{
  "article_slug": "neural-collapse-geometry",
  "figures": [
    {
      "id": "simplex-features",
      "kind": "constructed_scatter",
      "caption": "Author-constructed three-class unit simplex, four symmetric offsets per class, epsilon=.15. Feature positions are mathematical inputs, not learned embeddings.",
      "series": [
        {
          "name": "Class 1",
          "x": [
            1.15,
            0.85,
            1.0,
            1.0
          ],
          "y": [
            0.0,
            0.0,
            0.15,
            -0.15
          ]
        },
        {
          "name": "Class 2",
          "x": [
            -0.35,
            -0.65,
            -0.5,
            -0.5
          ],
          "y": [
            0.8660254037844386,
            0.8660254037844386,
            1.0160254037844385,
            0.7160254037844386
          ]
        },
        {
          "name": "Class 3",
          "x": [
            -0.35,
            -0.65,
            -0.5,
            -0.5
          ],
          "y": [
            -0.8660254037844386,
            -0.8660254037844386,
            -0.7160254037844386,
            -1.0160254037844385
          ]
        }
      ]
    },
    {
      "id": "scatter-ratio",
      "kind": "analytical_curve",
      "caption": "Exact author-constructed ratio tr(W)/tr(B)=epsilon², with centered simplex means fixed. Epsilon is not training time; no empirical benchmark.",
      "x": {
        "name": "Within-class offset epsilon",
        "values": [
          0,
          0.05,
          0.1,
          0.15,
          0.25,
          0.5
        ]
      },
      "series": [
        {
          "name": "Declared scatter ratio",
          "values": [
            0,
            0.0025000000000000005,
            0.010000000000000002,
            0.0225,
            0.0625,
            0.25
          ]
        }
      ]
    }
  ],
  "cover_prompt": "Premium scientific editorial cover, landscape 3:2, no lettering or chart axes. Three luminous jade-and-ivory sculptural clusters hover at the vertices of an exact equilateral triangle over dark emerald stone; each cluster is formed by tiny translucent fragments converging toward a golden center, with a fine brass triangular frame revealing equal spacing. Warm ivory rim light, precise balanced composition, elegant research-journal photography. Conceptual neural-collapse geometry, not a plot of measured embeddings.",
  "numeric_data_provenance": "Computed and checked by the corresponding standalone standard-library script; constructed, not empirical."
}