{
  "assessment": "verified",
  "claim": 2,
  "destructive_control": {
    "effect": "functions converge uniformly while derivative amplitudes diverge",
    "mutation": "drop shared semiconvexity: add sin(nx)/sqrt(n)",
    "result": [
      {
        "function_error": 0.25,
        "gradient_amplitude": 4.0,
        "n": 16
      },
      {
        "function_error": 0.125,
        "gradient_amplitude": 8.0,
        "n": 64
      },
      {
        "function_error": 0.0625,
        "gradient_amplitude": 16.0,
        "n": 256
      },
      {
        "function_error": 0.03125,
        "gradient_amplitude": 32.0,
        "n": 1024
      }
    ]
  },
  "gradient_error_vs_spacing_exponent": 1.0000000000000002,
  "gradient_grid_rows": [
    {
      "atoms": 5,
      "differentiable_cells_checked": 16,
      "exact_gradient_supremum_bound": 0.25,
      "sampled_max_gradient_error": 0.2495,
      "spacing": 0.5
    },
    {
      "atoms": 9,
      "differentiable_cells_checked": 32,
      "exact_gradient_supremum_bound": 0.125,
      "sampled_max_gradient_error": 0.12475,
      "spacing": 0.25
    },
    {
      "atoms": 17,
      "differentiable_cells_checked": 64,
      "exact_gradient_supremum_bound": 0.0625,
      "sampled_max_gradient_error": 0.062375,
      "spacing": 0.125
    },
    {
      "atoms": 33,
      "differentiable_cells_checked": 128,
      "exact_gradient_supremum_bound": 0.03125,
      "sampled_max_gradient_error": 0.0311875,
      "spacing": 0.0625
    },
    {
      "atoms": 65,
      "differentiable_cells_checked": 256,
      "exact_gradient_supremum_bound": 0.015625,
      "sampled_max_gradient_error": 0.01559375,
      "spacing": 0.03125
    },
    {
      "atoms": 129,
      "differentiable_cells_checked": 512,
      "exact_gradient_supremum_bound": 0.0078125,
      "sampled_max_gradient_error": 0.007796875,
      "spacing": 0.015625
    }
  ],
  "literal_claim": "Theorem 2 (Section V.A) shows that, under semiconvexity conditions on \u03a6, gradients of finitely \u1ef8-convex functions densely approximate gradients of all \u1ef8-convex functions.",
  "semiconvex_native_instance": "the same quadratic Phi and f; every finite transform shares semiconvexity constant K=0"
}