{
  "proof_locator": "IK.9, equations IK.30-IK.33",
  "normal_orientation": "w = s - t",
  "operator_coefficient": "a_j = 1/(j!)^2",
  "kernel": "K(w) = exp(1/w)/(2*pi*i*w)",
  "positive_real_slice": true,
  "kernel_plot_quantity": "log10(abs(2*pi*K(1/k))) = (k+log(k))/log(10)",
  "finite_partial_degrees": [
    8,
    32
  ],
  "coefficients": [
    {
      "j": 0,
      "a_j": {
        "numerator": 1,
        "denominator": 1
      },
      "j_factorial_a_j": {
        "numerator": 1,
        "denominator": 1
      }
    },
    {
      "j": 1,
      "a_j": {
        "numerator": 1,
        "denominator": 1
      },
      "j_factorial_a_j": {
        "numerator": 1,
        "denominator": 1
      }
    },
    {
      "j": 2,
      "a_j": {
        "numerator": 1,
        "denominator": 4
      },
      "j_factorial_a_j": {
        "numerator": 1,
        "denominator": 2
      }
    },
    {
      "j": 3,
      "a_j": {
        "numerator": 1,
        "denominator": 36
      },
      "j_factorial_a_j": {
        "numerator": 1,
        "denominator": 6
      }
    },
    {
      "j": 4,
      "a_j": {
        "numerator": 1,
        "denominator": 576
      },
      "j_factorial_a_j": {
        "numerator": 1,
        "denominator": 24
      }
    },
    {
      "j": 5,
      "a_j": {
        "numerator": 1,
        "denominator": 14400
      },
      "j_factorial_a_j": {
        "numerator": 1,
        "denominator": 120
      }
    },
    {
      "j": 6,
      "a_j": {
        "numerator": 1,
        "denominator": 518400
      },
      "j_factorial_a_j": {
        "numerator": 1,
        "denominator": 720
      }
    },
    {
      "j": 7,
      "a_j": {
        "numerator": 1,
        "denominator": 25401600
      },
      "j_factorial_a_j": {
        "numerator": 1,
        "denominator": 5040
      }
    },
    {
      "j": 8,
      "a_j": {
        "numerator": 1,
        "denominator": 1625702400
      },
      "j_factorial_a_j": {
        "numerator": 1,
        "denominator": 40320
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    },
    {
      "j": 9,
      "a_j": {
        "numerator": 1,
        "denominator": 131681894400
      },
      "j_factorial_a_j": {
        "numerator": 1,
        "denominator": 362880
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    },
    {
      "j": 10,
      "a_j": {
        "numerator": 1,
        "denominator": 13168189440000
      },
      "j_factorial_a_j": {
        "numerator": 1,
        "denominator": 3628800
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    },
    {
      "j": 11,
      "a_j": {
        "numerator": 1,
        "denominator": 1593350922240000
      },
      "j_factorial_a_j": {
        "numerator": 1,
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    },
    {
      "j": 12,
      "a_j": {
        "numerator": 1,
        "denominator": 229442532802560000
      },
      "j_factorial_a_j": {
        "numerator": 1,
        "denominator": 479001600
      }
    }
  ],
  "kernel_samples": [
    {
      "k": 2,
      "w": {
        "numerator": 1,
        "denominator": 2
      },
      "exact_log10_abs_2pi_K": "(k+log(k))/log(10)",
      "sample_log10_abs_2pi_K": 1.1696189594704849
    },
    {
      "k": 3,
      "w": {
        "numerator": 1,
        "denominator": 3
      },
      "exact_log10_abs_2pi_K": "(k+log(k))/log(10)",
      "sample_log10_abs_2pi_K": 1.780004700429418
    },
    {
      "k": 4,
      "w": {
        "numerator": 1,
        "denominator": 4
      },
      "exact_log10_abs_2pi_K": "(k+log(k))/log(10)",
      "sample_log10_abs_2pi_K": 2.3392379189409698
    },
    {
      "k": 5,
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        "numerator": 1,
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      },
      "exact_log10_abs_2pi_K": "(k+log(k))/log(10)",
      "sample_log10_abs_2pi_K": 2.870442413852278
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    {
      "k": 6,
      "w": {
        "numerator": 1,
        "denominator": 6
      },
      "exact_log10_abs_2pi_K": "(k+log(k))/log(10)",
      "sample_log10_abs_2pi_K": 3.383918141803154
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    {
      "k": 7,
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        "denominator": 7
      },
      "exact_log10_abs_2pi_K": "(k+log(k))/log(10)",
      "sample_log10_abs_2pi_K": 3.8851594133370195
    },
    {
      "k": 8,
      "w": {
        "numerator": 1,
        "denominator": 8
      },
      "exact_log10_abs_2pi_K": "(k+log(k))/log(10)",
      "sample_log10_abs_2pi_K": 4.377445842217958
    },
    {
      "k": 9,
      "w": {
        "numerator": 1,
        "denominator": 9
      },
      "exact_log10_abs_2pi_K": "(k+log(k))/log(10)",
      "sample_log10_abs_2pi_K": 4.862892846568591
    },
    {
      "k": 10,
      "w": {
        "numerator": 1,
        "denominator": 10
      },
      "exact_log10_abs_2pi_K": "(k+log(k))/log(10)",
      "sample_log10_abs_2pi_K": 5.342944819032518
