1[2 {"id":"C1","text":"Proposition A is false: normal marginals do not imply a normal sum.","results_keys":["statement_true"],"status":"supported","caveat":"Established by the explicit counterexample in argument.md; the simulation only illustrates it."},3 {"id":"C2","text":"With X~N(0,1), W an independent random sign and Y=WX, Y is N(0,1) but X+Y has an atom of mass 1/2 at 0 and variance 2, so it is not normal.","results_keys":["counterexample_provided","counterexample_values"],"status":"supported","caveat":"Proof in argument.md."},4 {"id":"C3","text":"In the simulated sample (n=100000, seed 20261029) the KS distance of x+y from a fitted normal is about 0.25, far above sampling noise, while the marginal moments match N(0,1).","results_keys":["counterexample_values","n_analysed","random_seed"],"status":"supported","caveat":"Numerical illustration only; KS computed with estimated parameters."},5 {"id":"C4","text":"Proposition B is true: the sum of independent normals is normal with summed means and variances.","results_keys":["statement_b_true"],"status":"supported","caveat":"Characteristic-function proof; degenerate (variance 0) cases are included."}6]7