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 "outer_radius": "1999/2000",
 "target_margin": "1/10000000000",
 "status": "Exact frozen candidate. See certificate files for verification status; specification alone makes no certification claim.",
 "rational_interpretation": "Every numerical string, including decimal and scientific notation, denotes an exact rational number.",
 "defining_formula": "P=(z-a)*h_L/h_L(a)+Delta0+X+H; Delta0=(z-a)^2*(K*R0+sum_{j=0}^3 free_j*(z-mu)^j); X=(z-a)^2*sum_{ell,k} extra[ell,k]*(z/(6/5))^k*q^ell; H=tail_phase*(gamma*q^N+q^(N-tail_blocks)*sum_{ell,k} tail[ell,k]*(z/(6/5))^k*q^ell); p(x)=(P(normalization_base+x)-P(normalization_base))/Pprime(normalization_base).",
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 "hermite_coefficient": {
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  "im": "0"
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 "annular_margin": "1/5000000",
 "primitive": false,
 "strict_smale_lower_bound": "10000000001/10000000000",
 "strict_annular_lower_bound": "5000001/5000000",
 "high_power_coefficient": "5535125673272137/9007199254740992",
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 "free_center_definition": "(correction_centers[0]+correction_centers[1])/2, exactly",
 "certificate_method": "Pairwise disjoint critical disks, each with one zero; count equals derivative degree.",
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 "tail_base_exponent": 1792,
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 ],
 "arithmetic_kernel": "Two independent certificates: (i) arb ball arithmetic at 1280 bits, Krawczyk boxes, Taylor enclosure with Cauchy bound (verify_ball_N1840.py); (ii) the Adamczewski-Colbrook verify.cpp (MPFR-directed interval bounds, Rouche disks), recompiled with N_PARAM=1840, TAIL_BLOCKS_PARAM=48.",
 "unused_legacy_metadata": "The inner/outer radii and annular margin are inherited initialization metadata, not a root-coverage or annular-inequality assertion in this certificate.",
 "tail_phase": {
  "re": "-63927535149896946731567739296577/63967149231154078900364511271745",
  "im": "2250873257077401008009918808064/63967149231154078900364511271745"
 },
 "construction_change": "Degree 29440 (N = 1840, L = 1838, 48 tail blocks, tail base exponent 1792). Same model, anchor, Blaschke parameter, centres and correction structure as the degree-29984 Adamczewski-Colbrook example; coefficient tables, amplitude, exact rational phase and normalization base re-chosen.",
 "normalization_base": {
  "re": "585354180009313/562949953421312",
  "im": "1358016153176321/4503599627370496"
 },
 "normalization_definition": "p(x)=(P(normalization_base+x)-P(normalization_base))/Pprime(normalization_base), exactly",
 "provenance": "Optimised by D. Niedbala Giraudin with Claude (Anthropic), starting from the Adamczewski-Colbrook certificate (arXiv:2610.36158)."
}