{"abstract":"The exact hensel simple root lift result violates the stated contract at lift correction residue.","category":"Numerics","checks":8,"contract":"Input [coefficients,p,r,k], p prime, r root mod p with nonzero derivative, k>=1; return unique root mod p^k congruent r mod p.","contract_signature":"x","evaluation_group":"s3-numerics-hensel-simple-root-lift","failed_approach":"The partial repair ((value//mod)*pow(derivative,-1,p))%p still violates the lift correction residue invariant.","family":"s3-numerics-hensel-simple-root-lift-lift-correction-residue","id":"FA-15501","implementations":{"attempt":{"sha256":"53ab56dd6002986f5eec176e8040345607169e39ad69cd7db1fb6d65516978f5","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,r,k=x\n    mod=p;r%=p\n    for _ in range(1,k):\n     value=sum(c*r**i for i,c in enumerate(coeff))\n     derivative=sum(i*coeff[i]*r**(i-1) for i in range(1,len(coeff)))\n     if derivative%p==0:return None\n     correction=((value//mod)*pow(derivative,-1,p))%p\n     r=r+mod*correction\n     mod=mod*p\n     r%=mod\n    return [r,mod]\ndef check(label, actual, expected):\n    observations.append({\"check\": label, \"actual\": actual, \"expected\": expected, \"passed\": actual == expected})\nfixtures = [[([[1, 1, 1], 7, 2, 2], [30, 49]), ([[1, 1, 1], 7, 2, 1], [2, 7]), ([[2, 3, 1], 13, 12, 5], [371292, 371293]), ([[1, 1, 1], 7, 2, 3], [324, 343]), ([[1, 1, 1], 7, 2, 4], [1353, 2401]), ([[1, 1, 1], 7, 2, 5], [1353, 16807]), ([[1, 1, 1], 7, 4, 1], [4, 7]), ([[1, 1, 1], 7, 4, 2], [18, 49])], [([[1, 1, 1], 7, 2, 3], [324, 343]), ([[1, 1, 1], 7, 2, 5], [1353, 16807]), ([[1, 1, 1], 7, 2, 1], [2, 7]), ([[2, 3, 1], 13, 12, 5], [371292, 371293]), ([[1, 1, 1], 13, 9, 3], [1036, 2197]), ([[1, 1, 1], 13, 9, 4], [7627, 28561]), ([[1, 1, 1], 13, 9, 5], [150432, 371293]), ([[-2, 0, 1], 7, 3, 1], [3, 7])], [([[1, 1, 1], 7, 2, 4], [1353, 2401]), ([[1, 1, 1], 7, 4, 4], [1047, 2401]), ([[1, 1, 1], 7, 2, 1], [2, 7]), ([[2, 3, 1], 13, 12, 5], [371292, 371293]), ([[-1, 1, 1], 11, 3, 5], [77234, 161051]), ([[-1, 1, 1], 11, 7, 1], [7, 11]), ([[-1, 1, 1], 11, 7, 2], [84, 121]), ([[-1, 1, 1], 11, 7, 3], [1294, 1331])], [([[1, 1, 1], 7, 2, 5], [1353, 16807]), ([[1, 1, 1], 13, 3, 3], [1160, 2197]), ([[1, 1, 1], 7, 2, 1], [2, 7]), ([[2, 3, 1], 13, 12, 5], [371292, 371293]), ([[-3, 1, 2], 7, 1, 2], [1, 49]), ([[-3, 1, 2], 7, 1, 3], [1, 343]), ([[-3, 1, 2], 7, 1, 4], [1, 2401]), ([[-3, 1, 2], 7, 1, 5], [1, 16807])], [([[1, 1, 1], 7, 4, 2], [18, 49]), ([[1, 1, 1], 13, 9, 2], [22, 169]), ([[1, 1, 1], 7, 2, 1], [2, 7]), ([[2, 3, 1], 13, 12, 5], [371292, 371293]), ([[-3, 1, 2], 11, 4, 4], [7319, 14641]), ([[-3, 1, 2], 11, 4, 5], [80524, 161051]), ([[-3, 1, 2], 13, 1, 1], [1, 13]), ([[-3, 1, 2], 13, 1, 2], [1, 169])]]\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":"09d4dfffa4736c1ac1408f3a563687d960203cbd63d96ce7ed0caf6ce6599e8f","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,r,k=x\n    