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    {
      "k": 11,
      "w": {
        "numerator": 1,
        "denominator": 11
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      "exact_log10_abs_2pi_K": "(k+log(k))/log(10)",
      "sample_log10_abs_2pi_K": 5.818631986093995
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    {
      "k": 12,
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      "exact_log10_abs_2pi_K": "(k+log(k))/log(10)",
      "sample_log10_abs_2pi_K": 6.290715028886646
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      "k": 13,
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      "exact_log10_abs_2pi_K": "(k+log(k))/log(10)",
      "sample_log10_abs_2pi_K": 6.759771617049109
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      "exact_log10_abs_2pi_K": "(k+log(k))/log(10)",
      "sample_log10_abs_2pi_K": 7.226250782323763
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      "k": 15,
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      },
      "exact_log10_abs_2pi_K": "(k+log(k))/log(10)",
      "sample_log10_abs_2pi_K": 7.6905084876044585
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      "k": 16,
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      "exact_log10_abs_2pi_K": "(k+log(k))/log(10)",
      "sample_log10_abs_2pi_K": 8.152831693107954
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      "k": 17,
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        "numerator": 1,
        "denominator": 17
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      "exact_log10_abs_2pi_K": "(k+log(k))/log(10)",
      "sample_log10_abs_2pi_K": 8.613455113733554
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      "k": 18,
      "w": {
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        "denominator": 18
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      "exact_log10_abs_2pi_K": "(k+log(k))/log(10)",
      "sample_log10_abs_2pi_K": 9.072573179361838
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    {
      "k": 19,
      "w": {
        "numerator": 1,
        "denominator": 19
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      "exact_log10_abs_2pi_K": "(k+log(k))/log(10)",
      "sample_log10_abs_2pi_K": 9.530348757114613
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    {
      "k": 20,
      "w": {
        "numerator": 1,
        "denominator": 20
      },
      "exact_log10_abs_2pi_K": "(k+log(k))/log(10)",
      "sample_log10_abs_2pi_K": 9.986919633729016
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    {
      "k": 21,
      "w": {
        "numerator": 1,
        "denominator": 21
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      "exact_log10_abs_2pi_K": "(k+log(k))/log(10)",
      "sample_log10_abs_2pi_K": 10.442403414702207
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    {
      "k": 22,
      "w": {
        "numerator": 1,
        "denominator": 22
      },
      "exact_log10_abs_2pi_K": "(k+log(k))/log(10)",
      "sample_log10_abs_2pi_K": 10.896901282693745
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    {
      "k": 23,
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        "numerator": 1,
        "denominator": 23
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      "exact_log10_abs_2pi_K": "(k+log(k))/log(10)",
      "sample_log10_abs_2pi_K": 11.350500919792383
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      "k": 24,
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        "numerator": 1,
        "denominator": 24
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      "exact_log10_abs_2pi_K": "(k+log(k))/log(10)",
      "sample_log10_abs_2pi_K": 11.803278807389649
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      "k": 25,
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      "exact_log10_abs_2pi_K": "(k+log(k))/log(10)",
      "sample_log10_abs_2pi_K": 12.255302056253331
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      "k": 26,
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        "numerator": 1,
        "denominator": 26
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      "exact_log10_abs_2pi_K": "(k+log(k))/log(10)",
      "sample_log10_abs_2pi_K": 12.706629877455365
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    {
      "k": 27,
      "w": {
        "numerator": 1,
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      "exact_log10_abs_2pi_K": "(k+log(k))/log(10)",
      "sample_log10_abs_2pi_K": 13.157314775546785
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    {
      "k": 28,
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        "numerator": 1,
        "denominator": 28
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      "exact_log10_abs_2pi_K": "(k+log(k))/log(10)",
      "sample_log10_abs_2pi_K": 13.607403524633268
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      "k": 29,
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      "exact_log10_abs_2pi_K": "(k+log(k))/log(10)",
