{
  "task_id": "F5-conf-03",
  "statement_true": false,
  "counterexample_provided": true,
  "counterexample_values": {
    "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).",
    "n": 100000,
    "seed": 20261029,
    "mean_x": -0.0033187424,
    "sd_x": 0.9982238691,
    "mean_y": -0.0002490922,
    "sd_y": 0.9982293549,
    "mean_sum": -0.0035678346,
    "sd_sum": 1.405886647,
    "ks_sum_vs_normal": 0.2508475723
  },
  "statement_b_true": true,
  "feasibility": "complete",
  "feasibility_reason": null,
  "data_issues": [],
  "n_input_rows": {
    "propositions.csv": 2,
    "design.csv": 2
  },
  "n_analysed": 100000,
  "random_seed": 20261029,
  "ks_x_vs_normal": 0.0020917798,
  "ks_y_vs_normal": 0.0019518221,
  "frac_sum_exactly_zero": 0.50047,
  "frac_w_equal_1": 0.49953,
  "corr_xy": -0.0082268898,
  "ks_critical_value_5pct_approx": 0.0042943731,
  "baseline_independent_mean_sum": -0.0154521314,
  "baseline_independent_sd_sum": 1.4124003044,
  "baseline_independent_ks_sum_vs_normal": 0.0020272536
}