1# Verification checks (P4)23- Clean re-run of out/analysis.py: results.json byte-identical: PASS4- x1 = a m/p1 = 16.8: PASS5- x2 = (1-a)m/p2 = 19.5: PASS6- |budget residual|<1e-9: PASS7- MRS == price ratio: PASS8- share1 == alpha: PASS9- |v - v_num|<1e-8: PASS10- all 3 numerical x1 within 1e-4 of closed form: PASS11- no numerical u exceeds closed-form v by >1e-12: PASS12- v = 0.35 ln16.8 + 0.65 ln19.5: PASS13- exclusion ledger 4-0=4 = n_analysed: PASS14- 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.15