mod=p;r%=p\n    for _ in range(1,k):\n     value=sum(c*r**i for i,c in enumerate(coeff))\n     derivative=sum(i*coeff[i]*r**(i-1) for i in range(1,len(coeff)))\n     if derivative%p==0:return None\n     correction=(-value*pow(derivative,-1,p))%p\n     r=r+mod*correction\n     mod=mod*p\n     r%=mod\n    return [r,mod]\ndef check(label, actual, expected):\n    observations.append({\"check\": label, \"actual\": actual, \"expected\": expected, \"passed\": actual == expected})\nfixtures = [[([[1, 1, 1], 7, 2, 2], [30, 49]), ([[1, 1, 1], 7, 2, 1], [2, 7]), ([[2, 3, 1], 13, 12, 5], [371292, 371293]), ([[1, 1, 1], 7, 2, 3], [324, 343]), ([[1, 1, 1], 7, 2, 4], [1353, 2401]), ([[1, 1, 1], 7, 2, 5], [1353, 16807]), ([[1, 1, 1], 7, 4, 1], [4, 7]), ([[1, 1, 1], 7, 4, 2], [18, 49])], [([[1, 1, 1], 7, 2, 3], [324, 343]), ([[1, 1, 1], 7, 2, 5], [1353, 16807]), ([[1, 1, 1], 7, 2, 1], [2, 7]), ([[2, 3, 1], 13, 12, 5], [371292, 371293]), ([[1, 1, 1], 13, 9, 3], [1036, 2197]), ([[1, 1, 1], 13, 9, 4], [7627, 28561]), ([[1, 1, 1], 13, 9, 5], [150432, 371293]), ([[-2, 0, 1], 7, 3, 1], [3, 7])], [([[1, 1, 1], 7, 2, 4], [1353, 2401]), ([[1, 1, 1], 7, 4, 4], [1047, 2401]), ([[1, 1, 1], 7, 2, 1], [2, 7]), ([[2, 3, 1], 13, 12, 5], [371292, 371293]), ([[-1, 1, 1], 11, 3, 5], [77234, 161051]), ([[-1, 1, 1], 11, 7, 1], [7, 11]), ([[-1, 1, 1], 11, 7, 2], [84, 121]), ([[-1, 1, 1], 11, 7, 3], [1294, 1331])], [([[1, 1, 1], 7, 2, 5], [1353, 16807]), ([[1, 1, 1], 13, 3, 3], [1160, 2197]), ([[1, 1, 1], 7, 2, 1], [2, 7]), ([[2, 3, 1], 13, 12, 5], [371292, 371293]), ([[-3, 1, 2], 7, 1, 2], [1, 49]), ([[-3, 1, 2], 7, 1, 3], [1, 343]), ([[-3, 1, 2], 7, 1, 4], [1, 2401]), ([[-3, 1, 2], 7, 1, 5], [1, 16807])], [([[1, 1, 1], 7, 4, 2], [18, 49]), ([[1, 1, 1], 13, 9, 2], [22, 169]), ([[1, 1, 1], 7, 2, 1], [2, 7]), ([[2, 3, 1], 13, 12, 5], [371292, 371293]), ([[-3, 1, 2], 11, 4, 4], [7319, 14641]), ([[-3, 1, 2], 11, 4, 5], [80524, 161051]), ([[-3, 1, 2], 13, 1, 1], [1, 13]), ([[-3, 1, 2], 13, 1, 2], [1, 169])]]\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-hensel-simple-root-lift-lift-correction-residue","generated_at":"2026-09-29T14:39:27.408805+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 lift correction residue step uses (-value*pow(derivative,-1,p))%p instead of (-(value//mod)*pow(derivative,-1,p))%p.","sha256":"e66126b28b8388818acd9f3de981d6cefdaf3be7a148f7c72df9b13227f12649","title":"Hensel simple root lift: lift correction residue · 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":42.925,"exit_code":1,"observations":[{"actual":[23,49],"check":"explicit oracle 0","expected":[30,49],"passed":false},{"actual":[2,7],"check":"explicit oracle 1","expected":[2,7],"passed":true},{"actual":[92975,371293],"check":"explicit oracle 2","expected":[371292,371293],"passed":false},{"actual":[268,343],"check":"explicit oracle 