[
 {"id":"C1","text":"Under 0<alpha<1 and positive prices and income, the unique optimum is x1*=alpha*m/p1, x2*=(1-alpha)*m/p2; at the given parameters x1*=16.8 and x2*=19.5.","results_keys":["x1_demand","x2_demand"],"status":"supported","caveat":"Analytic derivation; numerical evaluation at a single parameter point."},
 {"id":"C2","text":"Indirect utility at the closed-form demands is 2.9182520129 utils and the budget binds exactly (residual 0.0).","results_keys":["indirect_utility","budget_identity_residual"],"status":"supported","caveat":"Residual is subject to floating-point rounding; exactly 0.0 here."},
 {"id":"C3","text":"At the optimum the MRS equals the price ratio (0.625) and the expenditure share of good 1 equals alpha (0.35).","results_keys":["mrs_at_optimum","price_ratio","expenditure_share_good1"],"status":"supported","caveat":""},
 {"id":"C4","text":"Independent numerical maximisation (grid, Brent, SLSQP with inequality budget) reaches the same maximum utility as the closed form, with demand gaps at most ~1e-5.","results_keys":["indirect_utility_numerical","utility_gap_numerical_minus_closed_form","max_abs_demand_gap_x1","max_abs_demand_gap_x2","numerical_slsqp_budget_slack"],"status":"supported","caveat":"Validation covers only the single parameter point in data/parameters.csv."}
]