apply @plebhash review

#1

pplns-with-job-declaration.aux

@@ -18,12 +18,14 @@
 \citation{ocean}
 \citation{extension}
 \@writefile{toc}{\contentsline {section}{\numberline {1}Introduction}{1}{section.1}\protected@file@percent }
-\@writefile{toc}{\contentsline {section}{\numberline {2}Slices and windows}{1}{section.2}\protected@file@percent }
+\@writefile{toc}{\contentsline {section}{\numberline {2}Slices}{1}{section.2}\protected@file@percent }
 \@writefile{toc}{\contentsline {section}{\numberline {3}Scores}{2}{section.3}\protected@file@percent }
 \citation{ocean}
 \@writefile{toc}{\contentsline {subsection}{\numberline {3.1}Difficulty-based score}{3}{subsection.3.1}\protected@file@percent }
 \@writefile{toc}{\contentsline {subsection}{\numberline {3.2}Fee-based score}{3}{subsection.3.2}\protected@file@percent }
-\@writefile{toc}{\contentsline {subsection}{\numberline {3.3}Payout for each share}{4}{subsection.3.3}\protected@file@percent }
+\newlabel{properties-diff-based-score}{{3.2.1}{4}{Fee-based score}{remark.3.2.1}{}}
+\newlabel{addstoone}{{5}{4}{Fee-based score}{Item.7}{}}
+\@writefile{toc}{\contentsline {subsection}{\numberline {3.3}Payout for each share}{5}{subsection.3.3}\protected@file@percent }
 \@writefile{toc}{\contentsline {section}{\numberline {4}Shares' accountability}{5}{section.4}\protected@file@percent }
 \citation{demand}
 \citation{sv2}

pplns-with-job-declaration.log

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+Underfull \hbox (badness 10000) in paragraph at lines 181--183
 []\OT1/cmr/m/n/10.95 Extension on share ac-count-ing []$\OT1/cmtt/m/n/10.95 htt
 ps : / / github . com / demand-[]open-[]source /
  []
 
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pplns-with-job-declaration.out

@@ -1,5 +1,5 @@
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pplns-with-job-declaration.tex

@@ -13,6 +13,8 @@
 \author{Lorenzo Bonazzi, Filippo Merli}
 \date{August 28, 2024} % Activate to display a given date or no date (if empty),
          % otherwise the current date is printed 
+
+\newtheorem{remark}{Remark}[subsection]
 
 \begin{document}
 \maketitle
@@ -24,8 +26,10 @@ \section{Introduction}
 
 We define a fair payment schema that splits the share payout in two parts. The first comes from the coinbase and pays accordingly to the difficulty score of the share, in the same fashion as PPLNS. The second comes from the block fee and pays accordigly to share difficulty score and the share fee score in such a way that share with the highest fee in its group of shares (slice, see below) is paid at least $F_S/n$, where $F_S$ is the part of the block fee reserved to that group of shares $S$ and $n$ is the number of shares in $S$.
 
-\section{Slices and windows}
-In order to pay shares in a fair way with respect to fees, they must be scored somehow and the payout of the block fee must be redistributed accodingly to this score. Doing so, it is necessary to divide them in groups and compare scores only for shares belonging to the same group. Indeed, does not make any sense compare two shares taken randomly in the PPLNS window, because the mempool maximum extractable fees (MMEF) may change consistently. Indeed, it is difficult even to compare shares produced between a block and the following (we call this a \emph{mining round}).
+In the following, we will call the \emph{window} of PPLNS the shares that are paid once that a block is found by some pool's miner. So, the windows size is $N$, where $N$ is symbol used for the acronym "Pay Per Last $N$ Shares". We will call also \emph{mining round} the number of shares produced in the time between a Bitcoin block and the following one or, equivalently, all the shares \emph{with the same prev-hash}.
+
+\section{Slices}
+In order to pay shares in a fair way with respect to fees, they must be scored somehow and the payout of the block fee must be redistributed accodingly to this score. Doing so, it is necessary to divide them in groups and compare scores only for shares belonging to the same group. Indeed, does not make any sense compare two shares taken randomly in the PPLNS window, because the mempool maximum extractable fees (MMEF) may change consistently. Indeed, it is difficult even to compare shares produced in a single mining round.
 
 We call a group of shares whose fees can be scored (and the scores can be compared) a \emph{slice}, which is a portion of the stream of shares. As we seen in the previous paragraph, a slice must not cross two mining rounds. With other words, all shares in one slice must have the same prevhash. It is needed an introduction before giving the precise definition of a slice.
 
