1[2 {"id":"C1","text":"Proposition A is false.","results_keys":["statement_true"],"status":"supported","caveat":"Rests on the mathematical counterexample in argument.md, not on the simulation."},3 {"id":"C2","text":"A concrete counterexample to A is X ~ N(0,1), Y = S*X with S an independent Rademacher sign; Y is N(0,1) but X+Y is the mixture 1/2*delta_0 + 1/2*N(0,4).","results_keys":["counterexample_provided","counterexample_values"],"status":"supported","caveat":"X and Y are uncorrelated but dependent and not jointly normal."},4 {"id":"C3","text":"In the simulated sample (seed 20261029, n=100000) the marginals are close to N(0,1) while the KS distance of x+y to its fitted normal is about 0.25, matching the theoretical bound of 1/4; about half the sums are exactly 0.","results_keys":["counterexample_values","n_analysed","random_seed"],"status":"supported","caveat":"Numerical illustration only; finite-sample agreement with marginal normality cannot prove normality."},5 {"id":"C4","text":"Proposition B is true (proof via characteristic functions); the independent-sum simulation baseline has KS about 0.002, consistent with normality.","results_keys":["statement_b_true","baseline_independent_sum"],"status":"supported","caveat":"Simulation is consistent with but does not prove B."}6]7