{"abstract":"The exact binary field minimal polynomial result violates the stated contract at distinct conjugate stopping.","category":"Numerics","checks":8,"contract":"Input [a,irreducible bitmask]; return ascending 0/1 coefficients of the minimal polynomial of a over F2, through its distinct Frobenius orbit.","contract_signature":"x","evaluation_group":"s3-numerics-binary-field-minimal-polynomial","failed_approach":"The partial repair v==a still violates the distinct conjugate stopping invariant.","family":"s3-numerics-binary-field-minimal-polynomial-distinct-conjugate-stopping","id":"FA-15691","implementations":{"attempt":{"sha256":"f45b8b85b97a9fe2fb109d72b20f9e6ee0a30a50de1bdb29954d3cb1355d07b6","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    orbit=[];v=a\n    for _ in range(n+1):\n     if v==a:break\n     orbit.append(v);v=mul(v,v)\n    c=[1]\n    for root in orbit:\n     d=[0]*(len(c)+1)\n     for j,b in enumerate(c):\n      d[j]=d[j]^mul(b,root)\n      d[j+1]=d[j+1]^b\n     c=d\n    return c\ndef check(label, actual, expected):\n    observations.append({\"check\": label, \"actual\": actual, \"expected\": expected, \"passed\": actual == expected})\nfixtures = [[([0, 7], [0, 1]), ([31, 37], [1, 0, 0, 1, 0, 1]), ([1, 7], [1, 1]), ([2, 7], [1, 1, 1]), ([3, 7], [1, 1, 1]), ([0, 11], [0, 1]), ([1, 11], [1, 1]), ([2, 11], [1, 1, 0, 1])], [([1, 7], [1, 1]), ([3, 7], [1, 1, 1]), ([0, 7], [0, 1]), ([31, 37], [1, 0, 0, 1, 0, 1]), ([5, 19], [1, 1, 0, 0, 1]), ([6, 19], [1, 1, 1]), ([7, 19], [1, 1, 1]), ([8, 19], [1, 1, 1, 1, 1])], [([0, 11], [0, 1]), ([2, 11], [1, 1, 0, 1]), ([0, 7], [0, 1]), ([31, 37], [1, 0, 0, 1, 0, 1]), ([6, 25], [1, 1, 0, 0, 1]), ([7, 25], [1, 1, 0, 0, 1]), ([8, 25], [1, 1, 1, 1, 1]), ([9, 25], [1, 0, 0, 1, 1])], [([1, 11], [1, 1]), ([5, 11], [1, 0, 1, 1]), ([0, 7], [0, 1]), ([31, 37], [1, 0, 0, 1, 0, 1]), ([7, 31], [1, 1, 0, 0, 1]), ([8, 31], [1, 1, 1, 1, 1]), ([9, 31], [1, 0, 0, 1, 1]), ([10, 31], [1, 1, 0, 0, 1])], [([0, 19], [0, 1]), ([0, 7], [0, 1]), ([31, 37], [1, 0, 0, 1, 0, 1]), ([8, 37], [1, 0, 1, 1, 1, 1]), ([9, 37], [1, 0, 0, 1, 0, 1]), ([10, 37], [1, 0, 1, 1, 1, 1]), ([11, 37], [1, 0, 0, 1, 0, 1]), ([12, 37], [1, 1, 1, 0, 1, 1])]]\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":"98c7046bfbc855578121b1c09e71a4686893d5022776ece1ea8c3bfd2fee36d7","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    orbit=[];v=a\n    for _ in range(n+1):\n     if len(orbit)==n:break\n     orbit.append(v);v=mul(v,v)\n    c=[1]\n    for root in orbit:\n     d=[0]*(len(c)+1)\n     for j,b in enumerate(c):\n      d[j]=d[j]^mul(b,root)\n      d[j+1]=d[j+1]^b\n     c=d\n    return c\ndef check(label, actual, expected):\n    observations.append({\"check\": label, \"actual\": actual, \"expected\": expected, \"passed\": actual == expected})\nfixtures = [[([0, 7], [0, 1]), ([31, 37], [1, 0, 0, 1, 0, 1]), ([1, 7], [1, 1]), ([2, 7], [1, 1, 1]), ([3, 7], [1, 1, 1]), ([0, 11], [0, 1]), ([1, 11], [1, 1]), ([2, 11], [1, 1, 0, 1])], [([1, 7], [1, 1]), ([3, 7], [1, 1, 1]), ([0, 7], [0, 1]), ([31, 37], [1, 0, 0, 1, 0, 1]), ([5, 19], [1, 1, 0, 0, 1]), ([6, 19], [1, 1, 1]), ([7, 19], [1, 1, 1]), ([8, 19], [1, 1, 1, 1, 1])], [([0, 11], [0, 1]), ([2, 11], [1, 1, 0, 1]), ([0, 7], [0, 1]), ([31, 37], [1, 0, 0, 1, 0, 1]), ([6, 25], [1, 1, 0, 0, 1]), ([7, 25], [1, 1, 0, 0, 1]), ([8, 25], [1, 1, 1, 1, 1]), ([9, 25], [1, 0, 0, 1, 1])], [([1, 11], [1, 1]), ([5, 11], [1, 0, 1, 1]), ([0, 7], [0, 1]), ([31, 37], [1, 0, 0, 1, 0, 1]), ([7, 31], [1, 1, 0, 0, 1]), ([8, 31], [1, 1, 1, 1, 1]), ([9, 31], [1, 0, 0, 1, 1]), ([10, 31], [1, 1, 0, 0, 1])], [([0, 19], [0, 1]), ([0, 7], [0, 1]), ([31, 37], [1, 0, 0, 1, 0, 1]), ([8, 37], [1, 0, 1, 1, 1, 1]), ([9, 37], [1, 0, 0, 1, 0, 