{"abstract":"The exact exact ldl decomposition result violates the stated contract at lower factor diagonal divisor.","category":"Numerics","checks":8,"contract":"Input symmetric positive definite integer matrix; return unit lower L and diagonal D as rational pairs such that A=L*D*Ltranspose.","evaluation_group":"s3-numerics-exact-ldl-decomposition","failed_approach":"The partial repair D[j-1] if j else 1 still violates the lower factor diagonal divisor invariant.","family":"s3-numerics-exact-ldl-decomposition-lower-factor-diagonal-divisor","id":"FA-15461","implementations":{"attempt":{"sha256":"a7fa501ceb714cc76291f499d5039aefa97f5eac3ae742874a0cbf32e1faf05a","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=[list(map(Fraction,row)) for row in x];n=len(a);L=[[Fraction(int(i==j)) for j in range(n)] for i in range(n)];D=[Fraction(0)]*n\n    for j in range(n):\n     D[j]=a[j][j]-sum(L[j][k]**2*D[k] for k in range(j))\n     if not D[j]:return None\n     for i in range(j+1,n):\n      L[i][j]=(a[i][j]-sum(L[i][k]*L[j][k]*D[k] for k in range(j)))/(D[j-1] if j else 1)\n    return [[[[v.numerator,v.denominator] for v in row] for row in L],[[v.numerator,v.denominator] for v in D]]\ndef check(label, actual, expected):\n    observations.append({\"check\": label, \"actual\": actual, \"expected\": expected, \"passed\": actual == expected})\nfixtures = [[([[15, 14, 1], [14, 20, -1], [1, -1, 6]], [[[[1, 1], [0, 1], [0, 1]], [[14, 15], [1, 1], [0, 1]], [[1, 15], [-29, 104], [1, 1]]], [[15, 1], [104, 15], [561, 104]]]), ([[14, 2], [2, 2]], [[[[1, 1], [0, 1]], [[1, 7], [1, 1]]], [[14, 1], [12, 7]]]), ([[2]], [[[[1, 1]]], [[2, 1]]]), ([[20, -8, 15, -3], [-8, 8, -9, -7], [15, -9, 15, 4], [-3, -7, 4, 22]], [[[[1, 1], [0, 1], [0, 1], [0, 1]], [[-2, 5], [1, 1], [0, 1], [0, 1]], [[3, 4], [-5, 8], [1, 1], [0, 1]], [[-3, 20], [-41, 24], [3, 5], [1, 1]]], [[20, 1], [24, 5], [15, 8], [103, 15]]]), ([[5]], [[[[1, 1]]], [[5, 1]]]), ([[5]], [[[[1, 1]]], [[5, 1]]]), ([[2]], [[[[1, 1]]], [[2, 1]]]), ([[10]], [[[[1, 1]]], [[10, 1]]])], [([[11, 2, 6], [2, 14, 6], [6, 6, 9]], [[[[1, 1], [0, 1], [0, 1]], [[2, 11], [1, 1], [0, 1]], [[6, 11], [9, 25], [1, 1]]], [[11, 1], [150, 11], [99, 25]]]), ([[14, -12], [-12, 14]], [[[[1, 1], [0, 1]], [[-6, 7], [1, 1]]], [[14, 1], [26, 7]]]), ([[2]], [[[[1, 1]]], [[2, 1]]]), ([[20, -8, 15, -3], [-8, 8, -9, -7], [15, -9, 15, 4], [-3, -7, 4, 22]], [[[[1, 1], [0, 1], [0, 1], [0, 1]], [[-2, 5], [1, 1], [0, 1], [0, 1]], [[3, 4], [-5, 8], [1, 1], [0, 1]], [[-3, 20], [-41, 24], [3, 5], [1, 1]]], [[20, 1], [24, 5], [15, 8], [103, 15]]]), ([[10]], [[[[1, 1]]], [[10, 1]]]), ([[2]], [[[[1, 1]]], [[2, 1]]]), ([[10]], [[[[1, 1]]], [[10, 1]]]), ([[5]], [[[[1, 1]]], [[5, 1]]])], [([[20, -4, -6], [-4, 15, 11], [-6, 11, 15]], [[[[1, 1], [0, 1], [0, 1]], [[-1, 5], [1, 1], [0, 1]], [[-3, 10], [49, 71], [1, 1]]], [[20, 1], [71, 5], [457, 71]]]), ([[14, -1], [-1, 6]], [[[[1, 1], [0, 1]], [[-1, 14], [1, 1]]], [[14, 1], [83, 