| { |
| "author_commit": "85a5da444a146ea28945173e22e3d130163dae42", |
| "claims": [ |
| { |
| "assessment": "verified", |
| "claim": 1, |
| "destructive_control": { |
| "errors": [ |
| 0.5, |
| 0.5, |
| 0.5, |
| 0.5 |
| ], |
| "mutation": "negate the conjugate intercept sign" |
| }, |
| "error_vs_spacing_exponent": 2.0, |
| "exact_theorem_rate_for_instance": "||f-f_grid||_infinity=h\u00b2/8", |
| "finite_grid_rows": [ |
| { |
| "atoms": 5, |
| "cells_exhausted": 4, |
| "exact_uniform_error": "1/32", |
| "spacing": 0.5, |
| "uniform_error": 0.03125 |
| }, |
| { |
| "atoms": 9, |
| "cells_exhausted": 8, |
| "exact_uniform_error": "1/128", |
| "spacing": 0.25, |
| "uniform_error": 0.0078125 |
| }, |
| { |
| "atoms": 17, |
| "cells_exhausted": 16, |
| "exact_uniform_error": "1/512", |
| "spacing": 0.125, |
| "uniform_error": 0.001953125 |
| }, |
| { |
| "atoms": 33, |
| "cells_exhausted": 32, |
| "exact_uniform_error": "1/2048", |
| "spacing": 0.0625, |
| "uniform_error": 0.00048828125 |
| }, |
| { |
| "atoms": 65, |
| "cells_exhausted": 64, |
| "exact_uniform_error": "1/8192", |
| "spacing": 0.03125, |
| "uniform_error": 0.0001220703125 |
| }, |
| { |
| "atoms": 129, |
| "cells_exhausted": 128, |
| "exact_uniform_error": "1/32768", |
| "spacing": 0.015625, |
| "uniform_error": 3.0517578125e-05 |
| } |
| ], |
| "literal_claim": "Theorem 1 (Section V.A) establishes that finitely \u1ef8-convex functions form a dense subset of all \u1ef8-convex functions, giving a universal approximation property for generalized convex functions.", |
| "native_instance": "Phi(x,y)=xy on X=Y=[-1,1], f(x)=x\u00b2/2, f^X(y)=y\u00b2/2" |
| }, |
| { |
| "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" |
| }, |
| { |
| "assessment": "verified", |
| "claim": 3, |
| "convex_combinations_checked": 570, |
| "destructive_control": { |
| "effect": "the middle atom is never active, so the parameterization is not lean", |
| "minimum_domination_margin_on_129_points": 0.25, |
| "mutation": "raise only the y=0 intercept of r(y)=y\u00b2 by 1/2" |
| }, |
| "exact_witness_inequalities_checked": 14250, |
| "finite_Y": [ |
| "-1", |
| "-1/2", |
| "0", |
| "1/2", |
| "1" |
| ], |
| "kernel": "Phi(x,y)=xy", |
| "lean_parameterizations": 20, |
| "literal_claim": "Theorem 4 (Section V.C) proves that the lean subset of finitely \u1ef8-convex functions forms a convex parameter space, which is what allows bilevel objectives to be rewritten as single-level problems solvable with standard first-order optimization (Section I).", |
| "unordered_pairs": 190, |
| "witness_identity": "at x=2ay_i+b, line_i-line_j=a(y_i-y_j)^2 >= 0" |
| }, |
| { |
| "assessment": "falsified", |
| "boundary_control": "Replacing 'exactly' by 'within 0.001 per item at printed precision' makes all four available comparisons pass.", |
| "claim": 4, |
| "exact_match_failures": [ |
| 5, |
| 10 |
| ], |
| "literal_claim": "Table I reports multi-item auction experiments for n in {1,2,5,10,20} goods in which, for n up to 10, the learned mechanism's revenue matches the Straight-Jacket auction benchmark exactly.", |
| "literal_registered_word": "exactly", |
| "result": "The source table prints 0.314 vs 0.315 at n=5 and 0.346 vs 0.347 at n=10, so the conjunction is false as literally registered.", |
| "source_file_sha256": "021cdf77be6af063a4b420e8f4e068c8879ba1842c8ac4ed39e10efecc50f8ed", |
| "source_table_rows": [ |
| { |
| "absolute_per_item_gap": 0.0, |
| "exact_match": true, |
| "learned_profit_per_item": "0.250", |
| "n": 1, |
| "straight_jacket_per_item": "0.250" |
| }, |
| { |
| "absolute_per_item_gap": 0.0, |
| "exact_match": true, |
| "learned_profit_per_item": "0.274", |
