{"abstract":"The exact binary polynomial division result violates the stated contract at quotient remainder result positions.","category":"Numerics","checks":8,"contract":"Input [a,b] integer bitmasks for F2 polynomials with b>0; return [quotient,remainder] of carryless division. Bounds: a,b<2^32.","contract_signature":"x","evaluation_group":"s3-numerics-binary-polynomial-division","failed_approach":"The partial repair [q,a&1] still violates the quotient remainder result positions invariant.","family":"s3-numerics-binary-polynomial-division-quotient-remainder-result-positions","id":"FA-15636","implementations":{"attempt":{"sha256":"6286184836fcf355210383b6eb9947749789e6f0e7b7a0fcfb4c04cf8e9b9675","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=x;q=0\n    for _ in range(40):\n     if a.bit_length()<b.bit_length():break\n     shift=a.bit_length()-b.bit_length()\n     if shift<0:return None\n     q=q^(1<<shift)\n     a=a^(b<<shift)\n    return [q,a&1]\ndef check(label, actual, expected):\n    observations.append({\"check\": label, \"actual\": actual, \"expected\": expected, \"passed\": actual == expected})\nfixtures = [[([1, 1], [1, 0]), ([2, 4], [0, 2]), ([0, 1], [0, 0]), ([127, 31], [4, 3]), ([0, 2], [0, 0]), ([0, 3], [0, 0]), ([0, 4], [0, 0]), ([0, 5], [0, 0])], [([1, 2], [0, 1]), ([2, 7], [0, 2]), ([0, 1], [0, 0]), ([127, 31], [4, 3]), ([0, 18], [0, 0]), ([0, 19], [0, 0]), ([0, 20], [0, 0]), ([0, 21], [0, 0])], [([1, 3], [0, 1]), ([2, 10], [0, 2]), ([0, 1], [0, 0]), ([127, 31], [4, 3]), ([1, 4], [0, 1]), ([1, 5], [0, 1]), ([1, 6], [0, 1]), ([1, 7], [0, 1])], [([1, 4], [0, 1]), ([2, 13], [0, 2]), ([0, 1], [0, 0]), ([127, 31], [4, 3]), ([1, 21], [0, 1]), ([1, 22], [0, 1]), ([1, 23], [0, 1]), ([1, 24], [0, 1])], [([1, 5], [0, 1]), ([2, 16], [0, 2]), ([0, 1], [0, 0]), ([127, 31], [4, 3]), ([2, 7], [0, 2]), ([2, 8], [0, 2]), ([2, 9], [0, 2]), ([2, 10], [0, 2])]]\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":"5de021d5c13f4035d449bdde592e39dd3394192db17f26e7c37e16c7d70469f4","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=x;q=0\n    for _ in range(40):\n     if a.bit_length()<b.bit_length():break\n     shift=a.bit_length()-b.bit_length()\n     if shift<0:return None\n     q=q^(1<<shift)\n     a=a^(b<<shift)\n    return [a,q]\ndef check(label, actual, expected):\n    observations.append({\"check\": label, \"actual\": actual, \"expected\": expected, \"passed\": actual == expected})\nfixtures = [[([1, 1], [1, 0]), ([2, 4], [0, 2]), ([0, 1], [0, 0]), ([127, 31], [4, 3]), ([0, 2], [0, 0]), ([0, 3], [0, 0]), ([0, 4], [0, 0]), ([0, 5], [0, 0])], [([1, 2], [0, 1]), ([2, 7], [0, 2]), ([0, 1], [0, 0]), ([127, 31], [4, 3]), ([0, 18], [0, 0]), ([0, 19], [0, 0]), ([0, 20], [0, 0]), ([0, 21], [0, 0])], [([1, 3], [0, 1]), ([2, 10], [0, 2]), ([0, 1], [0, 0]), ([127, 31], [4, 3]), ([1, 4], [0, 1]), ([1, 5], [0, 1]), ([1, 6], [0, 1]), ([1, 7], [0, 1])], [([1, 4], [0, 1]), ([2, 13], [0, 2]), ([0, 1], [0, 0]), ([127, 31], [4, 3]), ([1, 21], [0, 1]), ([1, 22], [0, 1]), ([1, 23], [0, 1]), ([1, 24], [0, 1])], [([1, 5], [0, 1]), ([2, 16], [0, 2]), ([0, 1], [0, 0]), ([127, 31], [4, 3]), ([2, 7], [0, 2]), ([2, 8], [0, 2]), ([2, 9], [0, 2]), ([2, 10], [0, 2])]]\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-polynomial-division-quotient-remainder-result-positions","generated_at":"2026-09-29T14:39:28.776606+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 