{"abstract":"The exact binary field trace result violates the stated contract at Frobenius squaring transition.","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.","evaluation_group":"s3-numerics-binary-field-trace","failed_approach":"The partial repair v<<1 still violates the Frobenius squaring transition invariant.","family":"s3-numerics-binary-field-trace-Frobenius-squaring-transition","id":"FA-15681","implementations":{"attempt":{"sha256":"d1c61cc6209426b667bce259520d252faa3eac82f2937b6572dbd29df9cb58f0","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=v<<1\n    return out\ndef check(label, actual, expected):\n    observations.append({\"check\": label, \"actual\": actual, \"expected\": expected, \"passed\": actual == expected})\nfixtures = [[([2, 11], 0), ([1, 7], 0), ([0, 7], 0), ([63, 67], 1), ([2, 7], 1), ([3, 7], 1), ([0, 11], 0), ([1, 11], 1)], [([3, 11], 1), ([1, 11], 1), ([0, 7], 0), ([63, 67], 1), ([5, 19], 0), ([6, 19], 0), ([7, 19], 0), ([8, 19], 1)], [([4, 11], 0), ([0, 7], 0), ([63, 67], 1), ([6, 25], 0), ([7, 25], 0), ([8, 25], 1), ([9, 25], 1), ([10, 25], 0)], [([5, 11], 1), ([7, 11], 1), ([0, 7], 0), ([63, 67], 1), ([7, 31], 0), ([8, 31], 1), ([9, 31], 1), ([10, 31], 0)], [([6, 11], 0), ([3, 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":"1919b86124bb5ec529d31de65dc9985f7ca24086bbee81defe3ae20d1e1a795a","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,a)\n    return out\ndef check(label, actual, expected):\n    observations.append({\"check\": label, \"actual\": actual, \"expected\": expected, \"passed\": actual == expected})\nfixtures = [[([2, 11], 0), ([1, 7], 0), ([0, 7], 0), ([63, 67], 1), ([2, 7], 1), ([3, 7], 1), ([0, 11], 0), ([1, 11], 1)], [([3, 11], 1), ([1, 11], 1), ([0, 7], 0), ([63, 67], 1), ([5, 19], 0), ([6, 19], 0), ([7, 19], 0), ([8, 19], 1)], [([4, 11], 0), ([0, 7], 0), ([63, 67], 1), ([6, 25], 0), ([7, 25], 0), ([8, 25], 1), ([9, 25], 1), ([10, 25], 0)], [([5, 11], 1), ([7, 11], 1), ([0, 7], 0), ([63, 67], 1), ([7, 31], 0), ([8, 31], 1), ([9, 31], 1), ([10, 31], 0)], [([6, 11], 0), ([3, 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"},"fixed":{"sha256":"4569e44d2638fb4731aac02cb3202892764784f76b0e6daa2b24a140310f2be7","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 = [[([2, 11], 0), ([1, 7], 0), ([0, 7], 0), ([63, 67], 1), ([2, 7], 1), ([3, 7], 1), ([0, 11], 0), ([1, 11], 1)], [([3, 11], 1), ([1, 11], 1), ([0, 7], 0), ([63, 67], 1), ([5, 19], 0), ([6, 19], 0), ([7, 19], 0), ([8, 19], 1)], [([4, 11], 0), ([0, 7], 0), ([63, 67], 1), ([6, 25], 0), ([7, 25], 0), ([8, 25], 1), ([9, 25], 1), ([10, 25], 0)], [([5, 11], 1), ([7, 11], 1), ([0, 7], 0), ([63, 67], 1), ([7, 31], 0), ([8, 31], 1), ([9, 31], 1), ([10, 31], 0)], [([6, 11], 0), ([3, 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-Frobenius-squaring-transition","generated_at":"2026-09-29T14:39:29.288427+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 mul(v,v) at the Frobenius squaring transition step.","root_cause":"The Frobenius squaring transition step uses mul(v,a) instead of mul(v,v).","sha256":"b0f841eca6b50538de97a1d8490f4b240d51fba31c7163c87d0ea2cecc7f9ff7","title":"Binary field trace: Frobenius squaring transition · case 01","variant":1,"variant_policy":"Five numbered records share a model and may reuse boundary