      "sample_log10_abs_2pi_K": 14.056937973093259
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    {
      "k": 30,
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      "exact_log10_abs_2pi_K": "(k+log(k))/log(10)",
      "sample_log10_abs_2pi_K": 14.505955711817217
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      "k": 31,
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      "exact_log10_abs_2pi_K": "(k+log(k))/log(10)",
      "sample_log10_abs_2pi_K": 14.954490632835078
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    {
      "k": 32,
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      "exact_log10_abs_2pi_K": "(k+log(k))/log(10)",
      "sample_log10_abs_2pi_K": 15.402573399223964
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    {
      "k": 33,
      "w": {
        "numerator": 1,
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      "exact_log10_abs_2pi_K": "(k+log(k))/log(10)",
      "sample_log10_abs_2pi_K": 15.850231842685195
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    {
      "k": 34,
      "w": {
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      },
      "exact_log10_abs_2pi_K": "(k+log(k))/log(10)",
      "sample_log10_abs_2pi_K": 16.297491301752814
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    {
      "k": 35,
      "w": {
        "numerator": 1,
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      "exact_log10_abs_2pi_K": "(k+log(k))/log(10)",
      "sample_log10_abs_2pi_K": 16.744374910964087
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    {
      "k": 36,
      "w": {
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        "denominator": 36
      },
      "exact_log10_abs_2pi_K": "(k+log(k))/log(10)",
      "sample_log10_abs_2pi_K": 17.19090384928435
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    {
      "k": 37,
      "w": {
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      "exact_log10_abs_2pi_K": "(k+log(k))/log(10)",
      "sample_log10_abs_2pi_K": 17.637097554487312
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    {
      "k": 38,
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      "exact_log10_abs_2pi_K": "(k+log(k))/log(10)",
      "sample_log10_abs_2pi_K": 18.082973908940378
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    {
      "k": 39,
      "w": {
        "numerator": 1,
        "denominator": 39
      },
      "exact_log10_abs_2pi_K": "(k+log(k))/log(10)",
      "sample_log10_abs_2pi_K": 18.52854940125332
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    {
      "k": 40,
      "w": {
        "numerator": 1,
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      },
      "exact_log10_abs_2pi_K": "(k+log(k))/log(10)",
      "sample_log10_abs_2pi_K": 18.973839267458033
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  ],
  "monomial_evaluations": [
    {
      "m": 0,
      "full": {
        "numerator": 1,
        "denominator": 1
      },
      "partial_order_3": {
        "numerator": 1,
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      }
    },
    {
      "m": 1,
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        "numerator": 1,
        "denominator": 1
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      "partial_order_3": {
        "numerator": 1,
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    },
    {
      "m": 2,
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      },
      "partial_order_3": {
        "numerator": 1,
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      }
    },
    {
      "m": 3,
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      },
      "partial_order_3": {
        "numerator": 1,
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      }
    },
    {
      "m": 4,
      "full": {
        "numerator": 1,
        "denominator": 24
      },
      "partial_order_3": {
        "numerator": 0,
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      }
    },
    {
      "m": 5,
      "full": {
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      "partial_order_3": {
        "numerator": 0,
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    },
    {
      "m": 6,
      "full": {
        "numerator": 1,
        "denominator": 720
      },
      "partial_order_3": {
        "numerator": 0,
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      }
    },
    {
      "m": 7,
      "full": {
        "numerator": 1,
        "denominator": 5040
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      "partial_order_3": {
        "numerator": 0,
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    },
    {
      "m": 8,
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      "partial_order_3": {
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  ],
  "source_normalization": "Free HolIII III.2, printed p.883; full proof local.",
  "limitations": "Real numerical kernel samples; no distribution limit is claimed."
}