# Verification checks (P4)
| check | outcome |
|---|---|
| Clean re-run (results.json deleted, script run twice) gives byte-identical results.json | PASS (cmp identical) |
| Closed-form p vs brentq root of z1(p) | PASS: 0.875 vs 0.875, abs diff 1.1e-16 (`abs_diff_closed_vs_numerical`) |
| Market clearing good 1: x1_A+x1_B = 12 | PASS (`excess_demand_good1` = 0.0) |
| Market clearing good 2: x2_A+x2_B = 10 | PASS (`excess_demand_good2` = 0.0) |
| Walras' law p*z1+z2 = 0 | PASS (`walras_law_residual` = 0.0) |
| Budget constraints hold for A and B | PASS (residual 0.0 each) |
| MRS_A = MRS_B = p (Pareto efficiency, tangency) | PASS (0.875 both) |
| Interior solution: all consumptions > 0 | PASS |
| Hand calculation: p = (0.4*2+0.6*8)/(0.6*8+0.4*4) = 5.6/6.4 = 0.875 | PASS |
| Numerical root uses only z1(p) (brentq on bracket [1e-6, 1]), not the closed form | PASS (by code inspection) |
| n_raw(2) - removed(0) = n_analysed(2) | PASS |
| Data inconsistency (E1 14 vs 12) reported in data_issues | PASS |