14]]]), ([[2]], [[[[1, 1]]], [[2, 1]]]), ([[20, -8, 15, -3], [-8, 8, -9, -7], [15, -9, 15, 4], [-3, -7, 4, 22]], [[[[1, 1], [0, 1], [0, 1], [0, 1]], [[-2, 5], [1, 1], [0, 1], [0, 1]], [[3, 4], [-5, 8], [1, 1], [0, 1]], [[-3, 20], [-41, 24], [3, 5], [1, 1]]], [[20, 1], [24, 5], [15, 8], [103, 15]]]), ([[10]], [[[[1, 1]]], [[10, 1]]]), ([[5]], [[[[1, 1]]], [[5, 1]]]), ([[2]], [[[[1, 1]]], [[2, 1]]]), ([[10]], [[[[1, 1]]], [[10, 1]]])], [([[7, 2, -10], [2, 5, -4], [-10, -4, 18]], [[[[1, 1], [0, 1], [0, 1]], [[2, 7], [1, 1], [0, 1]], [[-10, 7], [-8, 31], [1, 1]]], [[7, 1], [31, 7], [106, 31]]]), ([[19, 15], [15, 14]], [[[[1, 1], [0, 1]], [[15, 19], [1, 1]]], [[19, 1], [41, 19]]]), ([[2]], [[[[1, 1]]], [[2, 1]]]), ([[20, -8, 15, -3], [-8, 8, -9, -7], [15, -9, 15, 4], [-3, -7, 4, 22]], [[[[1, 1], [0, 1], [0, 1], [0, 1]], [[-2, 5], [1, 1], [0, 1], [0, 1]], [[3, 4], [-5, 8], [1, 1], [0, 1]], [[-3, 20], [-41, 24], [3, 5], [1, 1]]], [[20, 1], [24, 5], [15, 8], [103, 15]]]), ([[5]], [[[[1, 1]]], [[5, 1]]]), ([[5]], [[[[1, 1]]], [[5, 1]]]), ([[5]], [[[[1, 1]]], [[5, 1]]]), ([[5]], [[[[1, 1]]], [[5, 1]]])], [([[18, -12, -2], [-12, 14, 2], [-2, 2, 13]], [[[[1, 1], [0, 1], [0, 1]], [[-2, 3], [1, 1], [0, 1]], [[-1, 9], [1, 9], [1, 1]]], [[18, 1], [6, 1], [343, 27]]]), ([[11, 6], [6, 5]], [[[[1, 1], [0, 1]], [[6, 11], [1, 1]]], [[11, 1], [19, 11]]]), ([[2]], [[[[1, 1]]], [[2, 1]]]), ([[20, -8, 15, -3], [-8, 8, -9, -7], [15, -9, 15, 4], [-3, -7, 4, 22]], [[[[1, 1], [0, 1], [0, 1], [0, 1]], [[-2, 5], [1, 1], [0, 1], [0, 1]], [[3, 4], [-5, 8], [1, 1], [0, 1]], [[-3, 20], [-41, 24], [3, 5], [1, 1]]], [[20, 1], [24, 5], [15, 8], [103, 15]]]), ([[1, 0], [0, 19]], [[[[1, 1], [0, 1]], [[0, 1], [1, 1]]], [[1, 1], [19, 1]]]), ([[3, 2], [2, 5]], [[[[1, 1], [0, 1]], [[2, 3], [1, 1]]], [[3, 1], [11, 3]]]), ([[9, 0], [0, 1]], [[[[1, 1], [0, 1]], [[0, 1], [1, 1]]], [[9, 1], [1, 1]]]), ([[6, 5], [5, 11]], [[[[1, 1], [0, 1]], [[5, 6], [1, 1]]], [[6, 1], [41, 6]]])]]\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":"b880a502a456a18890b50c396885cdb1332d202c19e6e770a31bff74ddbae6db","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=[list(map(Fraction,row)) for row in x];n=len(a);L=[[Fraction(int(i==j)) for j in range(n)] for i in range(n)];D=[Fraction(0)]*n\n    for j in range(n):\n     D[j]=a[j][j]-sum(L[j][k]**2*D[k] for k in range(j))\n     if not D[j]:return None\n     for i in range(j+1,n):\n      L[i][j]=(a[i][j]-sum(L[i][k]*L[j][k]*D[k] for k in range(j)))/(a[j][j])\n    return [[[[v.numerator,v.denominator] for v in row] for row in L],[[v.numerator,v.denominator] for v in D]]\ndef check(label, actual, expected):\n    observations.append({\"check\": label, \"actual\": actual, \"expected\": expected, \"passed\": actual == expected})\nfixtures = [[([[15, 14, 1], [14, 20, -1], [1, -1, 6]], [[[[1, 1], [0, 1], [0, 1]], [[14, 15], [1, 1], [0, 1]], [[1, 15], [-29, 104], [1, 1]]], [[15, 1], [104, 15], [561, 104]]]), ([[14, 2], [2, 