Verification checks (P4)
- Clean re-run: deleted out/results.json, re-ran out/analysis.py; SHA-256 of results.json identical to previous run. PASS
- Closed form c = (1-alpha*beta) y = 0.658 y at y=0.5,1,2. PASS
- Closed form verified against Bellman equation independently (scipy bounded maximisation of ln c + beta*V((y-c)^alpha) with V = A + B ln y): argmax equals 0.658y to <3e-8, and T V - V = 0.0 at y=0.5,1,2. PASS
- max_abs_error_policy equals max of the three |c_num - c_closed|. PASS
- savings_rate_closed_form = alpha*beta = 0.342. PASS
- Feasibility 0 < c_num < y at all eval points. PASS
- final_sup_change (9.997e-9) < vfi_tolerance (1e-8); n_iterations = 361. PASS
- No optimal choice at lower/upper grid bound (interior solution on grid). PASS
- |V_num(1) - V_closed(1)| = 3.4e-4 utils (< 1e-3; ~0.002% of |V|); exceeds the tolerance bound beta*tol/(1-beta)=1.9e-7, so the residual gap is discretisation error of the choice grid, not iteration error. PASS (as expected)
- Policy error 0.0064 < 1% of c at y=2; on-grid max policy error 0.025 (at large y, where step in y' maps to larger step in k) — consistent with discrete choice on a log grid. PASS
- n_analysed = grid_points_used = 400; exclusions: 0 removed (10 rows read = 10 used). PASS