# Contract (P0)
**Question.** Deterministic optimal growth, log utility, y' = (y-c)^alpha, full depreciation, A=1. Compare the closed-form policy/value with
value-function iteration (VFI) on the prescribed log-spaced grid (data/model.csv).

**Estimands.** c_closed(y) and c_num(y) at y = 0.5, 1, 2; max |c_num - c_closed| over these; V_closed(1), V_num(1);
closed-form savings rate (y-c)/y = alpha*beta; contraction modulus of the Bellman operator in sup norm (beta, via Blackwell,
valid on the compact grid where utility is bounded); VFI diagnostics (grid points, iterations, final sup change).
Assumptions: log utility, Cobb-Douglas, full depreciation, interior solution, 0<alpha<1, 0<beta<1.

**Outputs.** out/results.json (keys per TASK.md table: task_id, c_closed_y05/y1/y2, c_num_y05/y1/y2, max_abs_error_policy,
value_closed_y1, value_num_y1, savings_rate_closed_form, contraction_modulus, grid_points_used, n_iterations, final_sup_change,
feasibility, feasibility_reason, data_issues, n_input_rows, n_analysed, random_seed), out/report.md (6 sections), out/analysis.py,
out/claims.json, plus state files/exclusions ledger.

**Information rules.** Only data/model.csv; prescribed grid/parameters; numerical policy from VFI only; closed form derived
(guess-and-verify) and stated. Deviations recorded in data_issues.

**Invalidity.** Using closed form inside VFI; changing grid/tolerance without recording; VFI not converging; next-period
output leaving the grid such that the problem is mis-specified.