2]], [[[[1, 1], [0, 1]], [[1, 7], [1, 1]]], [[14, 1], [12, 7]]]), ([[2]], [[[[1, 1]]], [[2, 1]]]), ([[20, -8, 15, -3], [-8, 8, -9, -7], [15, -9, 15, 4], [-3, -7, 4, 22]], [[[[1, 1], [0, 1], [0, 1], [0, 1]], [[-2, 5], [1, 1], [0, 1], [0, 1]], [[3, 4], [-5, 8], [1, 1], [0, 1]], [[-3, 20], [-41, 24], [3, 5], [1, 1]]], [[20, 1], [24, 5], [15, 8], [103, 15]]]), ([[5]], [[[[1, 1]]], [[5, 1]]]), ([[5]], [[[[1, 1]]], [[5, 1]]]), ([[2]], [[[[1, 1]]], [[2, 1]]]), ([[10]], [[[[1, 1]]], [[10, 1]]])], [([[11, 2, 6], [2, 14, 6], [6, 6, 9]], [[[[1, 1], [0, 1], [0, 1]], [[2, 11], [1, 1], [0, 1]], [[6, 11], [9, 25], [1, 1]]], [[11, 1], [150, 11], [99, 25]]]), ([[14, -12], [-12, 14]], [[[[1, 1], [0, 1]], [[-6, 7], [1, 1]]], [[14, 1], [26, 7]]]), ([[2]], [[[[1, 1]]], [[2, 1]]]), ([[20, -8, 15, -3], [-8, 8, -9, -7], [15, -9, 15, 4], [-3, -7, 4, 22]], [[[[1, 1], [0, 1], [0, 1], [0, 1]], [[-2, 5], [1, 1], [0, 1], [0, 1]], [[3, 4], [-5, 8], [1, 1], [0, 1]], [[-3, 20], [-41, 24], [3, 5], [1, 1]]], [[20, 1], [24, 5], [15, 8], [103, 15]]]), ([[10]], [[[[1, 1]]], [[10, 1]]]), ([[2]], [[[[1, 1]]], [[2, 1]]]), ([[10]], [[[[1, 1]]], [[10, 1]]]), ([[5]], [[[[1, 1]]], [[5, 1]]])], [([[20, -4, -6], [-4, 15, 11], [-6, 11, 15]], [[[[1, 1], [0, 1], [0, 1]], [[-1, 5], [1, 1], [0, 1]], [[-3, 10], [49, 71], [1, 1]]], [[20, 1], [71, 5], [457, 71]]]), ([[14, -1], [-1, 6]], [[[[1, 1], [0, 1]], [[-1, 14], [1, 1]]], [[14, 1], [83, 14]]]), ([[2]], [[[[1, 1]]], [[2, 1]]]), ([[20, -8, 15, -3], [-8, 8, -9, -7], [15, -9, 15, 4], [-3, -7, 4, 22]], [[[[1, 1], [0, 1], [0, 1], [0, 1]], [[-2, 5], [1, 1], [0, 1], [0, 1]], [[3, 4], [-5, 8], [1, 1], [0, 1]], [[-3, 20], [-41, 24], [3, 5], [1, 1]]], [[20, 1], [24, 5], [15, 8], [103, 15]]]), ([[10]], [[[[1, 1]]], [[10, 1]]]), ([[5]], [[[[1, 1]]], [[5, 1]]]), ([[2]], [[[[1, 1]]], [[2, 1]]]), ([[10]], [[[[1, 1]]], [[10, 1]]])], [([[7, 2, -10], [2, 5, -4], [-10, -4, 18]], [[[[1, 1], [0, 1], [0, 1]], [[2, 7], [1, 1], [0, 1]], [[-10, 7], [-8, 31], [1, 1]]], [[7, 1], [31, 7], [106, 31]]]), ([[19, 15], [15, 14]], [[[[1, 1], [0, 1]], [[15, 19], [1, 1]]], [[19, 1], [41, 19]]]), ([[2]], [[[[1, 1]]], [[2, 1]]]), ([[20, -8, 15, -3], [-8, 8, -9, -7], [15, -9, 15, 4], [-3, -7, 4, 22]], [[[[1, 1], [0, 1], [0, 1], [0, 1]], [[-2, 5], [1, 1], [0, 1], [0, 1]], [[3, 4], [-5, 8], [1, 1], [0, 1]], [[-3, 20], [-41, 24], [3, 5], [1, 1]]], [[20, 1], [24, 5], [15, 8], [103, 15]]]), ([[5]], [[[[1, 1]]], [[5, 1]]]), ([[5]], [[[[1, 1]]], [[5, 1]]]), ([[5]], [[[[1, 1]]], [[5, 1]]]), ([[5]], [[[[1, 1]]], [[5, 1]]])], [([[18, -12, -2], [-12, 14, 2], [-2, 2, 13]], [[[[1, 1], [0, 1], [0, 1]], [[-2, 3], [1, 1], [0, 1]], [[-1, 9], [1, 9], [1, 1]]], [[18, 1], [6, 1], [343, 27]]]), ([[11, 6], [6, 5]], [[[[1, 1], [0, 1]], [[6, 11], [1, 1]]], [[11, 1], [19, 11]]]), ([[2]], [[[[1, 1]]], [[2, 1]]]), ([[20, -8, 15, -3], [-8, 8, -9, -7], [15, -9, 15, 4], [-3, -7, 4, 22]], [[[[1, 1], [0, 1], [0, 1], [0, 1]], [[-2, 5], [1, 1], [0, 1], [0, 1]], [[3, 4], [-5, 8], [1, 