# Verification checks

1. Re-run from clean state (results.json and counterexample.csv deleted): regenerated results.json byte-identical (cmp) — PASS.
2. counterexample.csv: 100000 rows = n = n_analysed = counterexample_values.n; no NaN — PASS.
3. Stats recomputed from CSV match results.json (sd_sum diff 0, mean_x diff ~1e-18) — PASS.
4. Fraction of x+y exactly 0 in CSV = 0.50047 ≈ 1/2 theory — PASS.
5. sd_sum 1.4059 vs theory sqrt(2)=1.4142; sd_x, sd_y ≈ 0.998 vs 1 — PASS (sampling error).
6. Marginal KS vs N(0,1): x 0.0028, y 0.0016, both below approx 5% critical 0.0043 — PASS (consistent with normal marginals).
7. ks_sum_vs_normal 0.2508 ≥ theoretical lower bound 0.25, vs baseline (independent) 0.0020 — PASS.
8. Gate runs: recorded in phases.json.