Minor corrections (gavofyork#61)

* Wrong formatting for H

* Safrole: Fix k_b to k_e

* Recents: Work Package hash is subscript h not p

* Statistics: Use e for epoch index, not m.

* Serialization: Include segment root in specification

text/merklization.tex

@@ -175,15 +175,15 @@ \subsubsection{Constant-Depth Tree}
 
 As above, we may assume a default value for $H$ of the Blake 2b hash function, $\mathcal{H}$.
 
-In all cases, a constancy preprocessor function $C$ is applied which hashes all data items with a fixed prefix and then pads them to the next power of two with the zero hash:
+In all cases, a constancy preprocessor function $C$ is applied which hashes all data items with a fixed prefix and then pads them to the next power of two with the zero hash $\mathbb{H}_0$:
 \begin{equation}
   C\colon\left\{\begin{aligned}
     (\seq{\Y}, \Y \to \H) &\to \seq{\H}\\
     (\mathbf{v}, H) &\mapsto \mathbf{v}' \ \where \left\{\; \begin{aligned}
       |\mathbf{v}'| &= 2^{\ceil{\log_2(|\mathbf{v}|)}}\\
       \mathbf{v}'_i &= \begin{cases}
         H(\token{\$leaf}\frown\mathbf{v}_i) &\when i < |\mathbf{v}|\\
-        H_0 &\otherwise \\
+        \mathbb{H}_0 &\otherwise \\
       \end{cases}
     \end{aligned}\right.
   \end{aligned}\right.

text/overview.tex

@@ -66,7 +66,7 @@ \subsubsection{State Transition Dependency Graph}
     \mathbf{C}
   \end{rcases*} &\prec (\xtassurances, \rho', \delta^\dagger, \chi, \iota, \varphi) \\
   \alpha' &\prec (\xtguarantees, \varphi', \alpha) \\
-  \pi' &\prec (\xtguarantees, \xtpreimages, \xtassurances, \xttickets, \tau, \tau', \pi)
+  \pi' &\prec (\xtguarantees, \xtpreimages, \xtassurances, \xttickets, \tau, \tau', \pi, \mathbf{H})
 \end{align}
 
 The only synchronous entanglements are visible through the intermediate components superscripted with a dagger and defined in equations \ref{eq:betadagger}, \ref{eq:deltadagger} and \ref{eq:rhoddagger}. The latter two mark a merge and join in the dependency graph and, concretely, imply that the preimage lookup extrinsic must be folded into state before the availability extrinsic may be fully processed and accumulation of work happen.

text/ratings.tex

@@ -32,9 +32,9 @@ \section{Validator Activity Statistics}\label{sec:bookkeeping}
 
 The objective statistics are updated in line with their description, formally:
 \begin{align}
-    \using m =\; &\ffrac{\tau}{\mathsf{E}} \ ,\quad m' = \ffrac{\tau'}{\mathsf{E}}\\
+    \using e =\; &\ffrac{\tau}{\mathsf{E}} \ ,\quad e' = \ffrac{\tau'}{\mathsf{E}}\\
     (\mathbf{a}, \pi'_1) \equiv\;&\begin{cases}
-        (\pi_0, \pi_1) &\when m' = m \\
+        (\pi_0, \pi_1) &\when e' = e \\
         (\sq{\tuple{0, \dots, [0, \dots]}, \dots}, \pi_0) &\otherwise
     \end{cases}\\
     \forall v \in \N_\mathsf{V} :&\; \left\{\begin{aligned}

text/recent_history.tex

@@ -18,7 +18,7 @@ \section{Recent History}\label{sec:recenthistory}
     \using r &= \mathcal{M}_B(\orderby{s}{\se_4(s)\frown\se(h) \mid (s, h) \in \mathbf{C}}, \mathcal{H}_K) \\
     \using \mathbf{b} &= \mathcal{A}(\text{last}(\sq{\sq{}} \concat \sq{x_\mathbf{b} \mid x \orderedin \beta}), r)\\
     \using n &= \tup{
-      \is{\mathbf{p}}{[((g_w)_s)_p \mid g \orderedin \xtguarantees]}\ts
+      \is{\mathbf{p}}{[((g_w)_s)_h \mid g \orderedin \xtguarantees]}\ts
       \is{h}{\mathcal{H}(\mathbf{H})}\ts\mathbf{b}\ts\is{s}{\H^0}
       }
     \end{aligned}

text/safrole.tex

@@ -99,7 +99,7 @@ \subsection{Key Rotation}\label{sec:keyrotation}
   \end{cases} \\
   \nonumber \where z &= \mathcal{O}([k_b \mid k \orderedin \gamma'_\mathbf{k}]) \\
   \label{eq:blacklistfilter} \Phi(\mathbf{k}) &\equiv \sq{
-    \begin{rcases}[0, 0, \dots] &\when k_b \in \offenders' \\ k &\otherwise \end{rcases}
+    \begin{rcases}[0, 0, \dots] &\when k_e \in \offenders' \\ k &\otherwise \end{rcases}
     \,\middle\mid\, k \orderedin \mathbf{k}
   }
 \end{align}

text/serialization.tex

@@ -133,7 +133,7 @@ \subsection{Block Serialization}
   \se(\mathbf{H}) &= \se_U(\mathbf{H})\concat\se(\mathbf{H}_s) \\
   \se_U(\mathbf{H}) &= \se(\mathbf{H}_p,\mathbf{H}_r,\mathbf{H}_x)\concat\se_4(\mathbf{H}_t)\concat\se(\maybe{\mathbf{H}_e},\maybe{\mathbf{H}_w},\var{\mathbf{H}_j},\var{\mathbf{H}_o},\se_2(\mathbf{H}_i),\mathbf{H}_v)\\
   \se(x \in \mathbb{X}) &\equiv \se(x_a, x_s, x_b, x_l)\concat\se_4(x_t)\concat\se(\maybe{x_p})\\
-  \se(x \in \mathbb{S}) &\equiv \se(x_h) \concat \se_4(x_l)\concat\se(x_u) \\
+  \se(x \in \mathbb{S}) &\equiv \se(x_h) \concat \se_4(x_l)\concat\se(x_u, x_e) \\
   \se(x \in \mathbb{L}) &\equiv \se_4(x_s)\concat\se(x_c, x_l)\concat\se_8(x_g)\concat\se(O(x_o))\\
   \se(x \in \mathbb{W}) &\equiv \se(x_a, x_c, \var{x_o}, x_x, x_s, \var{x_r}) \\
   \se(x \in \mathbb{P}) &\equiv \se(\var{x_\mathbf{j}}, \se_4(x_h), x_c, \var{x_\mathbf{p}}, x_\mathbf{x}, \var{x_\mathbf{i}}) \\