# Verification checks (P4)

All executed 2026-10-30; all PASS.

1. Clean re-run: deleted out/results.json and out/counterexample.csv, re-ran analysis.py; results.json byte-identical (cmp) and counterexample.csv SHA-256 identical. PASS
2. Row counts: counterexample.csv rows = counterexample_values.n = n_analysed = 100000 = design n. PASS
3. Seed: counterexample_values.seed = random_seed = 20261029 = design seed. PASS
4. All sample SDs positive. PASS
5. Construction identity |y| = |x| holds row-wise. PASS
6. Magnitude: sd_sum = 1.4059 vs theoretical sqrt(2)=1.4142 (Var(X+Y)=E[(1+S)^2]E[X^2]=2). PASS (within 0.02)
7. Theory check of KS: the law of X+Y has CDF 1/4 just below 0 and 3/4 at 0, while the fitted N(0,2) CDF is 1/2 at 0, so the population KS distance is 0.25; sample ks_sum_vs_normal = 0.2508. PASS
8. ks_sum_vs_normal (0.251) > 10 x the nominal 1% KS critical value 1.628/sqrt(n)=0.0051; ks of x, y, and baseline independent sum (0.0021, 0.0020, 0.0018) are all below it. PASS (note: with estimated parameters the nominal critical value is conservative; does not affect conclusion)
9. frac_sum_exactly_zero = 0.50047 ≈ 1/2. PASS
10. n_input_rows matches audit (2, 2); exclusion ledger: 2 - 0 = 2. PASS