{"abstract":"The exact singular hensel root tree result violates the stated contract at lifted polynomial congruence.","category":"Numerics","checks":8,"contract":"Input [polynomial coefficients,p,k], p prime and k>=1; enumerate all roots modulo p^k, including singular roots.","evaluation_group":"s3-numerics-singular-hensel-root-tree","failed_approach":"The partial repair sum(c*r**i for i,c in enumerate(coeff))%nextmod==0 still violates the lifted polynomial congruence invariant.","family":"s3-numerics-singular-hensel-root-tree-lifted-polynomial-congruence","id":"FA-15811","implementations":{"attempt":{"sha256":"098cdf87519c3633cf93f5129ec124f3682eb20eec6d221150c9343557be8c29","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    coeff,p,k=x\n    roots=[r for r in range(p) if sum(c*r**i for i,c in enumerate(coeff))%p==0];mod=p\n    for _ in range(1,k):\n     nextmod=mod*p\n     out=[]\n     for r in roots:\n      for digit in range(p):\n       s=r+digit*mod\n       if sum(c*r**i for i,c in enumerate(coeff))%nextmod==0:out.append(s)\n     roots=sorted(set(out));mod=nextmod\n    return [roots,mod]\ndef check(label, actual, expected):\n    observations.append({\"check\": label, \"actual\": actual, \"expected\": expected, \"passed\": actual == expected})\nfixtures = [[([[0, 0, 1], 2, 3], [[0, 4], 8]), ([[-1, 0, 1], 3, 2], [[1, 8], 9]), ([[0, 0, 1], 2, 1], [[0], 2]), ([[-2, 1, 1], 7, 4], [[1, 2399], 2401]), ([[0, 0, 1], 2, 2], [[0, 2], 4]), ([[0, 0, 1], 2, 4], [[0, 4, 8, 12], 16]), ([[0, 0, 1], 3, 1], [[0], 3]), ([[0, 0, 1], 3, 2], [[0, 3, 6], 9])], [([[0, 0, 1], 3, 3], [[0, 9, 18], 27]), ([[-1, 0, 1], 5, 2], [[1, 24], 25]), ([[0, 0, 1], 2, 1], [[0], 2]), ([[-2, 1, 1], 7, 4], [[1, 2399], 2401]), ([[-1, 0, 1], 2, 2], [[1, 3], 4]), ([[-1, 0, 1], 2, 3], [[1, 3, 5, 7], 8]), ([[-1, 0, 1], 2, 4], [[1, 7, 9, 15], 16]), ([[-1, 0, 1], 3, 1], [[1, 2], 3])], [([[0, 0, 1], 5, 3], [[0, 25, 50, 75, 100], 125]), ([[-1, 0, 1], 7, 2], [[1, 48], 49]), ([[0, 0, 1], 2, 1], [[0], 2]), ([[-2, 1, 1], 7, 4], [[1, 2399], 2401]), ([[0, 2, 1], 2, 3], [[0, 2, 4, 6], 8]), ([[0, 2, 1], 2, 4], [[0, 6, 8, 14], 16]), ([[0, 2, 1], 3, 1], [[0, 1], 3]), ([[0, 2, 1], 3, 2], [[0, 7], 9])], [([[0, 0, 1], 7, 3], [[0, 49, 98, 147, 196, 245, 294], 343]), ([[0, 2, 1], 3, 2], [[0, 7], 9]), ([[0, 0, 1], 2, 1], [[0], 2]), ([[-2, 1, 1], 7, 4], [[1, 2399], 2401]), ([[-4, 0, 1], 2, 4], [[2, 6, 10, 14], 16]), ([[-4, 0, 1], 3, 1], [[1, 2], 3]), ([[-4, 0, 1], 3, 2], [[2, 7], 9]), ([[-4, 0, 1], 3, 3], [[2, 25], 27])], [([[-1, 0, 1], 2, 4], [[1, 7, 9, 15], 16]), ([[0, 2, 1], 5, 2], [[0, 23], 25]), ([[0, 0, 1], 2, 1], [[0], 2]), ([[-2, 1, 1], 7, 4], [[1, 2399], 2401]), ([[0, 0, 0, 1], 3, 1], [[0], 3]), ([[0, 0, 0, 1], 3, 