{
  "licence": "CC0-1.0; font terms separate",
  "formula": "d_c(t)=2*exp(-c)/(1+exp(-c))^2*(1-cos(c*t))",
  "index_interval": [
    1,
    2
  ],
  "normalized_time": "r=t/(2*pi)",
  "plot_grid": {
    "r_points": 801,
    "c_points": 401,
    "r_range": [
      -1,
      1
    ],
    "c_range": [
      1,
      2
    ]
  },
  "zero_curves": "r*c belongs to Z; interior plotted branches r=0 and r=+/-1/c",
  "samples": [
    {
      "c": 1,
      "r": -1,
      "d_c(2*pi*r)": 1.1794826810678901e-32
    },
    {
      "c": 1,
      "r": -0.5,
      "d_c(2*pi*r)": 0.7864477329659275
    },
    {
      "c": 1,
      "r": 0,
      "d_c(2*pi*r)": 0.0
    },
    {
      "c": 1,
      "r": 0.5,
      "d_c(2*pi*r)": 0.7864477329659275
    },
    {
      "c": 1,
      "r": 1,
      "d_c(2*pi*r)": 1.1794826810678901e-32
    },
    {
      "c": 1.25,
      "r": -1,
      "d_c(2*pi*r)": 0.34620957398499397
    },
    {
      "c": 1.25,
      "r": -0.5,
      "d_c(2*pi*r)": 0.591016711461489
    },
    {
      "c": 1.25,
      "r": 0,
      "d_c(2*pi*r)": 0.0
    },
    {
      "c": 1.25,
      "r": 0.5,
      "d_c(2*pi*r)": 0.591016711461489
    },
    {
      "c": 1.25,
      "r": 1,
      "d_c(2*pi*r)": 0.34620957398499397
    },
    {
      "c": 1.5,
      "r": -1,
      "d_c(2*pi*r)": 0.5965858082813313
    },
    {
      "c": 1.5,
      "r": -0.5,
      "d_c(2*pi*r)": 0.2982929041406657
    },
    {
      "c": 1.5,
      "r": 0,
      "d_c(2*pi*r)": 0.0
    },
    {
      "c": 1.5,
      "r": 0.5,
      "d_c(2*pi*r)": 0.2982929041406657
    },
    {
      "c": 1.5,
      "r": 1,
      "d_c(2*pi*r)": 0.5965858082813313
    },
    {
      "c": 1.75,
      "r": -1,
      "d_c(2*pi*r)": 0.25225845037331057
    },
    {
      "c": 1.75,
      "r": -0.5,
      "d_c(2*pi*r)": 0.0738847895027325
    },
    {
      "c": 1.75,
      "r": 0,
      "d_c(2*pi*r)": 0.0
    },
    {
      "c": 1.75,
      "r": 0.5,
      "d_c(2*pi*r)": 0.0738847895027325
    },
    {
      "c": 1.75,
      "r": 1,
      "d_c(2*pi*r)": 0.25225845037331057
    },
    {
      "c": 2,
      "r": -1,
      "d_c(2*pi*r)": 2.5194425092104336e-32
    },
    {
      "c": 2,
      "r": -0.5,
      "d_c(2*pi*r)": 6.298606273026084e-33
    },
    {
      "c": 2,
      "r": 0,
      "d_c(2*pi*r)": 0.0
    },
    {
      "c": 2,
      "r": 0.5,
      "d_c(2*pi*r)": 6.298606273026084e-33
    },
    {
      "c": 2,
      "r": 1,
      "d_c(2*pi*r)": 2.5194425092104336e-32
    }
  ],
  "exact_conclusions": {
    "T(M)": "{0}",
    "Per(phi)": "{0}",
    "explicit_countable_detector": "Q intersect [1,2]"
  },
  "general_detection": "For continuous nonnegative d_i, rational-threshold open covers select countably many finite witness sets; their union J preserves the full extended sum at every real time.",
  "proof_locators": [
    "MIV11.a",
    "MIV11.b",
    "MIV12.b",
    "MIV12.e",
    "MIV13.a",
    "MIV13.b",
    "MIV13.c",
    "MIV13.e",
    "MIV13.f"
  ],
  "limits": "The heat map is a finite sampling of a scalar formula, not a proof of divergence. The diagram does not model cardinality geometrically or assert countable generation.",
  "programme_input": "OA-APPROX/infinite-tensor-products.html#theorem-4-1",
  "human_background": "H. Araki and E. J. Woods, A Classification of Factors, https://ems.press/journals/prims/articles/2441"
}