Ok, maybe I've figured this out. Hopefully somebody can correct me if this is wrong. Also, I'd still like to know a reference that writes this out in detail, if anybody has one.

I'll change the notation from the question slightly: $k$ is a perfect field of characteristic $p>0$, and $W=W(k)$. First let's recall the definition of Fontaine-Messing's syntomic sheaf:- we are working on the site $(\text{Spf }W)_{\text{NILSYN}}$. This is the site whose objects are $W$-schemes on which $p$ is locally nilpotent, and the topology is the syntomic topology. For each $n\in\mathbb{N}$, we have some sheaves on this site:- \begin{equation*} \mathcal{O}_{n}:X\mapsto\mathcal{O}_{X_{n}}(X_{n}) \\ \mathcal{O}_{n}^{\text{cris}}:X\mapsto H_{\text{cris}}^{0}(X_{1}/W_{n}) \\ J_{n}:=\ker\left(\mathcal{O}_{n}^{\text{cris}}\rightarrow\mathcal{O}_{n}\right) \end{equation*} Define $\tilde{J}_{n}^{[r]}:=\text{im}\left(J_{n+r}^{[r]}\rightarrow J_{n}^{[r]}\right)$. For $r<p$, Fontaine and Messing (2.3 on page 191) show that $\phi(\tilde{J}_{n}^{[r]})\subset p^{r}\mathcal{O}_{n}^{\text{cris}}$ where $\phi:\mathcal{O}_{n}^{\text{cris}}\rightarrow \mathcal{O}_{n}^{\text{cris}}$ is the Frobenius. So we have a map $\frac{\phi}{p^{r}}:\tilde{J}_{n}^{[r]}\rightarrow\mathcal{O}_{n}^{\text{cris}}$. The syntomic sheaf is then defined to be \begin{equation*} s_{n}(r)_{X}:=\ker\left(1-\frac{\phi}{p^{r}}:\tilde{J}_{n}^{[r]}\rightarrow\mathcal{O}_{n}^{\text{cris}}\right). \end{equation*} Now, for $X$ a $W$-scheme, let $u:X_{n,\text{SYN}}\rightarrow X_{n,\text{ET}}$ be the morphism of sites. The claim is that we have an isomorphism \begin{equation*} Ru_{\ast}s_{n}(r)_{X}\simeq\mathscr{S}_{n}(r)_{X}[-1] \end{equation*} By Remark 1.8 on page 213 of Kato's ``On p-adic Vanishing Cycles (Application of ideas of Fontaine-Messing)'', it suffices to prove the claim when $X$ is quasi-projective. But then we can find a closed immersion $X\hookrightarrow Z$ into a smooth $W$-scheme endowed with a Frobenius (e.g. $Z$ could be projective space). In the notation of the question, we are writing $D_{n}=D_{X_{n}}(Z_{n})$ for the PD-envelope of $X_{n}$ in $Z_{n}$ (wrt the PD-structure on $pW_{n}$). In this notation then, Theorem 7.2 of Berthelot-Ogus gives an isomorphism \begin{equation*} \mathbb{R}u_{\ast}J_{n}^{[r]}\simeq\mathbb{J}_{X_{n},Z_{n}}^{[r]} \end{equation*} (I think we also need the result of this question

Crystalline cohomology via the syntomic site

to make this part legit).

Now we simply use the definition of $s_{n}(r)_{X}$ as $\ker\left(1-\frac{\phi}{p^{r}}\right)$ and the triangle associated to mapping fibres to see the claim. (Note, this is where the shift by $-1$ comes in).

Ok, maybe I've figured this out. Hopefully somebody can correct me if this is wrong. Also, I'd still like to know a reference that writes this out in detail, if anybody has one.

