1[2 {"id":"C1","text":"Proposition A is false: with X~N(0,1), S an independent Rademacher sign, Y=S*X, both marginals are N(0,1) but X+Y has an atom of mass 1/2 at 0 and variance 2, so it is neither normal nor constant.","results_keys":["statement_true","counterexample_provided","counterexample_values.construction"],"status":"supported","caveat":"Supported by proof in out/argument.md; the simulation only illustrates it."},3 {"id":"C2","text":"Proposition B is true (characteristic-function proof).","results_keys":["statement_b_true"],"status":"supported","caveat":"Deductive; the independent-sum baseline KS is only an illustration."},4 {"id":"C3","text":"In the simulated sample (seed 20261029, n=100000) the marginals are consistent with N(0,1) (KS about 0.002) while x+y departs clearly from a fitted normal (KS 0.251, with 50.05% of rows exactly 0).","results_keys":["counterexample_values.ks_sum_vs_normal","supplementary_diagnostics.ks_x_vs_normal","supplementary_diagnostics.ks_y_vs_normal","supplementary_diagnostics.share_sum_exactly_zero","random_seed","n_analysed"],"status":"supported","caveat":"Numerical illustration with Monte Carlo error around 0.003; it is not a proof."},5 {"id":"C4","text":"X and Y in the counterexample are uncorrelated but dependent, so (X,Y) is not bivariate normal.","results_keys":["supplementary_diagnostics.corr_xy"],"status":"supported","caveat":"Zero covariance is proven analytically; the sample correlation of -0.008 is only consistent with it."}6]7