  Using log convergent topoi, %In the derived category of filtered complexes of
%sheaves of modules over %an isostructure we define two fundamental filtered
complexes $(E_{conv},P)$ and $(C_{conv},P)$ for the log scheme obtained by a
smooth scheme with a relative simple normal crossing divisor over a scheme of
characteristic $p>0$. Using $(C_{conv},P)$, we prove the $p$-adic purity. As a
corollary of it, we prove that $(E_{conv},P)$ and $(C_{conv},P)$ are
canonically isomorphic. These filtered complexes produce the weight spectral
sequence of the log convergent cohomology sheaf of the log scheme. We also give
the comparison theorem between the projections of $(E_{conv},P)$ and
$(C_{conv},P)$ to the derived category of bounded below filtered complexes of
sheaves of modules in the Zariski topos of the log scheme and the
weight-filtered isozariskian filtered complex $(E_{zar},P)_{Q}$ of the log
scheme defined in our previous book.