I'll change the notation from the question slightly: $k$ is a perfect field of characteristic $p>0$, and $W=W(k)$. First let's recall the definition of Fontaine-Messing's syntomic sheaf:- we are working on the site $(\text{Spf }W)_{\text{NILSYN}}$. This is the site whose objects are $W$-schemes on which $p$ is locally nilpotent, and the topology is the syntomic topology. For each $n\in\mathbb{N}$, we have some sheaves on this site:- \begin{equation*} \mathcal{O}_{n}:X\mapsto\mathcal{O}_{X_{n}}(X_{n}) \\ \mathcal{O}_{n}^{\text{cris}}:X\mapsto H_{\text{cris}}^{0}(X_{1}/W_{n}) \\ J_{n}:=\ker\left(\mathcal{O}_{n}^{\text{cris}}\rightarrow\mathcal{O}_{n}\right) \end{equation*} Define $\tilde{J}_{n}^{[r]}:=\text{im}\left(J_{n+r}^{[r]}\rightarrow J_{n}^{[r]}\right)$. For $r<p$, Fontaine and Messing (2.3 on page 191) show that $\phi(\tilde{J}_{n}^{[r]})\subset p^{r}\mathcal{O}_{n}^{\text{cris}}$ where $\phi:\mathcal{O}_{n}^{\text{cris}}\rightarrow \mathcal{O}_{n}^{\text{cris}}$ is the Frobenius. So we have a map $\frac{\phi}{p^{r}}:\tilde{J}_{n}^{[r]}\rightarrow\mathcal{O}_{n}^{\text{cris}}$. The syntomic sheaf is then defined to be \begin{equation*} s_{n}(r)_{X}:=\ker\left(1-\frac{\phi}{p^{r}}:\tilde{J}_{n}^{[r]}\rightarrow\mathcal{O}_{n}^{\text{cris}}\right). \end{equation*} Now, for $X$ a $W$-scheme, let $u:X_{n,\text{SYN}}\rightarrow X_{n,\text{ET}}$ be the morphism of sites. The claim is that we have an isomorphism \begin{equation*} Ru_{\ast}s_{n}(r)_{X}\simeq\mathscr{S}_{n}(r)_{X}[-1] \end{equation*} By Remark 1.8 on page 213 of Kato's ``On p-adic Vanishing Cycles (Application of ideas of Fontaine-Messing)'', it suffices to prove the claim when $X$ is quasi-projective. But then we can find a closed immersion $X\hookrightarrow Z$ into a smooth $W$-scheme endowed with a Frobenius (e.g. $Z$ could be projective space). In the notation of the question, we are writing $D_{n}=D_{X_{n}}(Z_{n})$ for the PD-envelope of $X_{n}$ in $Z_{n}$ (wrt the PD-structure on $pW_{n}$). In this notation then, Theorem 7.2 of Berthelot-Ogus gives an isomorphism \begin{equation*} \mathbb{R}u_{\ast}J_{n}^{[r]}\simeq\mathbb{J}_{X_{n},Z_{n}}^{[r]} \end{equation*} (I think we also need the result of this question

Crystalline cohomology via the syntomic site

to make this part legit).

Now we simply use the definition of $s_{n}(r)_{X}$ as $\ker\left(1-\frac{\phi}{p^{r}}\right)$ and the triangle associated to mapping fibres to see the claim. (Note, this is where the shift by $-1$ comes in).

Ok, maybe I've figured this out. Hopefully somebody can correct me if this is wrong. Also, I'd still like to know a reference that writes this out in detail, if anybody has one.

I'll change the notation from the question slightly: $k$ is a perfect field of characteristic $p>0$, and $W=W(k)$. First let's recall the definition of Fontaine-Messing's syntomic sheaf:- we are working on the site $(\text{Spf }W)_{\text{NILSYN}}$. This is the site whose objects are $W$-schemes on which $p$ is locally nilpotent, and the topology is the syntomic topology. For each $n\in\mathbb{N}$, we have some sheaves on this site:- \begin{equation*} \mathcal{O}_{n}:X\mapsto\mathcal{O}_{X_{n}}(X_{n}) \\ \mathcal{O}_{n}^{\text{cris}}:X\mapsto H_{\text{cris}}^{0}(X_{1}/W_{n}) \\ J_{n}:=\ker\left(\mathcal{O}_{n}^{\text{cris}}\rightarrow\mathcal{O}_{n}\right) \end{equation*} Define $\tilde{J}_{n}^{[r]}:=\text{im}\left(J_{n+r}^{[r]}\rightarrow J_{n}^{[r]}\right)$. For $r<p$, Fontaine and Messing (2.3 on page 191) show that $\phi(\tilde{J}_{n}^{[r]})\subset p^{r}\mathcal{O}_{n}^{\text{cris}}$ where $\phi:\mathcal{O}_{n}^{\text{cris}}\rightarrow \mathcal{O}_{n}^{\text{cris}}$ is the Frobenius. So we have a map $\frac{\phi}{p^{r}}:\tilde{J}_{n}^{[r]}\rightarrow\mathcal{O}_{n}^{\text{cris}}$. The syntomic sheaf is then defined to be \begin{equation*} s_{n}(r)_{x}:=\ker\left(1-\frac{\phi}{p^{r}}:\tilde{J}_{n}^{[r]}\rightarrow\mathcal{O}_{n}^{\text{cris}}\right). \end{equation*} Now, for $X$ a $W$-scheme, let $u:X_{n,\text{SYN}}\rightarrow X_{n,\text{ET}}$ be the morphism of sites. The claim is that we have an isomorphism \begin{equation*} Ru_{\ast}s_{n}(r)_{X}\simeq\mathscr{S}_{n}(r)_{X}[-1] \end{equation*} By Remark 1.8 on page 213 of Kato's ``On p-adic Vanishing Cycles (Application of ideas of Fontaine-Messing)'', it suffices to prove the claim when $X$ is quasi-projective. But then we can find a closed immersion $X\hookrightarrow Z$ into a smooth $W$-scheme endowed with a Frobenius (e.g. $Z$ could be projective space). In the notation of the question, we are writing $D_{n}=D_{X_{n}}(Z_{n})$ for the PD-envelope of $X_{n}$ in $Z_{n}$ (wrt the PD-structure on $pW_{n}$). In this notation then, Theorem 7.2 of Berthelot-Ogus gives an isomorphism \begin{equation*} \mathbb{R}u_{\ast}J_{n}^{[r]}\simeq\mathbb{J}_{X_{n},Z_{n}}^{[r]} \end{equation*} (I think we also need the result of this question

