Trouble understanding proof in "Aperiodicity in One-Dimensional Cellular Automata" paper by Erica Jen From the proof of page 7 in < (page 10 of the PDF): > The fact that $R$ is injective in its $(i+1)$-th component (the “mirror”’ argument holds for injectivity on the left) implies that the values of the two adjacent temporal sequences $W_{j-1}$ and $W_{j}$ determine the values (and hence the periodicity properties) of the temporal sequence $W_{j+1}$. Hence that temporal sequence is periodic with transience $T$, as is in fact every temporal sequence to the right of $W_{j}$. Why should $W_{j + 1}$ be periodic with transience $T$, instead of $T+1$? $W_{j + 1}$ at $t = 0$ is certainly not determined by $W_{j - 1}$ and $W_{j}$ and $W_{j + 1}$ at $t>0$ is determined by $W_{j}$ and $W_{j + 1}$ at $t-1$.

> $W_{j + 1}$ at $t = 0$ is certainly not determined by $W_{j - 1}$ and $W_{j}$ and $W_{j + 1}$ at $t>0$ is determined by $W_{j}$ and $W_{j + 1}$ at $t-1$.

This is wrong. The fact that R is injective means that $W_{j - 1}$, $W_{j}$ at $t-1$ and $W_{j}$ at $t$ uniquely determine $W_{j + 1}$ at $t-1$, which is why $W_{j + 1}$ is periodic with transience $T$.