{
  "proof_locators": [
    "equations (FEP.6)\u2013(FEP.15)",
    "equations (FEP.20)\u2013(FEP.21)",
    "proof sections FEA.1\u2013FEA.4, FEB.1\u2013FEB.5 and FEP.1\u2013FEP.10"
  ],
  "exact_parameters": {
    "base_radius_r_x": "1/2",
    "epsilon": "1/4",
    "x": "1/2",
    "tau": "1"
  },
  "candidate": "(1-exp(-x*tau))/x; holomorphic at x=0, excluded by the every-epsilon bound on every common base neighborhood",
  "finite_truncation": "sum_{j=1}^L (-1)^(j+1)*x^(j-1)*tau^j/j!",
  "residual": "(-1)^(L+1)*x^L*tau^L/L!",
  "all_L_identity": "b_L circle x = 1+r_L; P r_L=-1",
  "samples": [
    {
      "L": 1,
      "weighted_residual_norm": 2,
      "entire_residual_exact": "1/2",
      "signed_quotient_residual": -1
    },
    {
      "L": 2,
      "weighted_residual_norm": 4,
      "entire_residual_exact": "1/8",
      "signed_quotient_residual": -1
    },
    {
      "L": 3,
      "weighted_residual_norm": 8,
      "entire_residual_exact": "1/48",
      "signed_quotient_residual": -1
    },
    {
      "L": 4,
      "weighted_residual_norm": 16,
      "entire_residual_exact": "1/384",
      "signed_quotient_residual": -1
    },
    {
      "L": 5,
      "weighted_residual_norm": 32,
      "entire_residual_exact": "1/3840",
      "signed_quotient_residual": -1
    },
    {
      "L": 6,
      "weighted_residual_norm": 64,
      "entire_residual_exact": "1/46080",
      "signed_quotient_residual": -1
    },
    {
      "L": 7,
      "weighted_residual_norm": 128,
      "entire_residual_exact": "1/645120",
      "signed_quotient_residual": -1
    },
    {
      "L": 8,
      "weighted_residual_norm": 256,
      "entire_residual_exact": "1/10321920",
      "signed_quotient_residual": -1
    },
    {
      "L": 9,
      "weighted_residual_norm": 512,
      "entire_residual_exact": "1/185794560",
      "signed_quotient_residual": -1
    },
    {
      "L": 10,
      "weighted_residual_norm": 1024,
      "entire_residual_exact": "1/3715891200",
      "signed_quotient_residual": -1
    },
    {
      "L": 11,
      "weighted_residual_norm": 2048,
      "entire_residual_exact": "1/81749606400",
      "signed_quotient_residual": -1
    },
    {
      "L": 12,
      "weighted_residual_norm": 4096,
      "entire_residual_exact": "1/1961990553600",
      "signed_quotient_residual": -1
    },
    {
      "L": 13,
      "weighted_residual_norm": 8192,
      "entire_residual_exact": "1/51011754393600",
      "signed_quotient_residual": -1
    },
    {
      "L": 14,
      "weighted_residual_norm": 16384,
      "entire_residual_exact": "1/1428329123020800",
      "signed_quotient_residual": -1
    },
    {
      "L": 15,
      "weighted_residual_norm": 32768,
      "entire_residual_exact": "1/42849873690624000",
      "signed_quotient_residual": -1
    },
    {
      "L": 16,
      "weighted_residual_norm": 65536,
      "entire_residual_exact": "1/1371195958099968000",
      "signed_quotient_residual": -1
    },
    {
      "L": 17,
      "weighted_residual_norm": 131072,
      "entire_residual_exact": "1/46620662575398912000",
      "signed_quotient_residual": -1
    },
    {
      "L": 18,
      "weighted_residual_norm": 262144,
      "entire_residual_exact": "1/1678343852714360832000",
      "signed_quotient_residual": -1
    },
    {
      "L": 19,
      "weighted_residual_norm": 524288,
      "entire_residual_exact": "1/63777066403145711616000",
      "signed_quotient_residual": -1
    },
    {
      "L": 20,
      "weighted_residual_norm": 1048576,
      "entire_residual_exact": "1/2551082656125828464640000",
      "signed_quotient_residual": -1
    }
  ],
  "left_panel": "Exact operator identities; no geometric scale represented",
  "proper_ideal_proof_uses_flatness": false,
  "unit_injectivity_uses_right_flatness": true,
  "right_flatness_proof": "Section 5.47, proof sections MF.1\u2013MF.7"
}