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), Cov(X,Y)=0, and X+Y = (1+S)X is 0 with probability 1/2 and 2X ~ N(0,4) with probability 1/2 (a non-degenerate mixture with an atom at 0, 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 "diagnostic_ks_x_vs_normal": 0.0020917798156042666,28 "diagnostic_ks_y_vs_normal": 0.0019518220846299617,29 "diagnostic_frac_sum_exactly_zero": 0.50047,30 "diagnostic_ks_critical_value_alpha_0.05": 0.0042943730625086595,31 "baseline_independent_ks_sum_vs_normal": 0.0017196848465321768,32 "baseline_independent_seed_spec": "numpy default_rng([seed, 1])"33}