1], [0, 1]], [[-3, 20], [-41, 24], [3, 5], [1, 1]]], [[20, 1], [24, 5], [15, 8], [103, 15]]]), ([[1, 0], [0, 19]], [[[[1, 1], [0, 1]], [[0, 1], [1, 1]]], [[1, 1], [19, 1]]]), ([[3, 2], [2, 5]], [[[[1, 1], [0, 1]], [[2, 3], [1, 1]]], [[3, 1], [11, 3]]]), ([[9, 0], [0, 1]], [[[[1, 1], [0, 1]], [[0, 1], [1, 1]]], [[9, 1], [1, 1]]]), ([[6, 5], [5, 11]], [[[[1, 1], [0, 1]], [[5, 6], [1, 1]]], [[6, 1], [41, 6]]])]]\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":"6243c838e05296e5f84cbf2ea4b6e6f0c3eea924972cbcd3deb590a50eb23795","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=[list(map(Fraction,row)) for row in x];n=len(a);L=[[Fraction(int(i==j)) for j in range(n)] for i in range(n)];D=[Fraction(0)]*n\n    for j in range(n):\n     D[j]=a[j][j]-sum(L[j][k]**2*D[k] for k in range(j))\n     if not D[j]:return None\n     for i in range(j+1,n):\n      L[i][j]=(a[i][j]-sum(L[i][k]*L[j][k]*D[k] for k in range(j)))/(D[j])\n    return [[[[v.numerator,v.denominator] for v in row] for row in L],[[v.numerator,v.denominator] for v in D]]\ndef check(label, actual, expected):\n    observations.append({\"check\": label, \"actual\": actual, \"expected\": expected, \"passed\": actual == expected})\nfixtures = [[([[15, 14, 1], [14, 20, -1], [1, -1, 6]], [[[[1, 1], [0, 1], [0, 1]], [[14, 15], [1, 1], [0, 1]], [[1, 15], [-29, 104], [1, 1]]], [[15, 1], [104, 15], [561, 104]]]), ([[14, 2], [2, 2]], [[[[1, 1], [0, 1]], [[1, 7], [1, 1]]], [[14, 1], [12, 7]]]), ([[2]], [[[[1, 1]]], [[2, 1]]]), ([[20, -8, 15, -3], [-8, 8, -9, -7], [15, -9, 15, 4], [-3, -7, 4, 22]], [[[[1, 1], [0, 1], [0, 1], [0, 1]], [[-2, 5], [1, 1], [0, 1], [0, 1]], [[3, 4], [-5, 8], [1, 1], [0, 1]], [[-3, 20], [-41, 24], [3, 5], [1, 1]]], [[20, 1], [24, 5], [15, 8], [103, 15]]]), ([[5]], [[[[1, 1]]], [[5, 1]]]), ([[5]], [[[[1, 1]]], [[5, 1]]]), ([[2]], [[[[1, 1]]], [[2, 1]]]), ([[10]], [[[[1, 1]]], [[10, 1]]])], [([[11, 2, 6], [2, 14, 6], [6, 6, 9]], [[[[1, 1], [0, 1], [0, 1]], [[2, 11], [1, 1], [0, 1]], [[6, 11], [9, 25], [1, 1]]], [[11, 1], [150, 11], [99, 25]]]), ([[14, -12], [-12, 14]], [[[[1, 1], [0, 1]], [[-6, 7], [1, 1]]], [[14, 1], [26, 7]]]), ([[2]], [[[[1, 1]]], [[2, 1]]]), ([[20, -8, 15, -3], [-8, 8, -9, -7], [15, -9, 15, 4], [-3, -7, 4, 22]], [[[[1, 1], [0, 1], [0, 1], [0, 1]], [[-2, 5], [1, 1], [0, 1], [0, 1]], [[3, 4], [-5, 8], [1, 1], [0, 1]], [[-3, 20], [-41, 24], [3, 5], [1, 1]]], [[20, 1], [24, 5], [15, 8], [103, 15]]]), ([[10]], [[[[1, 1]]], [[10, 1]]]), ([[2]], [[[[1, 1]]], [[2, 1]]]), ([[10]], [[[[1, 1]]], [[10, 1]]]), ([[5]], [[[[1, 1]]], [[5, 1]]])], [([[20, -4, -6], [-4, 15, 11], [-6, 11, 15]], [[[[1, 1], [0, 1], [0, 1]], [[-1, 5], [1, 1], [0, 1]], [[-3, 10], [49, 71], [1, 1]]], [[20, 1], [71, 5], [457, 71]]]), ([[14, -1], [-1, 6]], [[[[1, 1], [0, 1]], [[-1, 14], [1, 1]]], [[14, 1], [83, 14]]]), ([[2]], [[[[1, 1]]], [[2, 1]]]), ([[20, -8, 15, -3], [-8, 8, -9, -7], [15, -9, 15, 4], [-3, -7, 4, 22]], [[[[1, 1], [0, 1], [0, 1], [0, 1]], [[-2, 