1[2 {"id":"C1","text":"Proposition A is false: normal marginals do not imply X+Y is normal.","results_keys":["statement_true"],"status":"supported","caveat":"Supported by the mathematical counterexample in argument.md; simulation is illustrative only."},3 {"id":"C2","text":"A concrete counterexample is X ~ N(0,1), S an independent random sign, Y = S X; Y ~ N(0,1) but X+Y has an atom of mass 1/2 at 0 and variance 2.","results_keys":["counterexample_provided","counterexample_values"],"status":"supported","caveat":"(X,Y) is not jointly normal."},4 {"id":"C3","text":"In the simulated sample (n=100000, seed 20261029) the marginals are consistent with N(0,1) while x+y is far from normal (KS distance 0.2508, about half of sums exactly 0).","results_keys":["counterexample_values","numerical_checks","n_analysed","random_seed"],"status":"supported","caveat":"Numerical illustration only; KS p-values with estimated parameters are approximate."},5 {"id":"C4","text":"Proposition B is true: the sum of independent normals is normal with added means and variances.","results_keys":["statement_b_true","numerical_checks"],"status":"supported","caveat":"Proof via characteristic functions; simulated baseline (KS 0.0024 vs N(0,2)) is only a consistency check."}6]7