# Verification checks (P4)

| check | outcome |
|---|---|
| Clean re-run of out/analysis.py (results.json deleted first) reproduces byte-identical results.json | PASS (cmp identical) |
| budget_identity_residual = 0 (|r| < 1e-9) | PASS (0.0) |
| mrs_at_optimum = price_ratio (tangency) | PASS (0.625 = 0.625) |
| expenditure_share_good1 = alpha | PASS (0.35) |
| indirect_utility equals α ln(αm/p1)+(1−α) ln((1−α)m/p2) | PASS |
| Numerical max does not exceed closed-form utility (beyond 1e-12) | PASS (gap 0.0) |
| Grid, Brent and SLSQP demands agree with closed form | PASS (max |Δx1| = 9.6e-6 from grid spacing; |Δx2| ≤ 4.4e-7) |
| SLSQP with inequality constraint finds binding budget (monotonicity) | PASS (slack 2.8e-14) |
| Counts: n_input_rows 4 − removed 0 = n_analysed 4 | PASS |
| Magnitude vs baseline: closed-form v vs numerical baseline v | PASS (identical to float precision) |