{ "seed": 938, "math_check": { "coefficient_max_abs_error": 3.3306690738754696e-16, "coefficient_rows": [ { "gamma": 0.1, "lambda": 0.0, "pred": 1.0, "fit": 1.0, "abs_err": 0.0 }, { "gamma": 0.1, "lambda": 0.5, "pred": 0.9523809523809523, "fit": 0.952380952380952, "abs_err": 3.3306690738754696e-16 }, { "gamma": 0.1, "lambda": 2.0, "pred": 0.8333333333333334, "fit": 0.8333333333333336, "abs_err": 2.220446049250313e-16 }, { "gamma": 0.1, "lambda": 8.0, "pred": 0.5555555555555556, "fit": 0.5555555555555555, "abs_err": 1.1102230246251565e-16 }, { "gamma": 0.3, "lambda": 0.0, "pred": 1.0, "fit": 1.0, "abs_err": 0.0 }, { "gamma": 0.3, "lambda": 0.5, "pred": 0.8695652173913044, "fit": 0.8695652173913043, "abs_err": 1.1102230246251565e-16 }, { "gamma": 0.3, "lambda": 2.0, "pred": 0.625, "fit": 0.6249999999999999, "abs_err": 1.1102230246251565e-16 }, { "gamma": 0.3, "lambda": 8.0, "pred": 0.29411764705882354, "fit": 0.29411764705882354, "abs_err": 0.0 }, { "gamma": 0.7, "lambda": 0.0, "pred": 1.0, "fit": 1.0, "abs_err": 0.0 }, { "gamma": 0.7, "lambda": 0.5, "pred": 0.7407407407407407, "fit": 0.7407407407407408, "abs_err": 1.1102230246251565e-16 }, { "gamma": 0.7, "lambda": 2.0, "pred": 0.4166666666666667, "fit": 0.4166666666666667, "abs_err": 0.0 }, { "gamma": 0.7, "lambda": 8.0, "pred": 0.15151515151515152, "fit": 0.15151515151515155, "abs_err": 2.7755575615628914e-17 }, { "gamma": 1.0, "lambda": 0.0, "pred": 1.0, "fit": 1.0, "abs_err": 0.0 }, { "gamma": 1.0, "lambda": 0.5, "pred": 0.6666666666666666, "fit": 0.6666666666666666, "abs_err": 0.0 }, { "gamma": 1.0, "lambda": 2.0, "pred": 0.3333333333333333, "fit": 0.3333333333333333, "abs_err": 0.0 }, { "gamma": 1.0, "lambda": 8.0, "pred": 0.1111111111111111, "fit": 0.11111111111111112, "abs_err": 1.3877787807814457e-17 } ], "mismatch_loglog_slope": 2.000000000000007, "mismatch_quadratic_prediction": 2.0, "lambda_zero_mismatch": 0.0, "stability_predicted_boundary": 2.0, "largest_stable_grid_a": 1.99, "smallest_unstable_grid_a": 2.01, "stability_rows": [ { "a": 0.5, "predicted_q": 0.25, "observed_final_abs": 3.552713678800501e-15, "stable_observed": true }, { "a": 1.0, "predicted_q": 0.5, "observed_final_abs": 5.960464477539063e-08, "stable_observed": true }, { "a": 1.8, "predicted_q": 0.9, "observed_final_abs": 0.07976644307687256, "stable_observed": true }, { "a": 1.99, "predicted_q": 0.995, "observed_final_abs": 0.8866535105013078, "stable_observed": true }, { "a": 2.01, "predicted_q": 1.005, "observed_final_abs": 1.1271597762053889, "stable_observed": false }, { "a": 2.2, "predicted_q": 1.1, "observed_final_abs": 9.84973267580763, "stable_observed": false }, { "a": 3.0, "predicted_q": 1.5, "observed_final_abs": 16834.112196028233, "stable_observed": false } ] }, "adaptation": { "a_star": 0.45454545454545453, "clean_mse_coefficient": 0.6860440633405669, "proximal_coefficient": 0.4545454545454545, "mixed_coefficient": 0.4800983002056744, "results": [ { "method": "clean_mse", "a": 0.6860440633405669, "final_F": 0.4410948547609531, "final_stationarity": 0.5887514386655519, "reconstruction_MSE_to_optimum": 0.058122213539468136, "prox_mismatch_on_states": 0.012912660641092063 }, { "method": "proximal_match", "a": 0.4545454545454545, "final_F": 0.35391153445175094, "final_stationarity": 2.0066435492405165e-16, "reconstruction_MSE_to_optimum": 8.116294220563262e-33, "prox_mismatch_on_states": 9.805814240963052e-34 }, { "method": "mixed", "a": 0.4800983002056744, "final_F": 0.35476819383213837, "final_stationarity": 0.05836054869135342, "reconstruction_MSE_to_optimum": 0.0005711062535916371, "prox_mismatch_on_states": 0.00015732491659826528 }, { "method": "ideal", "a": 0.45454545454545453, "final_F": 0.35391153445175094, "final_stationarity": 9.514546268907243e-17, "reconstruction_MSE_to_optimum": 2.2369317072984132e-33, "prox_mismatch_on_states": 0.0 } ] }, "interpretation": { "mechanism_confirmed": true, "practical_prox_better_than_clean": true, "note": "The toy teacher is exact and quadratic; this isolates the proposed proximal-mismatch mechanism, not image-denoising generalization." } }