5], [1, 1], [0, 1], [0, 1]], [[3, 4], [-5, 8], [1, 1], [0, 1]], [[-3, 20], [-41, 24], [3, 5], [1, 1]]], [[20, 1], [24, 5], [15, 8], [103, 15]]]), ([[10]], [[[[1, 1]]], [[10, 1]]]), ([[5]], [[[[1, 1]]], [[5, 1]]]), ([[2]], [[[[1, 1]]], [[2, 1]]]), ([[10]], [[[[1, 1]]], [[10, 1]]])], [([[7, 2, -10], [2, 5, -4], [-10, -4, 18]], [[[[1, 1], [0, 1], [0, 1]], [[2, 7], [1, 1], [0, 1]], [[-10, 7], [-8, 31], [1, 1]]], [[7, 1], [31, 7], [106, 31]]]), ([[19, 15], [15, 14]], [[[[1, 1], [0, 1]], [[15, 19], [1, 1]]], [[19, 1], [41, 19]]]), ([[2]], [[[[1, 1]]], [[2, 1]]]), ([[20, -8, 15, -3], [-8, 8, -9, -7], [15, -9, 15, 4], [-3, -7, 4, 22]], [[[[1, 1], [0, 1], [0, 1], [0, 1]], [[-2, 5], [1, 1], [0, 1], [0, 1]], [[3, 4], [-5, 8], [1, 1], [0, 1]], [[-3, 20], [-41, 24], [3, 5], [1, 1]]], [[20, 1], [24, 5], [15, 8], [103, 15]]]), ([[5]], [[[[1, 1]]], [[5, 1]]]), ([[5]], [[[[1, 1]]], [[5, 1]]]), ([[5]], [[[[1, 1]]], [[5, 1]]]), ([[5]], [[[[1, 1]]], [[5, 1]]])], [([[18, -12, -2], [-12, 14, 2], [-2, 2, 13]], [[[[1, 1], [0, 1], [0, 1]], [[-2, 3], [1, 1], [0, 1]], [[-1, 9], [1, 9], [1, 1]]], [[18, 1], [6, 1], [343, 27]]]), ([[11, 6], [6, 5]], [[[[1, 1], [0, 1]], [[6, 11], [1, 1]]], [[11, 1], [19, 11]]]), ([[2]], [[[[1, 1]]], [[2, 1]]]), ([[20, -8, 15, -3], [-8, 8, -9, -7], [15, -9, 15, 4], [-3, -7, 4, 22]], [[[[1, 1], [0, 1], [0, 1], [0, 1]], [[-2, 5], [1, 1], [0, 1], [0, 1]], [[3, 4], [-5, 8], [1, 1], [0, 1]], [[-3, 20], [-41, 24], [3, 5], [1, 1]]], [[20, 1], [24, 5], [15, 8], [103, 15]]]), ([[1, 0], [0, 19]], [[[[1, 1], [0, 1]], [[0, 1], [1, 1]]], [[1, 1], [19, 1]]]), ([[3, 2], [2, 5]], [[[[1, 1], [0, 1]], [[2, 3], [1, 1]]], [[3, 1], [11, 3]]]), ([[9, 0], [0, 1]], [[[[1, 1], [0, 1]], [[0, 1], [1, 1]]], [[9, 1], [1, 1]]]), ([[6, 5], [5, 11]], [[[[1, 1], [0, 1]], [[5, 6], [1, 1]]], [[6, 1], [41, 6]]])]]\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-exact-ldl-decomposition-lower-factor-diagonal-divisor","generated_at":"2026-09-29T14:39:26.993486+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 D[j] at the lower factor diagonal divisor step.","root_cause":"The lower factor diagonal divisor step uses a[j][j] instead of D[j].","sha256":"2c7d9578ffd4c55597c7db5ba94576aa792b60220a9bd6160cc50864adfb2db4","title":"Exact ldl decomposition: lower factor diagonal divisor · case 01","variant":1,"variant_policy":"Five numbered records share a model and may reuse boundary fixtures.","verification":{"attempt":{"elapsed_ms":45.512,"exit_code":1,"observations":[{"actual":[[[[1,1],[0,1],[0,1]],[[14,1],[1,1],[0,1]],[[1,1],[-211,15],[1,1]]],[[15,1],[-2920,1],[25999859,45]]],"check":"explicit oracle 0","expected":[[[[1,1],[0,1],[0,1]],[[14,15],[1,1],[0,1]],[[1,15],[-29,104],[1,1]]],[[15,1],[104,15],[561,104]]],"passed":false},{"actual":[[[[1,1],[0,1]],[[2,1],[1,1]]],[[14,1],[-54,1]]],"check":"explicit oracle 1","expected":[[[[1,1],[0,1]],[[1,7],[1,1]]],[[14,1],[12,7]]],"passed":false},{"actual":[[[[1,1]]],[[2,1]]],"check":"explicit oracle 