# Contract (P0)

Question. Deterministic optimal growth, log utility, y' = (y-c)^alpha, full depreciation, A=1. Compare the closed-form policy/value (guess-and-verify) with value-function iteration (VFI) on the prescribed log-spaced grid.

Estimands. c_closed(y) at y=0.5,1,2; c_num(y) from VFI at same points (interpolated if off-grid); max |c_num - c_closed| over the 3 points; 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 by Blackwell). Meaningful under: log utility, Cobb-Douglas technology, full depreciation, interior solution, 0<alpha<1, 0<beta<1; numerical quantities are conditional on grid [0.05,5], 400 log-spaced points, tolerance 1e-8.

Outputs. out/results.json (keys: 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 (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. Only data/model.csv; prescribed grid & parameters; deviations recorded in data_issues. Numerical policy must come from VFI, not the closed form. gate_spec.json is not an input and is not read.

Invalidity / impossibility. Missing/invalid parameters; optimal policy leaving the grid (corner solutions induced by grid bounds); VFI not converging; using closed form inside VFI.