# Verification checks (P4)

1. Clean re-run: deleted out/results.json and out/counterexample.csv, re-ran analysis.py; SHA-256 of both files identical to the first run. PASS
2. counterexample.csv has columns x,y and 100000 rows = counterexample_values.n = n_analysed = design n. PASS
3. random_seed = counterexample_values.seed = 20261029 (design). PASS
4. Construction identity |y| = |x| for every row (Y = ±X). PASS
5. mean_sum / sd_sum recomputed from CSV match results.json (|diff| < 1e-9). PASS
6. Nonzero sums equal 2x exactly; frac_sum_exactly_zero = 0.50047 ≈ 1/2 = P(W=-1); consistent with frac_w_equal_1 = 0.49953 (sum = 1). PASS
7. Magnitudes: sd_x, sd_y ≈ 1; sd_sum = 1.406 ≈ sqrt(2) (theory: Var(X+Y)=E[(1+W)^2]E[X^2]=2). corr_xy ≈ 0 (theory: 0). PASS
8. Theoretical KS distance between law of X+Y (0.5·δ0 + 0.5·N(0,4)) and N(0,2): 0.25, attained at 0 (CDF jumps 0.25→0.75 vs Φ(0)=0.5; difference decreases away from 0). Observed ks_sum_vs_normal = 0.2508. PASS
9. Baseline (independent normals, prop. B): KS = 0.0020, two orders of magnitude below 0.2508; marginal KS for x, y ≈ 0.002, same order as baseline → marginals look normal, sum does not. PASS
10. Note: ks_critical_value_5pct_approx = 1.358/sqrt(n) is the known-parameter KS critical value; with estimated parameters (Lilliefors) the correct value is smaller (~0.0028), so it is conservative for rejection. Reported as approximate only. NOTED