Multiplicative energy sumset bound ($E(A,B)\le 4\lceil \log |B|\rceil |A+A||B+B|$)

For finite sets of positive real numbers $A,B$ with $|B|\ge 2$, $E(A,B)\le 4\lceil \log |B|\rceil |A+A||B+B|$, with natural or binary logarithm. Partition ray occupancies of $A\times B$ into $\lceil \log |B|\rceil$ classes of multiplicative width $e$, or width $2$ for the binary convention. Injectivity of sums on two distinct rays and disjoint open planar sectors bound each class's sum of squared occupancies by $4|A+A||B+B|$. A singleton class contributes at most $|A||B|$. Combining this with the energy lower bound gives a sum-product inequality.