# Verification checks (P4)

- Clean re-run of out/analysis.py: results.json byte-identical: PASS
- x1 = a m/p1 = 16.8: PASS
- x2 = (1-a)m/p2 = 19.5: PASS
- |budget residual|<1e-9: PASS
- MRS == price ratio: PASS
- share1 == alpha: PASS
- |v - v_num|<1e-8: PASS
- all 3 numerical x1 within 1e-4 of closed form: PASS
- no numerical u exceeds closed-form v by >1e-12: PASS
- v = 0.35 ln16.8 + 0.65 ln19.5: PASS
- exclusion ledger 4-0=4 = n_analysed: PASS
- Note: SLSQP initially reported success=false with ftol=1e-15 (tolerance unattainable in float64); ftol set to 1e-12, success=true. Its tiny budget slack -5.7e-14 is floating-point rounding.