{
  "task_id": "F5-conf-03",
  "statement_true": false,
  "counterexample_provided": true,
  "counterexample_values": {
    "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) but X+Y = (1+S)X equals 0 w.p. 1/2 and 2X otherwise (mixture of a point mass at 0 and N(0,4)), which is not normal.",
    "n": 100000,
    "seed": 20261029,
    "mean_x": -0.0033187423835634258,
    "sd_x": 0.9982238690893596,
    "mean_y": -0.00024909222432809854,
    "sd_y": 0.9982293548745821,
    "mean_sum": -0.0035678346078915233,
    "sd_sum": 1.4058866469982523,
    "ks_sum_vs_normal": 0.2508475723268392,
    "ks_x_vs_std_normal": 0.0028297029959603037,
    "ks_y_vs_std_normal": 0.001648188194033784,
    "frac_sum_exactly_zero": 0.50047,
    "ks_critical_5pct_approx": 0.0042943730625086595
  },
  "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,
  "baseline_independent_sum": {
    "construction": "X, Y iid N(0,1), drawn from the same generator after the counterexample draws",
    "mean_sum": -0.015452131370751422,
    "sd_sum": 1.412400304390764,
    "ks_sum_vs_normal": 0.0020272536437521693
  }
}