# Verification checks (P4)

1. Clean re-run: deleted out/results.json and out/counterexample.csv, re-ran analysis.py; results.json byte-identical, counterexample.csv SHA-256 identical. PASS
2. rows(counterexample.csv) = counterexample_values.n = n_analysed = 100000 = design n. PASS
3. Recomputing mean_x and sd_sum from the CSV matches results.json. PASS
4. |x| == |y| holds for every row (follows from Y = S*X). PASS
5. Marginals: KS distance x vs fitted normal 0.0021, y 0.0020; KS p-values vs N(0,1) 0.40 and 0.95, consistent with exact N(0,1) marginals. PASS
6. Sum: share of exact zeros 0.50047 (theory 0.5); sd_sum 1.406 (theory sqrt(2)=1.414); KS distance vs fitted normal 0.2508 (theory: about 0.25, from the jump of 1/2 at 0 where Phi=1/2). PASS
7. Baseline for B: independent normals sum, KS vs N(0,2) 0.0024 (p=0.62), consistent with B being true. PASS
8. All sds are positive; seed in results equals design seed. PASS