2], [[0, 3, 6], 9]), ([[0, 0, 0, 1], 3, 3], [[0, 3, 6, 9, 12, 15, 18, 21, 24], 27]), ([[0, 0, 0, 1], 3, 4], [[0, 9, 18, 27, 36, 45, 54, 63, 72], 81])]]\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":"3e4c60a723ccc492271937b6bd5f230b0547153faf11febc01bcb15ce3ab4716","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    coeff,p,k=x\n    roots=[r for r in range(p) if sum(c*r**i for i,c in enumerate(coeff))%p==0];mod=p\n    for _ in range(1,k):\n     nextmod=mod*p\n     out=[]\n     for r in roots:\n      for digit in range(p):\n       s=r+digit*mod\n       if sum(c*s**i for i,c in enumerate(coeff))%mod==0:out.append(s)\n     roots=sorted(set(out));mod=nextmod\n    return [roots,mod]\ndef check(label, actual, expected):\n    observations.append({\"check\": label, \"actual\": actual, \"expected\": expected, \"passed\": actual == expected})\nfixtures = [[([[0, 0, 1], 2, 3], [[0, 4], 8]), ([[-1, 0, 1], 3, 2], [[1, 8], 9]), ([[0, 0, 1], 2, 1], [[0], 2]), ([[-2, 1, 1], 7, 4], [[1, 2399], 2401]), ([[0, 0, 1], 2, 2], [[0, 2], 4]), ([[0, 0, 1], 2, 4], [[0, 4, 8, 12], 16]), ([[0, 0, 1], 3, 1], [[0], 3]), ([[0, 0, 1], 3, 2], [[0, 3, 6], 9])], [([[0, 0, 1], 3, 3], [[0, 9, 18], 27]), ([[-1, 0, 1], 5, 2], [[1, 24], 25]), ([[0, 0, 1], 2, 1], [[0], 2]), ([[-2, 1, 1], 7, 4], [[1, 2399], 2401]), ([[-1, 0, 1], 2, 2], [[1, 3], 4]), ([[-1, 0, 1], 2, 3], [[1, 3, 5, 7], 8]), ([[-1, 0, 1], 2, 4], [[1, 7, 9, 15], 16]), ([[-1, 0, 1], 3, 1], [[1, 2], 3])], [([[0, 0, 1], 5, 3], [[0, 25, 50, 75, 100], 125]), ([[-1, 0, 1], 7, 2], [[1, 48], 49]), ([[0, 0, 1], 2, 1], [[0], 2]), ([[-2, 1, 1], 7, 4], [[1, 2399], 2401]), ([[0, 2, 1], 2, 3], [[0, 2, 4, 6], 8]), ([[0, 2, 1], 2, 4], [[0, 6, 8, 14], 16]), ([[0, 2, 1], 3, 1], [[0, 1], 3]), ([[0, 2, 1], 3, 2], [[0, 7], 9])], [([[0, 0, 1], 7, 3], [[0, 49, 98, 147, 196, 245, 294], 343]), ([[0, 2, 1], 3, 2], [[0, 7], 9]), ([[0, 0, 1], 2, 1], [[0], 2]), ([[-2, 1, 1], 7, 4], [[1, 2399], 2401]), ([[-4, 0, 1], 2, 4], [[2, 6, 10, 14], 16]), ([[-4, 0, 1], 3, 1], [[1, 2], 3]), ([[-4, 0, 1], 3, 2], [[2, 7], 9]), ([[-4, 0, 1], 3, 3], [[2, 25], 27])], [([[-1, 0, 1], 2, 4], [[1, 7, 9, 15], 16]), ([[0, 2, 1], 5, 2], [[0, 23], 25]), ([[0, 0, 1], 2, 1], [[0], 2]), ([[-2, 1, 1], 7, 4], [[1, 2399], 2401]), ([[0, 0, 0, 1], 3, 1], [[0], 3]), ([[0, 0, 0, 1], 3, 2], [[0, 3, 6], 9]), ([[0, 0, 0, 1], 3, 3], [[0, 3, 6, 9, 12, 15, 18, 21, 24], 27]), ([[0, 0, 0, 1], 3, 4], [[0, 9, 18, 27, 36, 45, 54, 63, 72], 81])]]\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":"ce3b93737c16846505da9bbbeef0130b571a975c7c6de47613c88b196a9dd595","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    coeff,p,k=x\n    roots=[r for r in range(p) if sum(c*r**i for i,c in