1[2 {"id": "C1", "text": "Proposition A is false: X ~ N(0,1), Y = S X with S an independent fair sign gives normal marginals but X+Y = (1+S)X, a mixture of a point mass at 0 (prob 1/2) and N(0,4), which is neither normal nor constant.", "results_keys": ["statement_true", "counterexample_provided", "counterexample_values"], "status": "supported", "caveat": "Supported by mathematical proof in argument.md; simulation is illustrative only."},3 {"id": "C2", "text": "Proposition B is true: the sum of independent normals is normal with summed means and variances (characteristic-function proof).", "results_keys": ["statement_b_true", "baseline_ks_sum_independent"], "status": "supported", "caveat": "Simulation baseline is illustrative only."},4 {"id": "C3", "text": "In the simulated sample (seed 20261029, n=100000) the marginals agree with N(0,1) (sample SDs ~0.998, KS ~0.002) while the sum has about half its mass exactly at 0 and KS distance ~0.251 from its fitted normal, matching the population value 0.25.", "results_keys": ["counterexample_values", "ks_x_vs_normal", "ks_y_vs_normal", "frac_sum_exactly_zero", "ks_critical_value_alpha_0_01", "n_analysed", "random_seed"], "status": "supported", "caveat": "Single seed; nominal KS critical value ignores parameter estimation (conservative)."}5]6