1{2 "task_id": "F5-conf-03",3 "statement_true": false,4 "counterexample_provided": true,5 "counterexample_values": {6 "construction": "X ~ N(0,1); S independent of X with P(S=1)=P(S=-1)=1/2; Y = S*X. Then Y ~ N(0,1) by symmetry of N(0,1), but X+Y = (1+S)X equals 0 with prob. 1/2 and 2X (N(0,4)) with prob. 1/2: a non-degenerate law with an atom at 0, hence not normal.",7 "n": 100000,8 "seed": 20261029,9 "mean_x": -0.0033187423835634258,10 "sd_x": 0.9982238690893596,11 "mean_y": -0.00024909222432809854,12 "sd_y": 0.9982293548745821,13 "mean_sum": -0.0035678346078915233,14 "sd_sum": 1.4058866469982523,15 "ks_sum_vs_normal": 0.250847572326839216 },17 "statement_b_true": true,18 "feasibility": "complete",19 "feasibility_reason": null,20 "data_issues": [],21 "n_input_rows": {22 "propositions.csv": 2,23 "design.csv": 224 },25 "n_analysed": 100000,26 "random_seed": 20261029,27 "counterexample_diagnostics": {28 "frac_sum_exactly_zero": 0.50047,29 "ks_x_vs_std_normal": 0.0028297029959603037,30 "ks_y_vs_std_normal": 0.001648188194033784,31 "ks_x_vs_fitted_normal": 0.0020917798156042666,32 "ks_y_vs_fitted_normal": 0.0019518220846299617,33 "corr_xy": -0.008226889835527693,34 "population_ks_sum_vs_N02": 0.25,35 "ks_95pct_critical_value_approx": 0.004294373062508659536 },37 "baseline_b_ks_sum_vs_normal": 0.0020272536437521693,38 "baseline_b_sd_sum": 1.41240030439076439}