1[2 {"id":"C1","text":"Proposition A is false: normal marginals do not imply a normal sum.","results_keys":["statement_true"],"status":"supported","caveat":"Rests on the mathematical counterexample in argument.md; the simulation is illustrative."},3 {"id":"C2","text":"With X~N(0,1), S an independent random sign, and Y=SX, both X and Y are N(0,1) but X+Y has an atom of mass 1/2 at 0 and so is not normal.","results_keys":["counterexample_provided","counterexample_values"],"status":"supported","caveat":"Marginal normality of Y is proved by symmetry; the sample (KS for x,y about 0.003 and 0.002) only illustrates it."},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 about 0.004 sampling noise.","results_keys":["counterexample_values"],"status":"supported","caveat":"The KS critical value is approximate, since parameters were estimated."},5 {"id":"C4","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":"Proof sketch only; this proposition was not evaluated numerically."}6]7