{
  "kind": "piecewise-objective-curves",
  "title": "Positive and negative advantages flatten on opposite sides",
  "description": "Plot min(rA,clip(r,0.8,1.2)A) for A=2 and A=−2 over r∈[0.4,1.6]. Mark r=0.8,1,1.2. Caption: objective geometry, not a bound on realized policy ratios.",
  "data": {
    "x_label": "new/old probability ratio",
    "x": [
      0.6,
      0.8,
      1,
      1.2,
      1.4
    ],
    "series": [
      {
        "name": "advantage +2",
        "values": [
          1.2,
          1.6,
          2,
          2.4,
          2.4
        ]
      },
      {
        "name": "advantage −2",
        "values": [
          -1.6,
          -1.6,
          -2,
          -2.4,
          -2.8
        ]
      }
    ],
    "clip_interval": [
      0.8,
      1.2
    ],
    "counterexample": {
      "old": [
        0.5,
        0.5
      ],
      "new": [
        0.9,
        0.1
      ],
      "ratio": 1.8,
      "kl_old_new": 0.5108256237659907
    }
  },
  "evidence": "Original educational calculation; no measured paper or hardware benchmark."
}