1{2 "task_id": "F5-conf-03",3 "statement_true": false,4 "counterexample_provided": true,5 "counterexample_values": {6 "construction": "X ~ N(0,1); W ~ Rademacher (P(W=1)=P(W=-1)=1/2) independent of X; Y = W*X. Then Y ~ N(0,1) but X+Y = (1+W)X is 0 with prob. 1/2 and 2X ~ N(0,4) otherwise (atom at 0).",7 "n": 100000,8 "seed": 20261029,9 "mean_x": -0.0033187424,10 "sd_x": 0.9982238691,11 "mean_y": -0.0002490922,12 "sd_y": 0.9982293549,13 "mean_sum": -0.0035678346,14 "sd_sum": 1.405886647,15 "ks_sum_vs_normal": 0.250847572316 },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 "ks_x_vs_normal": 0.0020917798,28 "ks_y_vs_normal": 0.0019518221,29 "frac_sum_exactly_zero": 0.50047,30 "frac_w_equal_1": 0.49953,31 "corr_xy": -0.0082268898,32 "ks_critical_value_5pct_approx": 0.0042943731,33 "baseline_independent_mean_sum": -0.0154521314,34 "baseline_independent_sd_sum": 1.4124003044,35 "baseline_independent_ks_sum_vs_normal": 0.002027253636}