected\": 1, \"passed\": false}, {\"check\": \"explicit oracle 3\", \"actual\": 5, \"expected\": 1, \"passed\": false}, {\"check\": \"explicit oracle 4\", \"actual\": 5, \"expected\": 1, \"passed\": false}, {\"check\": \"explicit oracle 5\", \"actual\": 0, \"expected\": 0, \"passed\": true}, {\"check\": \"explicit oracle 6\", \"actual\": 3, \"expected\": 1, \"passed\": false}, {\"check\": \"explicit oracle 7\", \"actual\": 12, \"expected\": 0, \"passed\": false}], \"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."}}