2","expected":[[[[1,1]]],[[2,1]]],"passed":true},{"actual":[[[[1,1],[0,1],[0,1],[0,1]],[[-8,1],[1,1],[0,1],[0,1]],[[15,1],[2391,20],[1,1],[0,1]],[[-3,1],[-487,20],[185097103,63600],[1,1]]],[[20,1],[-1272,1],[908759829,50],[-10378321195603879856981287,67416000000]]],"check":"explicit oracle 3","expected":[[[[1,1],[0,1],[0,1],[0,1]],[[-2,5],[1,1],[0,1],[0,1]],[[3,4],[-5,8],[1,1],[0,1]],[[-3,20],[-41,24],[3,5],[1,1]]],[[20,1],[24,5],[15,8],[103,15]]],"passed":false},{"actual":[[[[1,1]]],[[5,1]]],"check":"explicit oracle 4","expected":[[[[1,1]]],[[5,1]]],"passed":true},{"actual":[[[[1,1]]],[[5,1]]],"check":"explicit oracle 5","expected":[[[[1,1]]],[[5,1]]],"passed":true},{"actual":[[[[1,1]]],[[2,1]]],"check":"explicit oracle 6","expected":[[[[1,1]]],[[2,1]]],"passed":true},{"actual":[[[[1,1]]],[[10,1]]],"check":"explicit oracle 7","expected":[[[[1,1]]],[[10,1]]],"passed":true}],"passed":false,"stderr":"","stdout":"{\"observations\": [{\"check\": \"explicit oracle 0\", \"actual\": [[[[1, 1], [0, 1], [0, 1]], [[14, 1], [1, 1], [0, 1]], [[1, 1], [-211, 15], [1, 1]]], [[15, 1], [-2920, 1], [25999859, 45]]], \"expected\": [[[[1, 1], [0, 1], [0, 1]], [[14, 15], [1, 1], [0, 1]], [[1, 15], [-29, 104], [1, 1]]], [[15, 1], [104, 15], [561, 104]]], \"passed\": false}, {\"check\": \"explicit oracle 1\", \"actual\": [[[[1, 1], [0, 1]], [[2, 1], [1, 1]]], [[14, 1], [-54, 1]]], \"expected\": [[[[1, 1], [0, 1]], [[1, 7], [1, 1]]], [[14, 1], [12, 7]]], \"passed\": false}, {\"check\": \"explicit oracle 2\", \"actual\": [[[[1, 1]]], [[2, 1]]], \"expected\": [[[[1, 1]]], [[2, 1]]], \"passed\": true}, {\"check\": \"explicit oracle 3\", \"actual\": [[[[1, 1], [0, 1], [0, 1], [0, 1]], [[-8, 1], [1, 1], [0, 1], [0, 1]], [[15, 1], [2391, 20], [1, 1], [0, 1]], [[-3, 1], [-487, 20], [185097103, 63600], [1, 1]]], [[20, 1], [-1272, 1], [908759829, 50], [-10378321195603879856981287, 67416000000]]], \"expected\": [[[[1, 1], [0, 1], [0, 1], [0, 1]], [[-2, 5], [1, 1], [0, 1], [0, 1]], [[3, 4], [-5, 8], [1, 1], [0, 1]], [[-3, 20], [-41, 24], [3, 5], [1, 1]]], [[20, 1], [24, 5], [15, 8], [103, 15]]], \"passed\": false}, {\"check\": \"explicit oracle 4\", \"actual\": [[[[1, 1]]], [[5, 1]]], \"expected\": [[[[1, 1]]], [[5, 1]]], \"passed\": true}, {\"check\": \"explicit oracle 5\", \"actual\": [[[[1, 1]]], [[5, 1]]], \"expected\": [[[[1, 1]]], [[5, 1]]], \"passed\": true}, {\"check\": \"explicit oracle 6\", \"actual\": [[[[1, 1]]], [[2, 1]]], \"expected\": [[[[1, 1]]], [[2, 1]]], \"passed\": true}, {\"check\": \"explicit oracle 7\", \"actual\": [[[[1, 1]]], [[10, 1]]], \"expected\": [[[[1, 1]]], [[10, 1]]], \"passed\": true}], \"passed\": false}\n"},"broken":{"elapsed_ms":43.928,"exit_code":1,"observations":[{"actual":[[[[1,1],[0,1],[0,1]],[[14,15],[1,1],[0,1]],[[1,15],[-29,300],[1,1]]],[[15,1],[104,15],[990317,168750]]],"check":"explicit oracle 