1[2 {"id":"C1","text":"Closed-form Marshallian demands are x1* = alpha m/p1 = 16.8 and x2* = (1-alpha) m/p2 = 19.5, the unique optimum when 0<alpha<1 and p1,p2,m>0.","results_keys":["x1_demand","x2_demand"],"status":"supported","caveat":"Uniqueness relies on the stated parameter-range assumptions, which hold for the given inputs."},3 {"id":"C2","text":"Indirect utility at the optimum is 2.9182520128635305 utils and the budget binds exactly (residual 0).","results_keys":["indirect_utility","budget_identity_residual"],"status":"supported","caveat":"Float64 precision."},4 {"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":""},5 {"id":"C4","text":"Independent numerical maximisation of utility on the budget line reproduces the closed-form indirect utility to machine precision.","results_keys":["indirect_utility_numerical","indirect_utility","numerical_validation"],"status":"supported","caveat":"Validation covers only the single given parameter point."}6]7