# Contract (P0)
**Question.** Brock–Mirman optimal growth with log utility, y' = (y-c)^alpha, full depreciation, A=1. Compare the closed-form
policy/value with those obtained by discrete value-function iteration (VFI) on a prescribed log-spaced grid.

**Estimands.** Closed-form c(y) at y=0.5,1,2; VFI c(y) at the same points (interpolated if off-grid); max |c_num-c_closed|
over the 3 points; closed-form V(1) and VFI V(1); closed-form savings rate (alpha*beta); contraction modulus of the Bellman
operator in sup norm (beta, via Blackwell, on the bounded/compact grid problem). Meaningful under: log utility,
Cobb–Douglas technology, full depreciation, interior optimum, 0<alpha<1, 0<beta<1.

**Outputs.** out/results.json with keys task_id, c_closed_y05, c_closed_y1, c_closed_y2, c_num_y05, c_num_y1, c_num_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 (Question; Data and exclusions; Method; Results; Uncertainty; Limitations and open issues);
out/analysis.py; out/claims.json; plus protocol files (out/state/*, out/exclusions.csv).

**Information rules.** Use only data/model.csv; prescribed grid/parameters (deviations -> data_issues). Numerical policy
must come from VFI, never from the closed form; closed form derived by guess-and-verify and stated in report.

**Invalidating conditions.** Grid that cannot contain optimal choices (binding grid bounds), non-convergence of VFI,
infeasible consumption, use of closed form inside the VFI.