= Section induction for the Talagrand exponential moment
{title2=$\mathbb E e^{d_T(X,A)^2/4}\le1/\Pr(A)$}
Let the largest last-coordinate section have probability $a$, and another section probability $ar$. Mix mismatch vectors from these two sections with weights $\lambda,1-\lambda$. Their squared norm is at most $\lambda d_1^2+(1-\lambda)d_2^2+(1-\lambda)^2$. <Hölder's inequality> and induction bound that section's exponential integral by $a^{-1}\inf_\lambda r^{-\lambda}e^{(1-\lambda)^2/4}$. The <scalar estimate for Talagrand product induction> makes it at most $(2-r)/a$. Average over sections and use $m(2-m)\le1$ with $m=\Pr(A)/a$ to close the induction.
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