{
  "article_slug": "molecular-chirality-equivariance",
  "figures": [
    {
      "id": "signed-volume",
      "title": "Reflection preserves distances and reverses orientation",
      "kind": "analytical_curve",
      "x_label": "Third-vector coordinate z",
      "y_label": "Volume (constructed length units cubed)",
      "x": {
        "name": "Third-vector coordinate z",
        "values": [
          -1,
          -0.75,
          -0.5,
          -0.25,
          0,
          0.25,
          0.5,
          0.75,
          1
        ]
      },
      "series": [
        {
          "name": "Signed volume",
          "values": [
            -1,
            -0.75,
            -0.5,
            -0.25,
            0,
            0.25,
            0.5,
            0.75,
            1
          ]
        },
        {
          "name": "Unsigned volume",
          "values": [
            1,
            0.75,
            0.5,
            0.25,
            0,
            0.25,
            0.5,
            0.75,
            1
          ]
        }
      ],
      "caption": "Labeled vectors a=(1,0,0), b=(0,1,0), c=(0,0,z). Distances agree under z→-z. This geometric construction is not a chemical measurement; z=0 is degenerate."
    },
    {
      "id": "mirror-pair-accuracy",
      "title": "Class imbalance can hide orientation loss",
      "kind": "analytical_curve",
      "x_label": "Positive-orientation prevalence p",
      "y_label": "Optimal accuracy",
      "x": {
        "name": "Positive-orientation prevalence p",
        "values": [
          0.0,
          0.1,
          0.2,
          0.3,
          0.4,
          0.5,
          0.6,
          0.7,
          0.8,
          0.9,
          1.0
        ]
      },
      "series": [
        {
          "name": "Reflection-invariant representation",
          "values": [
            1.0,
            0.9,
            0.8,
            0.7,
            0.6,
            0.5,
            0.6,
            0.7,
            0.8,
            0.9,
            1.0
          ]
        },
        {
          "name": "Noiseless sign-aware representation",
          "values": [
            1,
            1,
            1,
            1,
            1,
            1,
            1,
            1,
            1,
            1,
            1
          ]
        }
      ],
      "caption": "Constructed mirror-pair classification. Invariant inputs identify both labels and can only exploit class prevalence; signed-volume inputs identify the noiseless nondegenerate orientation. No trained architecture."
    }
  ]
}