{
  "barrier": {
    "n": 2,
    "a": 1,
    "b": 3,
    "R": 3,
    "formula": "z^2-t^2+9",
    "proof_locators": [
      "S10",
      "S11",
      "S13",
      "S14"
    ]
  },
  "envelopes": {
    "periodic_example": {
      "k": 1,
      "alpha": "-1/4",
      "beta": "1/2",
      "M": "exp(-t)-t/4+1/2",
      "gamma": "-1/4"
    },
    "corner_example": {
      "M": "abs(t-2)",
      "gamma": 1
    },
    "secant_heights": [
      1,
      3
    ],
    "sample_domain": [
      "1/5",
      6
    ],
    "proof_locators": [
      "S15",
      "S18",
      "S20",
      "S21"
    ]
  },
  "plot_status": "Samples of explicit exact functions; no numerical claim substitutes for the proof",
  "source_text": "horizontal-envelopes-and-their-limiting-slope.md"
}