1[2 {"id":"C1","text":"Proposition A is false: X ~ N(0,1), Y = S X with an independent random sign S gives normal marginals but X+Y = (1+S)X has an atom of mass 1/2 at 0 and positive variance, so it is not normal (not even degenerate).","results_keys":["statement_true","counterexample_provided","counterexample_values"],"status":"supported","caveat":"Verdict rests on the proof in out/argument.md; simulation is illustrative."},3 {"id":"C2","text":"Proposition B is true: for independent normals the characteristic function of the sum is a normal characteristic function.","results_keys":["statement_b_true"],"status":"supported","caveat":"Proof by characteristic functions; the simulated baseline is only a numerical check."},4 {"id":"C3","text":"In the simulated counterexample sample (n=100000, seed 20261029), x and y have means near 0 and SDs near 1, while the sum has KS distance 0.2508 from its fitted normal, matching the theoretical 0.25, with about 50% of sums exactly 0.","results_keys":["counterexample_values","diagnostic_frac_sum_exactly_zero","diagnostic_ks_x_vs_normal","diagnostic_ks_y_vs_normal","n_analysed","random_seed"],"status":"supported","caveat":"Finite-sample illustration; does not by itself prove exact normality of the marginals."},5 {"id":"C4","text":"Under the same evaluation rule, the sum of independent normals has KS distance 0.0017, about 150 times smaller than the counterexample.","results_keys":["baseline_independent_ks_sum_vs_normal","diagnostic_ks_critical_value_alpha_0.05"],"status":"supported","caveat":"Critical value scale ignores parameter estimation (conservative)."}6]7