= Dyadic rigidity of a moment-generating function
{title2=$M(2t)=M(t)^4$}
Suppose a positive function $M$ satisfies the displayed fourth-power identity and $\log M(h)=h^2/2+o(h^2)$ at zero. Iteration gives $\log M(t)=4^m\log M(t/2^m)$, which tends to $t^2/2$. A related first-power symmetry step is: if $\psi(t)=\psi(t/2)^2$ and $\psi(h)=1+o(h^2)$, then $\log\psi(t)=2^m\log\psi(t/2^m)\to0$. These two different dyadic exponents must not be interchanged in the <Gaussian characterization by independent sum and difference>.
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