0","expected":[[[[1,1],[0,1],[0,1]],[[14,15],[1,1],[0,1]],[[1,15],[-29,104],[1,1]]],[[15,1],[104,15],[561,104]]],"passed":false},{"actual":[[[[1,1],[0,1]],[[1,7],[1,1]]],[[14,1],[12,7]]],"check":"explicit oracle 1","expected":[[[[1,1],[0,1]],[[1,7],[1,1]]],[[14,1],[12,7]]],"passed":true},{"actual":[[[[1,1]]],[[2,1]]],"check":"explicit oracle 2","expected":[[[[1,1]]],[[2,1]]],"passed":true},{"actual":[[[[1,1],[0,1],[0,1],[0,1]],[[-2,5],[1,1],[0,1],[0,1]],[[3,4],[-3,8],[1,1],[0,1]],[[-3,20],[-41,40],[881,3000],[1,1]]],[[20,1],[24,5],[123,40],[1949017399,120000000]]],"check":"explicit oracle 3","expected":[[[[1,1],[0,1],[0,1],[0,1]],[[-2,5],[1,1],[0,1],[0,1]],[[3,4],[-5,8],[1,1],[0,1]],[[-3,20],[-41,24],[3,5],[1,1]]],[[20,1],[24,5],[15,8],[103,15]]],"passed":false},{"actual":[[[[1,1]]],[[5,1]]],"check":"explicit oracle 4","expected":[[[[1,1]]],[[5,1]]],"passed":true},{"actual":[[[[1,1]]],[[5,1]]],"check":"explicit oracle 5","expected":[[[[1,1]]],[[5,1]]],"passed":true},{"actual":[[[[1,1]]],[[2,1]]],"check":"explicit oracle 6","expected":[[[[1,1]]],[[2,1]]],"passed":true},{"actual":[[[[1,1]]],[[10,1]]],"check":"explicit oracle 7","expected":[[[[1,1]]],[[10,1]]],"passed":true}],"passed":false,"stderr":"","stdout":"{\"observations\": [{\"check\": \"explicit oracle 0\", \"actual\": [[[[1, 1], [0, 1], [0, 1]], [[14, 15], [1, 1], [0, 1]], [[1, 15], [-29, 300], [1, 1]]], [[15, 1], [104, 15], [990317, 168750]]], \"expected\": [[[[1, 1], [0, 1], [0, 1]], [[14, 15], [1, 1], [0, 1]], [[1, 15], [-29, 104], [1, 1]]], [[15, 1], [104, 15], [561, 104]]], \"passed\": false}, {\"check\": \"explicit oracle 1\", \"actual\": [[[[1, 1], [0, 1]], [[1, 7], [1, 1]]], [[14, 1], [12, 7]]], \"expected\": [[[[1, 1], [0, 1]], [[1, 7], [1, 1]]], [[14, 1], [12, 7]]], \"passed\": true}, {\"check\": \"explicit oracle 2\", \"actual\": [[[[1, 1]]], [[2, 1]]], \"expected\": [[[[1, 1]]], [[2, 1]]], \"passed\": true}, {\"check\": \"explicit oracle 3\", \"actual\": [[[[1, 1], [0, 1], [0, 1], [0, 1]], [[-2, 5], [1, 1], [0, 1], [0, 1]], [[3, 4], [-3, 8], [1, 1], [0, 1]], [[-3, 20], [-41, 40], [881, 3000], [1, 1]]], [[20, 1], [24, 5], [123, 40], [1949017399, 120000000]]], \"expected\": [[[[1, 1], [0, 1], [0, 1], [0, 1]], [[-2, 5], [1, 1], [0, 1], [0, 1]], [[3, 4], [-5, 8], [1, 1], [0, 1]], [[-3, 20], [-41, 24], [3, 5], [1, 1]]], [[20, 1], [24, 5], [15, 8], [103, 15]]], \"passed\": false}, {\"check\": \"explicit oracle 4\", \"actual\": [[[[1, 1]]], [[5, 1]]], \"expected\": [[[[1, 1]]], [[5, 1]]], \"passed\": true}, {\"check\": \"explicit oracle 5\", \"actual\": [[[[1, 1]]], [[5, 1]]], \"expected\": [[[[1, 1]]], [[5, 1]]], \"passed\": true}, {\"check\": \"explicit oracle 6\", \"actual\": [[[[1, 1]]], [[2, 1]]], \"expected\": [[[[1, 1]]], [[2, 1]]], \"passed\": true}, {\"check\": \"explicit oracle 7\", \"actual\": [[[[1, 1]]], [[10, 1]]], \"expected\": [[[[1, 1]]], [[10, 1]]], \"passed\": true}], \"passed\": false}\n"},"fixed":{"elapsed_ms":45.611,"exit_code":0,"observations":[{"actual":[[[[1,1],[0,1],[0,1]],[[14,15],[1,1],[0,1]],[[1,15],[-29,104],[1,1]]],[[15,1],[104,15],[561,104]]],"check":"explicit