{
  "article_slug": "attention-rank-bottlenecks",
  "figures": [
    {
      "id": "contrast-rank",
      "title": "Rank-one scores, full-rank attention",
      "kind": "analytical_curve",
      "x_label": "Score scale a (dimensionless)",
      "y_label": "Determinant / contrast eigenvalue",
      "x": {
        "name": "Score scale a (dimensionless)",
        "values": [
          0,
          0.01,
          0.05,
          0.1,
          0.25,
          0.5,
          1,
          2,
          3,
          4
        ]
      },
      "series": [
        {
          "name": "tanh(a)",
          "values": [
            0.0,
            0.00999966667999946,
            0.04995837495787998,
            0.09966799462495582,
            0.24491866240370913,
            0.46211715726000974,
            0.7615941559557649,
            0.9640275800758169,
            0.9950547536867305,
            0.999329299739067
          ]
        }
      ],
      "caption": "Author-constructed two-token scalar head. Unmasked scores have rank one for a>0; normalized determinant equals tanh(a). No model training."
    },
    {
      "id": "saturation-tradeoff",
      "title": "Transmission and local score-scale sensitivity",
      "kind": "analytical_curve",
      "x_label": "Score scale a (dimensionless)",
      "y_label": "Dimensionless value",
      "x": {
        "name": "Score scale a (dimensionless)",
        "values": [
          0,
          0.01,
          0.05,
          0.1,
          0.25,
          0.5,
          1,
          2,
          3,
          4
        ]
      },
      "series": [
        {
          "name": "Contrast transmission",
          "values": [
            0.0,
            0.00999966667999946,
            0.04995837495787998,
            0.09966799462495582,
            0.24491866240370913,
            0.46211715726000974,
            0.7615941559557649,
            0.9640275800758169,
            0.9950547536867305,
            0.999329299739067
          ]
        },
        {
          "name": "Derivative with respect to a",
          "values": [
            1.0,
            0.9999000066662889,
            0.9975041607715679,
            0.9900662908474398,
            0.940014848806378,
            0.7864477329659274,
            0.41997434161402614,
            0.07065082485316443,
            0.009866037165440211,
            0.0013409506830258655
          ]
        }
      ],
      "caption": "Exact symmetric two-token control. Derivative concerns this scalar parameterization, not total neural training gradients."
    }
  ]
}