{
  "article_slug": "sharpness-parameter-symmetry",
  "figures": [
    {
      "id": "curvature-orbit",
      "kind": "analytical_curve",
      "caption": "Exact author-constructed nonzero Hessian eigenvalue c²+c⁻² at identical predictors f(x)=x. No measured generalization or deep-network benchmark.",
      "x": {
        "name": "Positive rescaling c",
        "values": [
          0.1,
          0.2,
          0.5,
          1.0,
          2.0,
          5.0,
          10.0
        ]
      },
      "series": [
        {
          "name": "Nonzero Hessian eigenvalue",
          "values": [
            100.00999999999999,
            25.039999999999996,
            4.25,
            2.0,
            4.25,
            25.04,
            100.01
          ]
        }
      ]
    },
    {
      "id": "perturbation-comparison",
      "kind": "analytical_grid",
      "caption": "Author-constructed exact nonlinear losses on 20,000 equally spaced angles, radius .1. Sampled maxima are lower bounds on continuous maxima. Transformed ellipses preserve every sampled loss; no trained-network measurement.",
      "x": {
        "name": "Positive rescaling c",
        "values": [
          0.1,
          0.2,
          0.5,
          1.0,
          2.0,
          5.0,
          10.0
        ]
      },
      "series": [
        {
          "name": "Ordinary circle sampled maximum",
          "values": [
            0.5001999534279226,
            0.12544946073884947,
            0.021775430288239094,
            0.010719606781186589,
            0.021775430288239094,
            0.12544946073884947,
            0.5001999534279226
          ]
        },
        {
          "name": "Corresponding ellipse sampled maximum",
          "values": [
            0.010719606781186589,
            0.010719606781186589,
            0.010719606781186589,
            0.010719606781186589,
            0.010719606781186589,
            0.010719606781186589,
            0.010719606781186589
          ]
        }
      ]
    }
  ],
  "cover_prompt": "Premium scientific editorial cover, landscape 3:2, no words, no axes. A single continuous jade ribbon passes through several brass coordinate frames that stretch tall and narrow then wide and shallow; its luminous golden centerline remains identical while translucent contour shells change shape. Warm ivory stone, deep emerald shadows, precise sculptural realism, restrained geometry, generous negative space. Conceptual parameter symmetry illustration, not experimental evidence.",
  "numeric_data_provenance": "Computed and checked by the corresponding standalone standard-library script; constructed, not empirical."
}