1[2 {"id":"C1","text":"Proposition A is false: X~N(0,1), Y=W X with independent Rademacher W has normal marginals but X+Y has an atom of mass 1/2 at 0 and is non-constant, hence not normal.","results_keys":["statement_true","counterexample_provided","counterexample_values.construction"],"status":"supported","caveat":"Supported by mathematical proof in argument.md; simulation is illustrative only."},3 {"id":"C2","text":"In the simulated sample (seed 20261029, n=100000) x and y are consistent with N(0,1) (KS 0.0028 and 0.0016) while x+y is far from normal (KS 0.251 vs fitted normal; 50.05% exact zeros).","results_keys":["counterexample_values.ks_sum_vs_normal","counterexample_values.ks_x_vs_std_normal","counterexample_values.ks_y_vs_std_normal","counterexample_values.fraction_sum_exactly_zero"],"status":"supported","caveat":"Finite-sample consistency with normality is not proof of normality."},4 {"id":"C3","text":"Proposition B is true: for independent normal X, Y, X+Y ~ N(mu1+mu2, s1^2+s2^2) by multiplication of characteristic functions and Levy uniqueness.","results_keys":["statement_b_true"],"status":"supported","caveat":"Proof sketch; not checked numerically."}5]6