quotient remainder result positions step uses [a,q] instead of [q,a].","sha256":"9b665794a0c7ad8f23e9afc375839093b38f0955fbefe70ae022b7f55d0a69e0","title":"Binary polynomial division: quotient remainder result positions · 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":43.023,"exit_code":1,"observations":[{"actual":[1,0],"check":"explicit oracle 0","expected":[1,0],"passed":true},{"actual":[0,0],"check":"explicit oracle 1","expected":[0,2],"passed":false},{"actual":[0,0],"check":"explicit oracle 2","expected":[0,0],"passed":true},{"actual":[4,1],"check":"explicit oracle 3","expected":[4,3],"passed":false},{"actual":[0,0],"check":"explicit oracle 4","expected":[0,0],"passed":true},{"actual":[0,0],"check":"explicit oracle 5","expected":[0,0],"passed":true},{"actual":[0,0],"check":"explicit oracle 6","expected":[0,0],"passed":true},{"actual":[0,0],"check":"explicit oracle 7","expected":[0,0],"passed":true}],"passed":false,"stderr":"","stdout":"{\"observations\": [{\"check\": \"explicit oracle 0\", \"actual\": [1, 0], \"expected\": [1, 0], \"passed\": true}, {\"check\": \"explicit oracle 1\", \"actual\": [0, 0], \"expected\": [0, 2], \"passed\": false}, {\"check\": \"explicit oracle 2\", \"actual\": [0, 0], \"expected\": [0, 0], \"passed\": true}, {\"check\": \"explicit oracle 3\", \"actual\": [4, 1], \"expected\": [4, 3], \"passed\": false}, {\"check\": \"explicit oracle 4\", \"actual\": [0, 0], \"expected\": [0, 0], \"passed\": true}, {\"check\": \"explicit oracle 5\", \"actual\": [0, 0], \"expected\": [0, 0], \"passed\": true}, {\"check\": \"explicit oracle 6\", \"actual\": [0, 0], \"expected\": [0, 0], \"passed\": true}, {\"check\": \"explicit oracle 7\", \"actual\": [0, 0], \"expected\": [0, 0], \"passed\": true}], \"passed\": false}\n"},"broken":{"elapsed_ms":44.422,"exit_code":1,"observations":[{"actual":[0,1],"check":"explicit oracle 0","expected":[1,0],"passed":false},{"actual":[2,0],"check":"explicit oracle 1","expected":[0,2],"passed":false},{"actual":[0,0],"check":"explicit oracle 2","expected":[0,0],"passed":true},{"actual":[3,4],"check":"explicit oracle 3","expected":[4,3],"passed":false},{"actual":[0,0],"check":"explicit oracle 4","expected":[0,0],"passed":true},{"actual":[0,0],"check":"explicit oracle 5","expected":[0,0],"passed":true},{"actual":[0,0],"check":"explicit oracle 6","expected":[0,0],"passed":true},{"actual":[0,0],"check":"explicit oracle 7","expected":[0,0],"passed":true}],"passed":false,"stderr":"","stdout":"{\"observations\": [{\"check\": \"explicit oracle 0\", \"actual\": [0, 1], \"expected\": [1, 0], \"passed\": false}, {\"check\": \"explicit oracle 1\", \"actual\": [2, 0], \"expected\": [0, 2], \"passed\": false}, {\"check\": \"explicit oracle 2\", \"actual\": [0, 0], \"expected\": [0, 0], \"passed\": true}, {\"check\": \"explicit oracle 3\", \"actual\": [3, 4], \"expected\": [4, 3], \"passed\": false}, {\"check\": \"explicit oracle 4\", \"actual\": [0, 0], \"expected\": [0, 0], \"passed\": true}, {\"check\": \"explicit oracle 5\", \"actual\": [0, 0], \"expected\": [0, 0], \"passed\": true}, {\"check\": \"explicit oracle 6\", \"actual\": [0, 0], \"expected\": [0, 0], \"passed\": true}, {\"check\": \"explicit oracle 7\", \"actual\": [0, 0], \"expected\": [0, 0], \"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."}}