{
  "schema": "AN02-L138-original-figure-geometry132/v1",
  "credit": "GPT-6.1 Sol (OpenAI), Ultra, October 2026; original diagrams CC0-1.0.",
  "frequency_budget": {
    "models": {
      "linear": "M_j=j",
      "quadratic": "M_j=j^2"
    },
    "frequency_interval": [
      1,
      32
    ],
    "frequency_samples": 500,
    "T_formula": "m*log(r)-alpha*log(m!), m=floor(r^(1/alpha))",
    "envelope": "exp(-T(r)); normalized C=1; algebraic bounds only",
    "finite_budget_identity": "integral_1^infinity T_N(r)/r^2 dr=sum_{j<=N}1/M_j",
    "plotted_budget_indices": [
      1,
      80
    ],
    "selected_exact_finite_budgets": [
      {
        "N": 1,
        "sum_1_over_j_exact": "1",
        "sum_1_over_j_squared_exact": "1"
      },
      {
        "N": 2,
        "sum_1_over_j_exact": "3/2",
        "sum_1_over_j_squared_exact": "5/4"
      },
      {
        "N": 4,
        "sum_1_over_j_exact": "25/12",
        "sum_1_over_j_squared_exact": "205/144"
      },
      {
        "N": 8,
        "sum_1_over_j_exact": "761/280",
        "sum_1_over_j_squared_exact": "1077749/705600"
      },
      {
        "N": 16,
        "sum_1_over_j_exact": "2436559/720720",
        "sum_1_over_j_squared_exact": "822968714749/519437318400"
      },
      {
        "N": 32,
        "sum_1_over_j_exact": "586061125622639/144403552893600",
        "sum_1_over_j_squared_exact": "33659238975573797429256624061/20852386088294732932920960000"
      },
      {
        "N": 64,
        "sum_1_over_j_exact": "623171679694215690971693339/131362987122535807501262400",
        "sum_1_over_j_squared_exact": "41409901935572392721815151397221689468404267553321859/25413727004476294175336659010085214676345183380992000"
      },
      {
        "N": 80,
        "sum_1_over_j_exact": "4880292608058024066886120358155997/982844219842241906412811281988800",
        "sum_1_over_j_squared_exact": "132102260735512757102403108631555952871934530025556285576624991549233/80919632781521945886764511097550761980389250669911461872887800320000"
      }
    ],
    "quadratic_sum_proved_bound": 2,
    "proof_locators": [
      "CN6.2",
      "CN6.3",
      "CN7.3",
      "CN9.1"
    ]
  },
  "trace_and_bump": {
    "local_zero_model": "F(z)=z^2; local factor only, not the global half-plane class",
    "multiplicity": 2,
    "trace_interval": [
      -1,
      1
    ],
    "plotted_heights_exact": [
      "1/2",
      "1/4",
      "1/16"
    ],
    "interval_error": "2*log(1+y^2)+4*y*atan(1/y)",
    "whole_line_bound": "2*pi*y",
    "bump": "exp(-2/(1-x^2)) for |x|<1, else0",
    "support_exact": [
      -1,
      1
    ],
    "value_at_zero": "exp(-2)",
    "derivative_bound": "18^k*(k!)^2, k>=1",
    "thresholds": "M_j=18*j^2",
    "reciprocal_sum_proved_bound": "1/9",
    "proof_locators": [
      "CN4.2",
      "CN4.3",
      "CN9.3",
      "CN9.4",
      "CN9.5",
      "CN9.6"
    ]
  }
}