enumerate(coeff))%p==0];mod=p\n    for _ in range(1,k):\n     nextmod=mod*p\n     out=[]\n     for r in roots:\n      for digit in range(p):\n       s=r+digit*mod\n       if sum(c*s**i for i,c in enumerate(coeff))%nextmod==0:out.append(s)\n     roots=sorted(set(out));mod=nextmod\n    return [roots,mod]\ndef check(label, actual, expected):\n    observations.append({\"check\": label, \"actual\": actual, \"expected\": expected, \"passed\": actual == expected})\nfixtures = [[([[0, 0, 1], 2, 3], [[0, 4], 8]), ([[-1, 0, 1], 3, 2], [[1, 8], 9]), ([[0, 0, 1], 2, 1], [[0], 2]), ([[-2, 1, 1], 7, 4], [[1, 2399], 2401]), ([[0, 0, 1], 2, 2], [[0, 2], 4]), ([[0, 0, 1], 2, 4], [[0, 4, 8, 12], 16]), ([[0, 0, 1], 3, 1], [[0], 3]), ([[0, 0, 1], 3, 2], [[0, 3, 6], 9])], [([[0, 0, 1], 3, 3], [[0, 9, 18], 27]), ([[-1, 0, 1], 5, 2], [[1, 24], 25]), ([[0, 0, 1], 2, 1], [[0], 2]), ([[-2, 1, 1], 7, 4], [[1, 2399], 2401]), ([[-1, 0, 1], 2, 2], [[1, 3], 4]), ([[-1, 0, 1], 2, 3], [[1, 3, 5, 7], 8]), ([[-1, 0, 1], 2, 4], [[1, 7, 9, 15], 16]), ([[-1, 0, 1], 3, 1], [[1, 2], 3])], [([[0, 0, 1], 5, 3], [[0, 25, 50, 75, 100], 125]), ([[-1, 0, 1], 7, 2], [[1, 48], 49]), ([[0, 0, 1], 2, 1], [[0], 2]), ([[-2, 1, 1], 7, 4], [[1, 2399], 2401]), ([[0, 2, 1], 2, 3], [[0, 2, 4, 6], 8]), ([[0, 2, 1], 2, 4], [[0, 6, 8, 14], 16]), ([[0, 2, 1], 3, 1], [[0, 1], 3]), ([[0, 2, 1], 3, 2], [[0, 7], 9])], [([[0, 0, 1], 7, 3], [[0, 49, 98, 147, 196, 245, 294], 343]), ([[0, 2, 1], 3, 2], [[0, 7], 9]), ([[0, 0, 1], 2, 1], [[0], 2]), ([[-2, 1, 1], 7, 4], [[1, 2399], 2401]), ([[-4, 0, 1], 2, 4], [[2, 6, 10, 14], 16]), ([[-4, 0, 1], 3, 1], [[1, 2], 3]), ([[-4, 0, 1], 3, 2], [[2, 7], 9]), ([[-4, 0, 1], 3, 3], [[2, 25], 27])], [([[-1, 0, 1], 2, 4], [[1, 7, 9, 15], 16]), ([[0, 2, 1], 5, 2], [[0, 23], 25]), ([[0, 0, 1], 2, 1], [[0], 2]), ([[-2, 1, 1], 7, 4], [[1, 2399], 2401]), ([[0, 0, 0, 1], 3, 1], [[0], 3]), ([[0, 0, 0, 1], 3, 2], [[0, 3, 6], 9]), ([[0, 0, 0, 1], 3, 3], [[0, 3, 6, 9, 12, 15, 18, 21, 24], 27]), ([[0, 0, 0, 1], 3, 4], [[0, 9, 18, 27, 36, 45, 54, 63, 72], 81])]]\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-singular-hensel-root-tree-lifted-polynomial-congruence","generated_at":"2026-09-29T14:39:30.446728+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 sum(c*s**i for i,c in enumerate(coeff))%nextmod==0 at the lifted polynomial congruence step.","root_cause":"The lifted polynomial congruence step uses sum(c*s**i for i,c in enumerate(coeff))%mod==0 instead of sum(c*s**i for i,c in enumerate(coeff))%nextmod==0.","sha256":"a6633c6f6de6f10a18738a681cef29c048da6ca867352253c8d1cdc092e19bb9","title":"Singular hensel root tree: lifted polynomial congruence · case 01","variant":1,"variant_policy":"Five numbered records