1[2 {"id":"C1","text":"Proposition A is false: marginal normality of X and Y does not imply normality of X+Y.","results_keys":["statement_true"],"status":"supported","caveat":"Supported by the mathematical counterexample in argument.md; the simulation is illustrative only."},3 {"id":"C2","text":"Counterexample: X ~ N(0,1), W Rademacher independent of X, Y = W*X; Y ~ N(0,1) but X+Y = 0.5*delta_0 + 0.5*N(0,4) is not normal (atom of mass 1/2 at 0).","results_keys":["counterexample_provided","counterexample_values"],"status":"supported","caveat":"Y = -X is not a valid counterexample because constants count as degenerate normals."},4 {"id":"C3","text":"Proposition B is true: the sum of independent normals is normal with summed means and variances.","results_keys":["statement_b_true","baseline_independent_ks_sum_vs_normal"],"status":"supported","caveat":"Proof via characteristic functions; the baseline simulation only illustrates it."},5 {"id":"C4","text":"In the simulated sample (seed 20261029, n=100000) the marginals look normal (KS about 0.002, the same as the independent baseline) while the sum departs clearly (KS 0.2508, about 1/4 as theory predicts; about 50% of the sums are exactly zero).","results_keys":["counterexample_values","ks_x_vs_normal","ks_y_vs_normal","baseline_independent_ks_sum_vs_normal","frac_sum_exactly_zero","n_analysed","random_seed"],"status":"supported","caveat":"Finite-sample illustration; not a proof. The KS critical value used is approximate (known-parameter formula)."},6 {"id":"C5","text":"X and Y in the counterexample are uncorrelated but dependent, so (X,Y) is not jointly Gaussian.","results_keys":["corr_xy"],"status":"supported","caveat":"Zero correlation is exact in theory; the sample value is -0.0082."}7]8