1[2 {"id":"C1","text":"Proposition A is false: with X~N(0,1), S an independent Rademacher sign and Y=S*X, both marginals are N(0,1) but X+Y ~ 0.5*delta_0 + 0.5*N(0,4) is neither normal nor constant.","results_keys":["statement_true","counterexample_provided","counterexample_values"],"status":"supported","caveat":"Established by proof; simulation is illustrative only."},3 {"id":"C2","text":"Proposition B is true: the sum of independent normals is N(mu1+mu2, s1^2+s2^2), by multiplying characteristic functions.","results_keys":["statement_b_true"],"status":"supported","caveat":"Proof-based; no numerical check of B was run."},4 {"id":"C3","text":"In the simulated sample (n=100000, seed 20261029) the marginals are close to normal (KS ~0.002) while x+y is far from normal (KS distance 0.251, 50.0% of sums exactly zero).","results_keys":["counterexample_values","n_analysed","random_seed"],"status":"supported","caveat":"KS uses fitted parameters; the p-values are approximate (Lilliefors issue)."}5]6