3","expected":[324,343],"passed":false},{"actual":[268,2401],"check":"explicit oracle 4","expected":[1353,2401],"passed":false},{"actual":[14674,16807],"check":"explicit oracle 5","expected":[1353,16807],"passed":false},{"actual":[4,7],"check":"explicit oracle 6","expected":[4,7],"passed":true},{"actual":[39,49],"check":"explicit oracle 7","expected":[18,49],"passed":false}],"passed":false,"stderr":"","stdout":"{\"observations\": [{\"check\": \"explicit oracle 0\", \"actual\": [23, 49], \"expected\": [30, 49], \"passed\": false}, {\"check\": \"explicit oracle 1\", \"actual\": [2, 7], \"expected\": [2, 7], \"passed\": true}, {\"check\": \"explicit oracle 2\", \"actual\": [92975, 371293], \"expected\": [371292, 371293], \"passed\": false}, {\"check\": \"explicit oracle 3\", \"actual\": [268, 343], \"expected\": [324, 343], \"passed\": false}, {\"check\": \"explicit oracle 4\", \"actual\": [268, 2401], \"expected\": [1353, 2401], \"passed\": false}, {\"check\": \"explicit oracle 5\", \"actual\": [14674, 16807], \"expected\": [1353, 16807], \"passed\": false}, {\"check\": \"explicit oracle 6\", \"actual\": [4, 7], \"expected\": [4, 7], \"passed\": true}, {\"check\": \"explicit oracle 7\", \"actual\": [39, 49], \"expected\": [18, 49], \"passed\": false}], \"passed\": false}\n"},"broken":{"elapsed_ms":43.784,"exit_code":1,"observations":[{"actual":[2,49],"check":"explicit oracle 0","expected":[30,49],"passed":false},{"actual":[2,7],"check":"explicit oracle 1","expected":[2,7],"passed":true},{"actual":[12,371293],"check":"explicit oracle 2","expected":[371292,371293],"passed":false},{"actual":[2,343],"check":"explicit oracle 3","expected":[324,343],"passed":false},{"actual":[2,2401],"check":"explicit oracle 4","expected":[1353,2401],"passed":false},{"actual":[2,16807],"check":"explicit oracle 5","expected":[1353,16807],"passed":false},{"actual":[4,7],"check":"explicit oracle 6","expected":[4,7],"passed":true},{"actual":[4,49],"check":"explicit oracle 7","expected":[18,49],"passed":false}],"passed":false,"stderr":"","stdout":"{\"observations\": [{\"check\": \"explicit oracle 0\", \"actual\": [2, 49], \"expected\": [30, 49], \"passed\": false}, {\"check\": \"explicit oracle 1\", \"actual\": [2, 7], \"expected\": [2, 7], \"passed\": true}, {\"check\": \"explicit oracle 2\", \"actual\": [12, 371293], \"expected\": [371292, 371293], \"passed\": false}, {\"check\": \"explicit oracle 3\", \"actual\": [2, 343], \"expected\": [324, 343], \"passed\": false}, {\"check\": \"explicit oracle 4\", \"actual\": [2, 2401], \"expected\": [1353, 2401], \"passed\": false}, {\"check\": \"explicit oracle 5\", \"actual\": [2, 16807], \"expected\": [1353, 16807], \"passed\": false}, {\"check\": \"explicit oracle 6\", \"actual\": [4, 7], \"expected\": [4, 7], \"passed\": true}, {\"check\": \"explicit oracle 7\", \"actual\": [4, 49], \"expected\": [18, 49], \"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."}}