# Verification checks (P4)
1. Determinism: re-ran analysis.py and byte-compared results.json with the previous run: IDENTICAL. PASS
2. Cross-check vs statsmodels OLS (nonrobust and HC1) on replications 0, 1, 1999 of the same draw: β̂, se_conv and se_HC1 agree to ~1e-15. PASS
3. Exact vs MC SD: sd_beta_exact 0.26798 vs sd_beta_mc 0.26965. The relative difference is 0.6%, which is inside the MC sampling SD of a sample SD (≈ 1/sqrt(2·1999) ≈ 1.6%). PASS
4. mean_se_conventional 0.19026 vs sqrt(E[s²]·[(X'X)⁻¹]₂₂) = 0.19318 (an upper bound on E[se] by Jensen). The MC mean is just below it, as expected. PASS
5. mean_se_conventional (0.190) < sd_beta_exact (0.268): the conventional SE is biased downward, which agrees with the undercoverage. PASS
6. Positive SEs/SDs; coverages lie in [0,1]; mean β̂ = 0.4962 ≈ 0.5 (MCSE ≈ 0.006). PASS
7. Undercoverage: 0.844 < threshold 0.93538 (nominal-p MCSE). It also holds with the estimated-p MCSE. The conclusion is robust to that choice. PASS
8. Exclusion ledger: 30 − 0 = 30 = n_input_rows. PASS