# Verification checks (P4)

1. Re-ran out/analysis.py from scratch; results.json byte-identical (cmp) -> PASS.
2. Cross-check vs statsmodels OLS (nonrobust and cov_type=HC1) for replications 1-5 using same eps draws: beta_hat, se_conv, se_HC1 agree to ~1e-16 -> PASS.
3. Exclusion ledger: 30 raw - 0 removed = 30 analysed = n_input_rows -> PASS.
4. Positive SDs/SEs: sd_beta_exact=0.2680, sd_beta_mc=0.2696, mean_se_conventional=0.1903 > 0 -> PASS.
5. sd_beta_mc vs sd_beta_exact: rel diff 0.6%; the MC SE of a sample SD at R=2000 is ~sd/sqrt(2(R-1)) ~ 0.0042, diff 0.0017 -> consistent, PASS.
6. mean_beta_hat=0.4962, |0.4962-0.5| = 0.0038 < 2*mcse 0.012 -> unbiasedness consistent, PASS.
7. mean_se_conventional (0.190) close to homoskedastic-formula SD sqrt(mean sigma^2/Sxx)=0.196 (below it, as expected from E[sqrt]<sqrt E and the leverage-weighting of residual variance) and well below sd_beta_exact 0.268 -> explains undercoverage, PASS.
8. Undercoverage rule: threshold 0.95 - 30.004873 = 0.9354; coverage_conventional 0.844 < threshold -> true. Also robust to using estimate-based MCSE (0.95-30.0081=0.926) -> same decision, PASS.
9. coverage values in [0,1]; n_replications=2000, n_obs=30, seed recorded -> PASS.