{
 "sector": "S2",
 "note": "Left null vectors v of the response matrices A(eps) of Eq. (27) (v^T A = 0), i.e. the flow-free directions of Table 3, normalized to 1 on one support row.  The relation among master integrals is sum_row weight[row] * M[row] = 0 at each eps.  Rows are response-matrix rows (field response_matrix_row of the masters above).  Weights that are rational and independent of eps are given exactly; the others are given at the twelve eps values of the computation.",
 "eps_grid": [
  "0.0003114995735603784084585420144685587089594",
  "0.0003104656021947690595436839176405119643851",
  "0.0003084681229037284362826933778519755147468",
  "0.0003056432605996104618809126321647189760712",
  "0.0003021835249805230940456316082359488035615",
  "0.0002983246913110740122426265998887033945465",
  "0.0002943297327289927657206455203116304952699",
  "0.0002904708990595436839176405119643850862549",
  "0.0002870111634404563160823594880356149137451",
  "0.0002841863011363383416805787423483583750696",
  "0.0002821888218452977184195882025598219254313",
  "0.0002811548504796883695047301057317751808570"
 ],
 "left_null_vectors": [
  {
   "support": [
    "I_r0",
    "I_r2",
    "I_r3"
   ],
   "rows": [
    0,
    2,
    3
   ],
   "weights_exact": [
    "-1/6",
    "1",
    "-8/9"
   ],
   "eps_dependence": "none (spread 2.4e-83 over the twelve eps values)",
   "relation": "-(1/6) I_r0 + I_r2 - (8/9) I_r3 = 0 (the same relation as in S_1 after relabeling)"
  },
  {
   "support": [
    "I_r1",
    "I_r5",
    "I_r6"
   ],
   "rows": [
    1,
    5,
    6
   ],
   "normalized_row": 6,
   "per_eps": [
    {
     "eps": "0.0003114995735603784084585420144685587089594",
     "weights": {
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       "im": "3.6776704130771185410963479453287253095504492099046e-80"
      },
      "I_r5": {
       "re": "0.29629629949674378391667389185574243589193619697742",
       "im": "4.5765223674786107762903475453440305159321914590415e-84"
      },
      "I_r6": {
       "re": "1.0",
       "im": "0"
      }
     },
     "max_off_support_weight": "8.59e-78",
     "residual_vT_A": "1.21e-77"
    },
    {
     "eps": "0.0003104656021947690595436839176405119643851",
     "weights": {
      "I_r1": {
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       "im": "3.7338568232810923535796414180851822050027304655869e-80"
      },
      "I_r5": {
       "re": "0.29629629947551259111934507895308643545470306089137",
       "im": "4.6027992014555288792451064589892025808295550497523e-84"
      },
      "I_r6": {
       "re": "1.0",
       "im": "0"
      }
     },
     "max_off_support_weight": "8.68e-78",
     "residual_vT_A": "1.22e-77"
    },
    {
     "eps": "0.0003084681229037284362826933778519755147468",
     "weights": {
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       "im": "3.8453865795105779002822851870698586264357892761146e-80"
      },
      "I_r5": {
       "re": "0.29629629943469754237251177912212870178955938412736",
       "im": "4.6542310720393251911334988509469887123120226195803e-84"
      },
      "I_r6": {
       "re": "1.0",
       "im": "0"
      }
     },
     "max_off_support_weight": "8.86e-78",
     "residual_vT_A": "1.25e-77"
    },
    {
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     "weights": {
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       "im": "4.0101150341760402266456186327553967412028334848634e-80"
      },
      "I_r5": {
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       "im": "4.7285071123368355292826347523960833403399817469113e-84"
      },
      "I_r6": {
       "re": "1.0",
       "im": "0"
      }
     },
     "max_off_support_weight": "9.12e-78",
     "residual_vT_A": "1.29e-77"
    },
    {
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     "weights": {
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      },
      "I_r5": {
       "re": "0.29629629930800567005475121688470551937178914941331",
       "im": "4.8220180507591340937316461972447653377437068234426e-84"
      },
      "I_r6": {
       "re": "1.0",
       "im": "0"
      }
     },
     "max_off_support_weight": "9.46e-78",
     "residual_vT_A": "1.33e-77"
    },
    {
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     "weights": {
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       "im": "4.4785270781626891972017812602170544923493934225527e-80"
      },
      "I_r5": {
       "re": "0.2962962992315107018202137981753855397075555866031",
       "im": "4.9297558531679185235426587249761924797130113092934e-84"
      },
      "I_r6": {
       "re": "1.0",
       "im": "0"
      }
     },
     "max_off_support_weight": "9.85e-78",
     "residual_vT_A": "1.39e-77"
    },
    {
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     "weights": {