1]), ([10, 37], [1, 0, 1, 1, 1, 1]), ([11, 37], [1, 0, 0, 1, 0, 1]), ([12, 37], [1, 1, 1, 0, 1, 1])]]\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-minimal-polynomial-distinct-conjugate-stopping","generated_at":"2026-09-29T14:39:29.427707+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 distinct conjugate stopping step uses len(orbit)==n instead of v in orbit.","sha256":"c064d23743db6b5302b9c2eb70592969c4133f801559e07dfea34320108735de","title":"Binary field minimal polynomial: distinct conjugate stopping · 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":45.441,"exit_code":1,"observations":[{"actual":[1],"check":"explicit oracle 0","expected":[0,1],"passed":false},{"actual":[1],"check":"explicit oracle 1","expected":[1,0,0,1,0,1],"passed":false},{"actual":[1],"check":"explicit oracle 2","expected":[1,1],"passed":false},{"actual":[1],"check":"explicit oracle 3","expected":[1,1,1],"passed":false},{"actual":[1],"check":"explicit oracle 4","expected":[1,1,1],"passed":false},{"actual":[1],"check":"explicit oracle 5","expected":[0,1],"passed":false},{"actual":[1],"check":"explicit oracle 6","expected":[1,1],"passed":false},{"actual":[1],"check":"explicit oracle 7","expected":[1,1,0,1],"passed":false}],"passed":false,"stderr":"","stdout":"{\"observations\": [{\"check\": \"explicit oracle 0\", \"actual\": [1], \"expected\": [0, 1], \"passed\": false}, {\"check\": \"explicit oracle 1\", \"actual\": [1], \"expected\": [1, 0, 0, 1, 0, 1], \"passed\": false}, {\"check\": \"explicit oracle 2\", \"actual\": [1], \"expected\": [1, 1], \"passed\": false}, {\"check\": \"explicit oracle 3\", \"actual\": [1], \"expected\": [1, 1, 1], \"passed\": false}, {\"check\": \"explicit oracle 4\", \"actual\": [1], \"expected\": [1, 1, 1], \"passed\": false}, {\"check\": \"explicit oracle 5\", \"actual\": [1], \"expected\": [0, 1], \"passed\": false}, {\"check\": \"explicit oracle 6\", \"actual\": [1], \"expected\": [1, 1], \"passed\": false}, {\"check\": \"explicit oracle 7\", \"actual\": [1], \"expected\": [1, 1, 0, 1], \"passed\": false}], \"passed\": false}\n"},"broken":{"elapsed_ms":44.616,"exit_code":1,"observations":[{"actual":[0,0,1],"check":"explicit oracle 0","expected":[0,1],"passed":false},{"actual":[1,0,0,1,0,1],"check":"explicit oracle 1","expected":[1,0,0,1,0,1],"passed":true},{"actual":[1,0,1],"check":"explicit oracle 2","expected":[1,1],"passed":false},{"actual":[1,1,1],"check":"explicit oracle 3","expected":[1,1,1],"passed":true},{"actual":[1,1,1],"check":"explicit oracle 4","expected":[1,1,1],"passed":true},{"actual":[0,0,0,1],"check":"explicit oracle 5","expected":[0,1],"passed":false},{"actual":[1,1,1,1],"check":"explicit oracle 6","expected":[1,1],"passed":false},{"actual":[1,1,0,1],"check":"explicit oracle 7","expected":[1,1,0,1],"passed":true}],"passed":false,"stderr":"","stdout":"{\"observations\": [{\"check\": \"explicit oracle 0\", \"actual\": [0, 0, 1], \"expected\": [0, 1], \"passed\": false}, {\"check\": \"explicit oracle 1\", \"actual\": [1, 0, 0, 1, 0, 1], \"expected\": [1, 0, 0, 1, 0, 1], \"passed\": true}, {\"check\": \"explicit oracle 2\", \"actual\": [1, 0, 1], \"expected\": [1, 1], \"passed\": false}, {\"check\": \"explicit oracle 3\", \"actual\": [1, 1, 1], \"expected\": [1, 1, 1], \"passed\": true}, {\"check\": \"explicit oracle 4\", \"actual\": [1, 1, 1], \"expected\": [1, 1, 1], \"passed\": true}, {\"check\": \"explicit oracle 5\", \"actual\": [0, 0, 0, 1], \"expected\": [0, 1], \"passed\": false}, {\"check\": \"explicit oracle 6\", \"actual\": [1, 1, 1, 1], \"expected\": [1, 1], \"passed\": false}, {\"check\": \"explicit oracle 7\", \"actual\": [1, 1, 0, 1], \"expected\": [1, 1, 0, 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."}}