fixtures.","verification":{"attempt":{"elapsed_ms":42.402,"exit_code":1,"observations":[{"actual":14,"check":"explicit oracle 0","expected":0,"passed":false},{"actual":3,"check":"explicit oracle 1","expected":0,"passed":false},{"actual":0,"check":"explicit oracle 2","expected":0,"passed":true},{"actual":1365,"check":"explicit oracle 3","expected":1,"passed":false},{"actual":6,"check":"explicit oracle 4","expected":1,"passed":false},{"actual":5,"check":"explicit oracle 5","expected":1,"passed":false},{"actual":0,"check":"explicit oracle 6","expected":0,"passed":true},{"actual":7,"check":"explicit oracle 7","expected":1,"passed":false}],"passed":false,"stderr":"","stdout":"{\"observations\": [{\"check\": \"explicit oracle 0\", \"actual\": 14, \"expected\": 0, \"passed\": false}, {\"check\": \"explicit oracle 1\", \"actual\": 3, \"expected\": 0, \"passed\": false}, {\"check\": \"explicit oracle 2\", \"actual\": 0, \"expected\": 0, \"passed\": true}, {\"check\": \"explicit oracle 3\", \"actual\": 1365, \"expected\": 1, \"passed\": false}, {\"check\": \"explicit oracle 4\", \"actual\": 6, \"expected\": 1, \"passed\": false}, {\"check\": \"explicit oracle 5\", \"actual\": 5, \"expected\": 1, \"passed\": false}, {\"check\": \"explicit oracle 6\", \"actual\": 0, \"expected\": 0, \"passed\": true}, {\"check\": \"explicit oracle 7\", \"actual\": 7, \"expected\": 1, \"passed\": false}], \"passed\": false}\n"},"broken":{"elapsed_ms":44.352,"exit_code":1,"observations":[{"actual":5,"check":"explicit oracle 0","expected":0,"passed":false},{"actual":0,"check":"explicit oracle 1","expected":0,"passed":true},{"actual":0,"check":"explicit oracle 2","expected":0,"passed":true},{"actual":38,"check":"explicit oracle 3","expected":1,"passed":false},{"actual":1,"check":"explicit oracle 4","expected":1,"passed":true},{"actual":1,"check":"explicit oracle 5","expected":1,"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\": 0, \"passed\": false}, {\"check\": \"explicit oracle 1\", \"actual\": 0, \"expected\": 0, \"passed\": true}, {\"check\": \"explicit oracle 2\", \"actual\": 0, \"expected\": 0, \"passed\": true}, {\"check\": \"explicit oracle 3\", \"actual\": 38, \"expected\": 1, \"passed\": false}, {\"check\": \"explicit oracle 4\", \"actual\": 1, \"expected\": 1, \"passed\": true}, {\"check\": \"explicit oracle 5\", \"actual\": 1, \"expected\": 1, \"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":44.728,"exit_code":0,"observations":[{"actual":0,"check":"explicit oracle 0","expected":0,"passed":true},{"actual":0,"check":"explicit oracle 1","expected":0,"passed":true},{"actual":0,"check":"explicit oracle 2","expected":0,"passed":true},{"actual":1,"check":"explicit oracle 3","expected":1,"passed":true},{"actual":1,"check":"explicit oracle 4","expected":1,"passed":true},{"actual":1,"check":"explicit oracle 5","expected":1,"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\": 0, \"expected\": 0, \"passed\": true}, {\"check\": \"explicit oracle 1\", \"actual\": 0, \"expected\": 0, \"passed\": true}, {\"check\": \"explicit oracle 2\", \"actual\": 0, \"expected\": 0, \"passed\": true}, {\"check\": \"explicit oracle 3\", \"actual\": 1, \"expected\": 1, \"passed\": true}, {\"check\": \"explicit oracle 4\", \"actual\": 1, \"expected\": 1, \"passed\": true}, {\"check\": \"explicit oracle 5\", \"actual\": 1, \"expected\": 1, \"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"}