oracle 0","expected":[[[[1,1],[0,1],[0,1]],[[14,15],[1,1],[0,1]],[[1,15],[-29,104],[1,1]]],[[15,1],[104,15],[561,104]]],"passed":true},{"actual":[[[[1,1],[0,1]],[[1,7],[1,1]]],[[14,1],[12,7]]],"check":"explicit oracle 1","expected":[[[[1,1],[0,1]],[[1,7],[1,1]]],[[14,1],[12,7]]],"passed":true},{"actual":[[[[1,1]]],[[2,1]]],"check":"explicit oracle 2","expected":[[[[1,1]]],[[2,1]]],"passed":true},{"actual":[[[[1,1],[0,1],[0,1],[0,1]],[[-2,5],[1,1],[0,1],[0,1]],[[3,4],[-5,8],[1,1],[0,1]],[[-3,20],[-41,24],[3,5],[1,1]]],[[20,1],[24,5],[15,8],[103,15]]],"check":"explicit oracle 3","expected":[[[[1,1],[0,1],[0,1],[0,1]],[[-2,5],[1,1],[0,1],[0,1]],[[3,4],[-5,8],[1,1],[0,1]],[[-3,20],[-41,24],[3,5],[1,1]]],[[20,1],[24,5],[15,8],[103,15]]],"passed":true},{"actual":[[[[1,1]]],[[5,1]]],"check":"explicit oracle 4","expected":[[[[1,1]]],[[5,1]]],"passed":true},{"actual":[[[[1,1]]],[[5,1]]],"check":"explicit oracle 5","expected":[[[[1,1]]],[[5,1]]],"passed":true},{"actual":[[[[1,1]]],[[2,1]]],"check":"explicit oracle 6","expected":[[[[1,1]]],[[2,1]]],"passed":true},{"actual":[[[[1,1]]],[[10,1]]],"check":"explicit oracle 7","expected":[[[[1,1]]],[[10,1]]],"passed":true}],"passed":true,"stderr":"","stdout":"{\"observations\": [{\"check\": \"explicit oracle 0\", \"actual\": [[[[1, 1], [0, 1], [0, 1]], [[14, 15], [1, 1], [0, 1]], [[1, 15], [-29, 104], [1, 1]]], [[15, 1], [104, 15], [561, 104]]], \"expected\": [[[[1, 1], [0, 1], [0, 1]], [[14, 15], [1, 1], [0, 1]], [[1, 15], [-29, 104], [1, 1]]], [[15, 1], [104, 15], [561, 104]]], \"passed\": true}, {\"check\": \"explicit oracle 1\", \"actual\": [[[[1, 1], [0, 1]], [[1, 7], [1, 1]]], [[14, 1], [12, 7]]], \"expected\": [[[[1, 1], [0, 1]], [[1, 7], [1, 1]]], [[14, 1], [12, 7]]], \"passed\": true}, {\"check\": \"explicit oracle 2\", \"actual\": [[[[1, 1]]], [[2, 1]]], \"expected\": [[[[1, 1]]], [[2, 1]]], \"passed\": true}, {\"check\": \"explicit oracle 3\", \"actual\": [[[[1, 1], [0, 1], [0, 1], [0, 1]], [[-2, 5], [1, 1], [0, 1], [0, 1]], [[3, 4], [-5, 8], [1, 1], [0, 1]], [[-3, 20], [-41, 24], [3, 5], [1, 1]]], [[20, 1], [24, 5], [15, 8], [103, 15]]], \"expected\": [[[[1, 1], [0, 1], [0, 1], [0, 1]], [[-2, 5], [1, 1], [0, 1], [0, 1]], [[3, 4], [-5, 8], [1, 1], [0, 1]], [[-3, 20], [-41, 24], [3, 5], [1, 1]]], [[20, 1], [24, 5], [15, 8], [103, 15]]], \"passed\": true}, {\"check\": \"explicit oracle 4\", \"actual\": [[[[1, 1]]], [[5, 1]]], \"expected\": [[[[1, 1]]], [[5, 1]]], \"passed\": true}, {\"check\": \"explicit oracle 5\", \"actual\": [[[[1, 1]]], [[5, 1]]], \"expected\": [[[[1, 1]]], [[5, 1]]], \"passed\": true}, {\"check\": \"explicit oracle 6\", \"actual\": [[[[1, 1]]], [[2, 1]]], \"expected\": [[[[1, 1]]], [[2, 1]]], \"passed\": true}, {\"check\": \"explicit oracle 7\", \"actual\": [[[[1, 1]]], [[10, 1]]], \"expected\": [[[[1, 1]]], [[10, 1]]], \"passed\": true}], \"passed\": true}\n"}},"verified":true,"visibility":"public"}