{"abstract":"The exact binary field minimal polynomial result violates the stated contract at monic polynomial product identity.","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 [a] still violates the monic polynomial product identity invariant.","family":"s3-numerics-binary-field-minimal-polynomial-monic-polynomial-product-identity","id":"FA-15701","implementations":{"attempt":{"sha256":"82ccdbdff37887d4dd04d9e2b8742b038462554bea0c21f3b77df8e18729e7d0","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 in orbit:break\n     orbit.append(v);v=mul(v,v)\n    c=[a]\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]), ([0, 11], [0, 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])], [([2, 7], [1, 1, 1]), ([4, 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])], [([3, 7], [1, 1, 1]), ([7, 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, 11], [0, 1]), ([3, 19], [1, 1, 0, 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])]]\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":"8b2e2d04bb2486b1cca16ebd536e3cb19d239296ad941595539b1e2f2785597a","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 in orbit:break\n     orbit.append(v);v=mul(v,v)\n    c=[0]\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]), ([0, 11], [0, 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])], [([2, 7], [1, 1, 1]), ([4, 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])], [([3, 7], [1, 1, 1]), ([7, 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, 11], [0, 1]), ([3, 19], [1, 1, 0, 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])]]\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-monic-polynomial-product-identity","generated_at":"2026-09-29T14:39:29.427902+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 monic polynomial product identity step uses [0] instead of [1].","sha256":"2394026961a57739615b7aba8d0c93bad876f4ecfbf3035de6ce83047627c180","title":"Binary field minimal polynomial: monic polynomial product identity · 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":44.451,"exit_code":1,"observations":[{"actual":[0,0],"check":"explicit oracle 0","expected":[0,1],"passed":false},{"actual":[31,0,0,31,0,31],"check":"explicit oracle 1","expected":[1,0,0,1,0,1],"passed":false},{"actual":[1,1],"check":"explicit oracle 2","expected":[1,1],"passed":true},{"actual":[2,2,2],"check":"explicit oracle 3","expected":[1,1,1],"passed":false},{"actual":[3,3,3],"check":"explicit oracle 4","expected":[1,1,1],"passed":false},{"actual":[0,0],"check":"explicit oracle 5","expected":[0,1],"passed":false},{"actual":[1,1],"check":"explicit oracle 6","expected":[1,1],"passed":true},{"actual":[2,2,0,2],"check":"explicit oracle 7","expected":[1,1,0,1],"passed":false}],"passed":false,"stderr":"","stdout":"{\"observations\": [{\"check\": \"explicit oracle 0\", \"actual\": [0, 0], \"expected\": [0, 1], \"passed\": false}, {\"check\": \"explicit oracle 1\", \"actual\": [31, 0, 0, 31, 0, 31], \"expected\": [1, 0, 0, 1, 0, 1], \"passed\": false}, {\"check\": \"explicit oracle 2\", \"actual\": [1, 1], \"expected\": [1, 1], \"passed\": true}, {\"check\": \"explicit oracle 3\", \"actual\": [2, 2, 2], \"expected\": [1, 1, 1], \"passed\": false}, {\"check\": \"explicit oracle 4\", \"actual\": [3, 3, 3], \"expected\": [1, 1, 1], \"passed\": false}, {\"check\": \"explicit oracle 5\", \"actual\": [0, 0], \"expected\": [0, 1], \"passed\": false}, {\"check\": \"explicit oracle 6\", \"actual\": [1, 1], \"expected\": [1, 1], \"passed\": true}, {\"check\": \"explicit oracle 7\", \"actual\": [2, 2, 0, 2], \"expected\": [1, 1, 0, 1], \"passed\": false}], \"passed\": false}\n"},"broken":{"elapsed_ms":42.864,"exit_code":1,"observations":[{"actual":[0,0],"check":"explicit oracle 0","expected":[0,1],"passed":false},{"actual":[0,0,0,0,0,0],"check":"explicit oracle 1","expected":[1,0,0,1,0,1],"passed":false},{"actual":[0,0],"check":"explicit oracle 2","expected":[1,1],"passed":false},{"actual":[0,0,0],"check":"explicit oracle 3","expected":[1,1,1],"passed":false},{"actual":[0,0,0],"check":"explicit oracle 4","expected":[1,1,1],"passed":false},{"actual":[0,0],"check":"explicit oracle 5","expected":[0,1],"passed":false},{"actual":[0,0],"check":"explicit oracle 6","expected":[1,1],"passed":false},{"actual":[0,0,0,0],"check":"explicit oracle 7","expected":[1,1,0,1],"passed":false}],"passed":false,"stderr":"","stdout":"{\"observations\": [{\"check\": \"explicit oracle 0\", \"actual\": [0, 0], \"expected\": [0, 1], \"passed\": false}, {\"check\": \"explicit oracle 1\", \"actual\": [0, 0, 0, 0, 0, 0], \"expected\": [1, 0, 0, 1, 0, 1], \"passed\": false}, {\"check\": \"explicit oracle 2\", \"actual\": [0, 0], \"expected\": [1, 1], \"passed\": false}, {\"check\": \"explicit oracle 3\", \"actual\": [0, 0, 0], \"expected\": [1, 1, 1], \"passed\": false}, {\"check\": \"explicit oracle 4\", \"actual\": [0, 0, 0], \"expected\": [1, 1, 1], \"passed\": false}, {\"check\": \"explicit oracle 5\", \"actual\": [0, 0], \"expected\": [0, 1], \"passed\": false}, {\"check\": \"explicit oracle 6\", \"actual\": [0, 0], \"expected\": [1, 1], \"passed\": false}, {\"check\": \"explicit oracle 7\", \"actual\": [0, 0, 0, 0], \"expected\": [1, 1, 0, 1], \"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."}}