# Verification checks (P4)
All executed 2026-10-30 against out/results.json produced by out/analysis.py.
1. Clean re-run (results.json deleted, script re-run) -> byte-identical results.json: PASS.
2. c_closed = (1-alpha*beta)*y at y=0.5,1,2: PASS.
3. max_abs_error_policy equals max of the three |c_num - c_closed|: PASS.
4. Closed form satisfies the Bellman equation: independent bounded scalar maximisation (scipy) of ln c + beta*V_closed((y-c)^alpha) at y=0.3,1,3 recovers c=(1-ab)y (|diff|<1e-6) and V_closed(y) (|diff|<1e-9): PASS.
5. savings_rate_closed_form = alpha*beta = 0.342 in (0,1): PASS.
6. 0 < final_sup_change < vfi_tolerance (1e-8): PASS.
7. |value_num_y1 - value_closed_y1| < 1e-3 (observed ~3.4e-4): PASS.
8. VFI policy never chooses the lowest or highest grid node (no truncation by grid bounds): PASS.
9. 0 < c_num < y at each evaluation point (feasibility): PASS.
10. Error at eval points <= max error over all grid nodes (0.0251): PASS.
11. n_analysed == grid_points_used == 400; contraction_modulus == beta: PASS.
12. Magnitude vs baseline: implied numerical savings rates 1 - c_num/y = 0.3424, 0.3410, 0.3452 vs closed-form 0.342: consistent.
Note: refinement diagnostic error is non-monotone in N (200 pts: 0.0059, 400: 0.0064, 800: 0.0014) - expected for a discrete choice grid where the error depends on node alignment with the true optimum; reported, not a failure.