1[2 {"id":"C1","text":"Proposition A is false: X~N(0,1), Y=S*X with S an independent Rademacher sign gives normal marginals but X+Y with an atom of mass 1/2 at 0, neither constant nor normal.","results_keys":["statement_true","counterexample_provided","counterexample_values"],"status":"supported","caveat":"Truth rests on the proof in out/argument.md; simulation is illustrative."},3 {"id":"C2","text":"Proposition B is true: the sum of independent normals is normal with summed means and variances (characteristic functions).","results_keys":["statement_b_true","baseline_indep_ks_sum_vs_normal"],"status":"supported","caveat":"Proof-based; baseline simulation only illustrates."},4 {"id":"C3","text":"In the seeded simulation (n=100000), x and y have KS distances about 0.002 to fitted normals, while x+y has KS distance about 0.25 and about 50% exact zeros.","results_keys":["counterexample_values","diagnostics","n_analysed","random_seed"],"status":"supported","caveat":"Monte Carlo error of order 1/sqrt(n); KS with estimated parameters is not used for p-values."},5 {"id":"C4","text":"X and Y in the counterexample are uncorrelated but not independent, so (X,Y) is not jointly Gaussian.","results_keys":["diagnostics"],"status":"supported","caveat":"Zero covariance is exact by theory; sample correlation is -0.008."}6]7