# Verification checks (P4)

1. Clean re-run of out/analysis.py regenerates byte-identical out/results.json (cmp) — PASS.
2. c_closed values equal (1-alpha*beta)*y at y=0.5,1,2 — PASS.
3. max_abs_error_policy equals max |c_num - c_closed| over the three points — PASS.
4. Closed form verified as Bellman fixed point by independent continuous maximisation (scipy bounded) at y=0.3,1,3: argmax matches (1-alpha*beta)y to ~2e-8, |TV - V| <= 4e-15 — PASS.
5. final_sup_change (9.997e-9) < vfi_tolerance (1e-8) — PASS.
6. Choice-grid bounds never bind (diag_choice_bound_binds = 0) — PASS.
7. Magnitudes: relative policy errors -0.06%, +0.15%, -0.48%; value gap at y=1 = -3.4e-4 utils (≈ diag_max_abs_value_error_grid order), mean numerical savings rate on grid 0.34202 vs baseline 0.342 — PASS (consistent with grid-discretisation error, log step 0.0115).
8. contraction_modulus = beta = 0.95 in (0,1) — PASS.
9. Exclusion ledger: 10 rows in, 0 removed, 10 out — PASS.