Crystalline cohomology via the syntomic site

to make this part legit).

Now we simply use the definition of $s_{n}(r)_{X}$ as $\ker\left(1-\frac{\phi}{p^{r}}\right)$ and the triangle associated to mapping fibres to see the claim. (Note, this is where the shift by $-1$ comes in).

Ok, maybe I've figured this out. Hopefully somebody can correct me if this is wrong. Also, I'd still like to know a reference that writes this out in detail, if anybody has one.

I'll change the notation from the question slightly: $k$ is a perfect field of characteristic $p>0$, and $W=W(k)$. First let's recall the definition of Fontaine-Messing's syntomic sheaf:- we are working on the site $(\text{Spf }W)_{\text{NILSYN}}$. This is the site whose objects are $W$-schemes on which $p$ is locally nilpotent, and the topology is the syntomic topology. For each $n\in\mathbb{N}$, we have some sheaves on this site:- \begin{equation*} \mathcal{O}_{n}:X\mapsto\mathcal{O}_{X_{n}}(X_{n}) \\ \mathcal{O}_{n}^{\text{cris}}:X\mapsto H_{\text{cris}}^{0}(X_{1}/W_{n}) \\ J_{n}:=\ker\left(\mathcal{O}_{n}^{\text{cris}}\rightarrow\mathcal{O}_{n}\right) \end{equation*} Define $\tilde{J}_{n}^{[r]}:=\text{im}\left(J_{n+r}^{[r]}\rightarrow J_{n}^{[r]}\right)$. For $r<p$, Fontaine and Messing (2.3 on page 191) show that $\phi(\tilde{J}_{n}^{[r]})\subset p^{r}\mathcal{O}_{n}^{\text{cris}}$ where $\phi:\mathcal{O}_{n}^{\text{cris}}\rightarrow \mathcal{O}_{n}^{\text{cris}}$ is the Frobenius. So we have a map $\frac{\phi}{p^{r}}:\tilde{J}_{n}^{[r]}\rightarrow\mathcal{O}_{n}^{\text{cris}}$. The syntomic sheaf is then defined to be \begin{equation*} s_{n}(r)_{X}:=\ker\left(1-\frac{\phi}{p^{r}}:\tilde{J}_{n}^{[r]}\rightarrow\mathcal{O}_{n}^{\text{cris}}\right). \end{equation*} Now, for $X$ a $W$-scheme, let $u:X_{n,\text{SYN}}\rightarrow X_{n,\text{ET}}$ be the morphism of sites. The claim is that we have an isomorphism \begin{equation*} Ru_{\ast}s_{n}(r)_{X}\simeq\mathscr{S}_{n}(r)_{X}[-1] \end{equation*} By Remark 1.8 on page 213 of Kato's ``On p-adic Vanishing Cycles (Application of ideas of Fontaine-Messing)'', it suffices to prove the claim when $X$ is quasi-projective. But then we can find a closed immersion $X\hookrightarrow Z$ into a smooth $W$-scheme endowed with a Frobenius (e.g. $Z$ could be projective space). In the notation of the question, we are writing $D_{n}=D_{X_{n}}(Z_{n})$ for the PD-envelope of $X_{n}$ in $Z_{n}$ (wrt the PD-structure on $pW_{n}$). In this notation then, Theorem 7.2 of Berthelot-Ogus gives an isomorphism \begin{equation*} \mathbb{R}u_{\ast}J_{n}^{[r]}\simeq\mathbb{J}_{X_{n},Z_{n}}^{[r]} \end{equation*} (I think we also need the result of this question

Crystalline cohomology via the syntomic site

to make this part legit).

Now we simply use the definition of $s_{n}(r)_{X}$ as $\ker\left(1-\frac{\phi}{p^{r}}\right)$ and the triangle associated to mapping fibres to see the claim. (Note, this is where the shift by $-1$ comes in).