| { |
| "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" |
| } |
|
|