| "n": 2, |
| "straight_jacket_per_item": "0.274" |
| }, |
| { |
| "absolute_per_item_gap": 0.001, |
| "exact_match": false, |
| "learned_profit_per_item": "0.314", |
| "n": 5, |
| "straight_jacket_per_item": "0.315" |
| }, |
| { |
| "absolute_per_item_gap": 0.001, |
| "exact_match": false, |
| "learned_profit_per_item": "0.346", |
| "n": 10, |
| "straight_jacket_per_item": "0.347" |
| }, |
| { |
| "learned_profit_per_item": "0.377", |
| "n": 20, |
| "straight_jacket_per_item": "---" |
| } |
| ] |
| }, |
| { |
| "assessment": "verified", |
| "author_mechanism_execution": { |
| "configured_posted_price": 0.5, |
| "grid_points": 4001, |
| "max_allocation_below_0.499": 0.0, |
| "min_allocation_above_0.501": 1.0 |
| }, |
| "author_repository_commit": "85a5da444a146ea28945173e22e3d130163dae42", |
| "benchmark_match": "Table I prints 0.274 learned and 0.274 SJa per item for n=2", |
| "claim": 5, |
| "destructive_control": { |
| "effect": "moving the threshold to 0.35 creates a 0.0225 revenue gap", |
| "posted_price": 0.35, |
| "revenue_gap": 0.0225 |
| }, |
| "literal_claim": "Figures 2-3 show the learned parametrization recovers the known optimal posted price of 0.5 in the single-item auction case and finds a mixed-bundling pricing scheme matching theoretical benchmarks in the two-item case.", |
| "mixed_menu_test": "both heatmaps contain large exact zero/full regions plus nonzero intermediate-allocation regions", |
| "single_item_analytic_oracle": { |
| "exact_optimum_price": 0.5, |
| "figure_learned_price": 0.495, |
| "revenue_gap_to_optimum": 2.5e-05, |
| "revenue_identity": "R(p)=p(1-p)" |
| }, |
| "two_item_source_heatmap_a1": { |
| "full_allocation_pixels": 253352, |
| "intermediate_allocation_pixels": 3171, |
| "sha256": "30e8063e8981a67f2456327c4578bc5417e528ecb3aa8aa539cdce35953df8a2", |
| "size": [ |
| 874, |
| 748 |
| ], |
| "zero_allocation_pixels": 126997 |
| }, |
| "two_item_source_heatmap_a2": { |
| "full_allocation_pixels": 230527, |
| "intermediate_allocation_pixels": 26051, |
| "sha256": "61678a82a62e4d95a4e097f4cb197e2e557b8a299e27ea1ea1248b2d3c51c7da", |
| "size": [ |
| 874, |
| 748 |
| ], |
| "zero_allocation_pixels": 127015 |
| } |
| }, |
| { |
| "assessment": "verified", |
| "claim": 6, |
| "couplings_exhausted": 40320, |
| "destructive_control": { |
| "mutation": "reverse the coupling", |
| "optimality_gap": 0.21428571428571427, |
| "surplus": 0.14285714285714285 |
| }, |
| "dual_constraints_checked": 64, |
| "dual_potentials": "phi(x)=x\u00b2/2 and psi(y)=y\u00b2/2; phi is the Y-transform of psi", |
| "dual_value": 0.35714285714285715, |
| "exact_duality_gap": "0/1", |
| "grid": [ |
| 0.0, |
| 0.14285714285714285, |
| 0.2857142857142857, |
| 0.42857142857142855, |
| 0.5714285714285714, |
| 0.7142857142857143, |
| 0.8571428571428571, |
| 1.0 |
| ], |
| "literal_claim": "Kantorovich dual solutions for optimal transport are characterized as \u1ef8-convex functions in Section IV, so gradients of the learned parametrization directly yield optimal transport maps via a diffeomorphism condition.", |
| "map_mismatches": 0, |
| "minimum_dual_slack": 0.0, |
| "optimal_permutation": [ |
| 0, |
| 1, |
| 2, |
| 3, |
| 4, |
| 5, |
| 6, |
| 7 |
| ], |
| "primal_value": 0.35714285714285715, |
| "surplus": "Phi(x,y)=xy", |
| "twist_gradient": "nabla_x Phi(x,y)=y is a diffeomorphism; inverse applied to nabla phi(x)=x yields gamma(x)=x" |
| } |
| ], |
| "paper_id": "63o9EmYHXt", |
| "source_sha256": "5a70d3abdde7cbe40605525514c111084b81015fc57d828bea5c2fb47e337255", |
| "summary": { |
| "falsified": 1, |
| "inconclusive": 0, |
| "registered_claims": 6, |
| "toy": 0, |
| "verified": 5 |
| } |
| } |
|
|
