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 ~ Rademacher (P(S=1)=P(S=-1)=1/2) independent of X; Y = S*X. Then Y ~ N(0,1) by symmetry, but X+Y = (1+S)X equals 0 with probability 1/2 and 2X otherwise (Var = 2), so X+Y has an atom at 0 and is neither normal nor constant.",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": 0.2508475723268392,16 "supplementary": {17 "share_sum_exactly_zero": 0.50047,18 "ks_x_vs_fitted_normal": 0.0020917798156042666,19 "ks_x_pvalue": 0.77318862336257,20 "ks_y_vs_fitted_normal": 0.0019518220846299617,21 "ks_y_pvalue": 0.8399489498054262,22 "corr_xy": -0.008226889835527693,23 "excess_kurtosis_sum": 2.989132293240780724 }25 },26 "statement_b_true": true,27 "feasibility": "complete",28 "feasibility_reason": null,29 "data_issues": [],30 "n_input_rows": {31 "propositions.csv": 2,32 "design.csv": 233 },34 "n_analysed": 100000,35 "random_seed": 2026102936}