Dyadic rigidity of a moment-generating function ($M(2t)=M(t{)}^4$)

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 $\log \psi (t)=2^m\log \psi (t/2^m)\to 0$. These two different dyadic exponents must not be interchanged in the Gaussian characterization by independent sum and difference.