share a model and may reuse boundary fixtures.","verification":{"attempt":{"elapsed_ms":41.157,"exit_code":1,"observations":[{"actual":[[0,4],8],"check":"explicit oracle 0","expected":[[0,4],8],"passed":true},{"actual":[[1,4,7],9],"check":"explicit oracle 1","expected":[[1,8],9],"passed":false},{"actual":[[0],2],"check":"explicit oracle 2","expected":[[0],2],"passed":true},{"actual":[[1,344,687,1030,1373,1716,2059],2401],"check":"explicit oracle 3","expected":[[1,2399],2401],"passed":false},{"actual":[[0,2],4],"check":"explicit oracle 4","expected":[[0,2],4],"passed":true},{"actual":[[0,4,8,12],16],"check":"explicit oracle 5","expected":[[0,4,8,12],16],"passed":true},{"actual":[[0],3],"check":"explicit oracle 6","expected":[[0],3],"passed":true},{"actual":[[0,3,6],9],"check":"explicit oracle 7","expected":[[0,3,6],9],"passed":true}],"passed":false,"stderr":"","stdout":"{\"observations\": [{\"check\": \"explicit oracle 0\", \"actual\": [[0, 4], 8], \"expected\": [[0, 4], 8], \"passed\": true}, {\"check\": \"explicit oracle 1\", \"actual\": [[1, 4, 7], 9], \"expected\": [[1, 8], 9], \"passed\": false}, {\"check\": \"explicit oracle 2\", \"actual\": [[0], 2], \"expected\": [[0], 2], \"passed\": true}, {\"check\": \"explicit oracle 3\", \"actual\": [[1, 344, 687, 1030, 1373, 1716, 2059], 2401], \"expected\": [[1, 2399], 2401], \"passed\": false}, {\"check\": \"explicit oracle 4\", \"actual\": [[0, 2], 4], \"expected\": [[0, 2], 4], \"passed\": true}, {\"check\": \"explicit oracle 5\", \"actual\": [[0, 4, 8, 12], 16], \"expected\": [[0, 4, 8, 12], 16], \"passed\": true}, {\"check\": \"explicit oracle 6\", \"actual\": [[0], 3], \"expected\": [[0], 3], \"passed\": true}, {\"check\": \"explicit oracle 7\", \"actual\": [[0, 3, 6], 9], \"expected\": [[0, 3, 6], 9], \"passed\": true}], \"passed\": false}\n"},"broken":{"elapsed_ms":41.943,"exit_code":1,"observations":[{"actual":[[0,2,4,6],8],"check":"explicit oracle 0","expected":[[0,4],8],"passed":false},{"actual":[[1,2,4,5,7,8],9],"check":"explicit oracle 1","expected":[[1,8],9],"passed":false},{"actual":[[0],2],"check":"explicit oracle 2","expected":[[0],2],"passed":true},{"actual":[[1,341,344,684,687,1027,1030,1370,1373,1713,1716,2056,2059,2399],2401],"check":"explicit oracle 3","expected":[[1,2399],2401],"passed":false},{"actual":[[0,2],4],"check":"explicit oracle 4","expected":[[0,2],4],"passed":true},{"actual":[[0,4,8,12],16],"check":"explicit oracle 5","expected":[[0,4,8,12],16],"passed":true},{"actual":[[0],3],"check":"explicit oracle 6","expected":[[0],3],"passed":true},{"actual":[[0,3,6],9],"check":"explicit oracle 7","expected":[[0,3,6],9],"passed":true}],"passed":false,"stderr":"","stdout":"{\"observations\": [{\"check\": \"explicit