@@ -54,16 +58,19 @@ \subsection{Difficulty-based score}
 If the window contains $N$ shares, then for the $i$-th share $s_i$ we define its score
 \[score_d(s_i) = \frac{d_i}{\sum_{j=1}^Nd_j},\]
 where $d_j$ is the difficulty of the share $s_j$ (for example, \cite{ocean} suggest to adopt $N$ such that the sum at the denominator eight times the bitcoin difficulty).\newline
-\textbf{Remark.} 
+\begin{remark}
+	Note that
+	\[\sum_{j=1}^N score_d(s_i) = 1\] and that this score depends only on difficulty.
+\end{remark}
 \[\sum_{j=1}^N score_d(s_i) = 1\] and that this score depends only on difficulty.
 \subsection{Fee-based score}
 Suppose we want to score the $i$-th share $s_i$, and suppose that this share belongs to a slice $S$ that contains all the shares from the $k_1$-th to the $k_2$-th.
 For the $i$-th share, call $\bar d_i = score_d(s_i)$ its score difficulty (calculated as above) and $f_i$ the fees of the share (which depends on the transactions chosen in the template). For this share, define the score relative to the fee
 \[
 score_f(s_i) = \frac{\bar d_if_i}{\sum_{j=k_1}^{k_2}\bar d_j f_j}.
 \]
-
-\textbf{Remarks.} \begin{enumerate} 
+\begin{remark}\label{properties-diff-based-score}
+\begin{enumerate} 
 \item As remaked earlier, the dependence on difficulty is intentional. Otherwise, if there are two shares with the same fee, the one with lower difficulty is paid the same as the other, while requiring less computational effort to be produced.
 \item Suppose that the slice $S$ contains $n$ shares, all of the same difficulty score. Then, if $f_i = f_{max}$ is the max fee of every share in the slice $S$, then the payout for this share is at least $score_f(s_i) \ge 1/n$. This sentence is proven by contraddiction. So, assume that 
 \[score_f(s_i) < 1/n.\]
@@ -75,16 +82,18 @@ \subsection{Fee-based score}
 which is impossible.
 \item if there is $c>0$ such that $f_i = cf_j$ for some $s_i, s_j$ two shares in the slice $S$, then $payout_f(s_i) = c \cdot payout_f(s_j)$. This means that if a miner becomes $c$ times more capable of finding a more profitable template, his shares will be paid proportionally.
 \item similarly, if $\bar d_i = c \bar d_j$, then $payout_f(s_i) = c \cdot payout_f(s_j)$ this similarly represents the fact that a miner is paid proportionally with respect his hashpower.
-\item similarly to difficulty score, if the slice $S$ contains shares from $k_1$-th to $k_2$-th, we have \[\sum_{i=k_1}^{k_2} score_f(s_i) = 1. \]
+\item similarly to difficulty score, if the slice $S$ contains shares from $k_1$-th to $k_2$-th, we have \[\sum_{i=k_1}^{k_2} score_f(s_i) = 1. \] \label{addstoone}
 \end{enumerate}
+
+\end{remark}
 \subsection{Payout for each share}
 Suppose that we want to pay the $i$-th share. This share will belong to a slice $S$, to which is reserved a portion of block fees subsidy $F_S$. Then the payout is
 
 \[payout(s_i) = r\cdot score_d(s_i) + F_S \cdot score_f(s_i).\]
  The redistribution of the coinbase reward necessarly depend only on $d_i$, because it is guaranteed even in empty blocks, so it is independent from work selection.
-We can see that the payouts of all shares add up to the right amount. Suppose that the slices are $S_1, \dots, S_t$, and suppose that there are $1=k_1\le k_2\le \dots \le k_t$ such that the $m$-th slice contains shares $s_{k_m} \dots s_{k_{m+1}}$. Note also that $F_S = F/t$. So:
+We can see that the payouts of all shares add up to the right amount. Suppose that $N$ is the size of PPLNS window and that the slices are $S_1, \dots, S_t$, and suppose that there are $k_0=1<k_1< k_2< \dots < k_t=N$ such that the $m$-th slice contains shares $s_{k_{m-1}+1}, \dots, s_{k_{m}}$. Note also that $F_S = F/t$. So, by point \ref{addstoone} of Remark5 \ref{properties-diff-based-score}:
 \begin{align*} 
-\sum_{i=1} ^N payout(s_i) &= r\sum_{i=1}^N score_d(s_i) +F_{S}\left(\sum_{j=k_1}^{k_2} score_f(s_j)+\dots +\sum_{j=k_{t-1}}^{k_t} score_f(s_j)\right) \\
+\sum_{i=1} ^N payout(s_i) &= r\sum_{i=1}^N score_d(s_i) +F_{S}\left(\sum_{j=k_0}^{k_1} score_f(s_j)+\dots +\sum_{j=k_{t-1}+1}^{k_t} score_f(s_j)\right) \\
 &= r +F_S\cdot t\\ &= r+F \\ &= R
 \end{align*}