1[2 {"id":"C1","text":"Proposition A is false: marginal normality of X and Y does not imply X+Y is normal, even counting constants as degenerate normals.","results_keys":["statement_true","counterexample_provided"],"status":"supported","caveat":"Supported by proof in out/argument.md; simulation is illustrative only."},3 {"id":"C2","text":"Counterexample: X ~ N(0,1), W independent Rademacher, Y = W X; Y ~ N(0,1) and X+Y has an atom of mass 1/2 at 0 and is non-constant, hence not normal.","results_keys":["counterexample_values.construction","supplementary.share_sum_exactly_zero"],"status":"supported","caveat":"Y=-X would not be a valid counterexample under A's convention."},4 {"id":"C3","text":"Proposition B is true: the sum of independent normals is N(mu1+mu2, s1^2+s2^2), by characteristic functions.","results_keys":["statement_b_true"],"status":"supported","caveat":"Mathematical proof; not established by simulation."},5 {"id":"C4","text":"In the simulated sample (seed 20261029, n=100000) the marginals are close to normal (KS ~0.002) while the sum is clearly non-normal (KS 0.2508, near the theoretical lower bound 0.25).","results_keys":["counterexample_values.ks_sum_vs_normal","supplementary.ks_x_vs_normal","supplementary.ks_y_vs_normal","supplementary.baseline_ks_independent_sum_vs_normal","counterexample_values.n","counterexample_values.seed"],"status":"supported","caveat":"KS with estimated parameters; distances only, no p-values."},6 {"id":"C5","text":"Sample moments match theory: means near 0, sd_x and sd_y near 1, sd_sum near sqrt(2); x and y are uncorrelated (corr near 0) yet dependent.","results_keys":["counterexample_values.mean_x","counterexample_values.sd_x","counterexample_values.mean_y","counterexample_values.sd_y","counterexample_values.mean_sum","counterexample_values.sd_sum","supplementary.corr_x_y"],"status":"supported","caveat":"Agreement within sampling error of order 1/sqrt(n)."}7]8