Worked solution: Marcus–Spielman–Srivastava proof via interlacing families (2013)

Putting the last three steps together: (1) the mixed characteristic polynomial $\mu[A_1,\ldots,A_m](x)$ equals the expected characteristic polynomial of the independent random sum and is real-rooted (step 4); (2) interlacing selects an outcome whose largest root is no larger than the average's largest root (step 5); (3) the barrier computation bounds that average root by $u^*=(1+\sqrt{2\delta})^2/2$ (step 6). Applying the paper's $r=2$ direct-sum lift to these random vectors yields complementary sets $S_1,S_2$ with $\left\|\sum_{i\in S_j}v_iv_i^*\right\|\le u^*$ for both $j$, proving $KS_2$.