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       "im": "4.7623001160954403135930908639723346809467664686287e-80"
      },
      "I_r5": {
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       "im": "5.0452973671717289366102639794802732794230832169324e-84"
      },
      "I_r6": {
       "re": "1.0",
       "im": "0"
      }
     },
     "max_off_support_weight": "1.03e-77",
     "residual_vT_A": "1.45e-77"
    },
    {
     "eps": "0.0002904708990595436839176405119643850862549",
     "weights": {
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       "im": "5.0574359582193522470207603593685470728607713003495e-80"
      },
      "I_r5": {
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       "im": "5.1609613022412635279558851947053677788837667304783e-84"
      },
      "I_r6": {
       "re": "1.0",
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      }
     },
     "max_off_support_weight": "3.91e-78",
     "residual_vT_A": "1.51e-77"
    },
    {
     "eps": "0.0002870111634404563160823594880356149137451",
     "weights": {
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       "im": "5.3411627226984673699644057120982903471543634667804e-80"
      },
      "I_r5": {
       "re": "0.29629629901292010175209303853825427074069605182856",
       "im": "5.2682195107502822091820170493441761211285465351056e-84"
      },
      "I_r6": {
       "re": "1.0",
       "im": "0"
      }
     },
     "max_off_support_weight": "1.11e-77",
     "residual_vT_A": "1.57e-77"
    },
    {
     "eps": "0.0002841863011363383416805787423483583750696",
     "weights": {
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       "im": "5.5872901159623170055893568199564952570047817675537e-80"
      },
      "I_r5": {
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       "im": "5.3583992554387596191086304344123514867306448960894e-84"
      },
      "I_r6": {
       "re": "1.0",
       "im": "0"
      }
     },
     "max_off_support_weight": "1.15e-77",
     "residual_vT_A": "1.62e-77"
    },
    {
     "eps": "0.0002821888218452977184195882025598219254313",
     "weights": {
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      },
      "I_r5": {
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       "im": "5.4236299413867521057161808284384548035518559192599e-84"
      },
      "I_r6": {
       "re": "1.0",
       "im": "0"
      }
     },
     "max_off_support_weight": "4.46e-78",
     "residual_vT_A": "1.66e-77"
    },
    {
     "eps": "0.0002811548504796883695047301057317751808570",
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       "im": "5.4578844534748845829914581945222177561836700185562e-84"
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      "I_r6": {
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     },
     "max_off_support_weight": "4.53e-78",
     "residual_vT_A": "1.67e-77"
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   ],
   "weight_digits": "extracted at 120-digit working precision from response-matrix entries with radii below 1e-107; at least 70 digits",
   "weights_at_eps_0": "(-7/54, 8/27, 1) (rational values reached in the limit eps -> 0; Table 5 of the paper)"
  },
  {
   "support": [
    "I_r1",
    "I_r5",
    "I_r7"
   ],
   "rows": [
    1,
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    7
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   "normalized_row": 7,
   "per_eps": [
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     "max_off_support_weight": "8.68e-78",
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    {
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   ],
   "weight_digits": "extracted at 120-digit working precision from response-matrix entries with radii below 1e-107; at least 70 digits",
   "weights_at_eps_0": "(-2/27, 2/27, 1) (rational values reached in the limit eps -> 0; Table 5 of the paper)"
  },
  {
   "support": [
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    "I_r18",
    "I_r19"
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   ],
   "weight_digits": "extracted at 120-digit working precision from response-matrix entries with radii below 1e-107; at least 70 digits"
  }
 ],
 "source_sha256": {
  "S2_values": {
   "description": "assembled S_2 boundary values I_r0..I_r7 with exact forms and routing structure",
   "sha256": "a79c2960dc2f5b72a016b851c139f015ce8527f8b84a4994ae1ee622b2f0e0f8"
  },
  "S2_nullvectors": {
   "description": "left null vectors of the S_2 response matrix at twelve values of epsilon (120-digit extraction)",
   "sha256": "4876a592e46a1798c008cfd925dce52cec36082b6d8647f94f37db29178a999a"
  },
  "S2_response_matrix": {
   "description": "response matrix A(epsilon) of S_2 (source of the epsilon grid, common to S_1, S_2, S_3)",
   "sha256": "c6255a2eca5bf90cd765e1657a128287c48aa2bf412d3446ceec23e99b8cccb5"
  }
 }
}