oracle 0\", \"actual\": [[0, 2, 4, 6], 8], \"expected\": [[0, 4], 8], \"passed\": false}, {\"check\": \"explicit oracle 1\", \"actual\": [[1, 2, 4, 5, 7, 8], 9], \"expected\": [[1, 8], 9], \"passed\": false}, {\"check\": \"explicit oracle 2\", \"actual\": [[0], 2], \"expected\": [[0], 2], \"passed\": true}, {\"check\": \"explicit oracle 3\", \"actual\": [[1, 341, 344, 684, 687, 1027, 1030, 1370, 1373, 1713, 1716, 2056, 2059, 2399], 2401], \"expected\": [[1, 2399], 2401], \"passed\": false}, {\"check\": \"explicit oracle 4\", \"actual\": [[0, 2], 4], \"expected\": [[0, 2], 4], \"passed\": true}, {\"check\": \"explicit oracle 5\", \"actual\": [[0, 4, 8, 12], 16], \"expected\": [[0, 4, 8, 12], 16], \"passed\": true}, {\"check\": \"explicit oracle 6\", \"actual\": [[0], 3], \"expected\": [[0], 3], \"passed\": true}, {\"check\": \"explicit oracle 7\", \"actual\": [[0, 3, 6], 9], \"expected\": [[0, 3, 6], 9], \"passed\": true}], \"passed\": false}\n"},"fixed":{"elapsed_ms":44.203,"exit_code":0,"observations":[{"actual":[[0,4],8],"check":"explicit oracle 0","expected":[[0,4],8],"passed":true},{"actual":[[1,8],9],"check":"explicit oracle 1","expected":[[1,8],9],"passed":true},{"actual":[[0],2],"check":"explicit oracle 2","expected":[[0],2],"passed":true},{"actual":[[1,2399],2401],"check":"explicit oracle 3","expected":[[1,2399],2401],"passed":true},{"actual":[[0,2],4],"check":"explicit oracle 4","expected":[[0,2],4],"passed":true},{"actual":[[0,4,8,12],16],"check":"explicit oracle 5","expected":[[0,4,8,12],16],"passed":true},{"actual":[[0],3],"check":"explicit oracle 6","expected":[[0],3],"passed":true},{"actual":[[0,3,6],9],"check":"explicit oracle 7","expected":[[0,3,6],9],"passed":true}],"passed":true,"stderr":"","stdout":"{\"observations\": [{\"check\": \"explicit oracle 0\", \"actual\": [[0, 4], 8], \"expected\": [[0, 4], 8], \"passed\": true}, {\"check\": \"explicit oracle 1\", \"actual\": [[1, 8], 9], \"expected\": [[1, 8], 9], \"passed\": true}, {\"check\": \"explicit oracle 2\", \"actual\": [[0], 2], \"expected\": [[0], 2], \"passed\": true}, {\"check\": \"explicit oracle 3\", \"actual\": [[1, 2399], 2401], \"expected\": [[1, 2399], 2401], \"passed\": true}, {\"check\": \"explicit oracle 4\", \"actual\": [[0, 2], 4], \"expected\": [[0, 2], 4], \"passed\": true}, {\"check\": \"explicit oracle 5\", \"actual\": [[0, 4, 8, 12], 16], \"expected\": [[0, 4, 8, 12], 16], \"passed\": true}, {\"check\": \"explicit oracle 6\", \"actual\": [[0], 3], \"expected\": [[0], 3], \"passed\": true}, {\"check\": \"explicit oracle 7\", \"actual\": [[0, 3, 6], 9], \"expected\": [[0, 3, 6], 9], \"passed\": true}], \"passed\": true}\n"}},"verified":true,"visibility":"public"}