papers/lux-flare-protocol/sections/06-comparison-with-wave.tex

\section{Comparison with Wave}

\label{sec:comparison}

\subsection{Finality Semantics}

Wave provides \emph{soft finality}: a transaction is considered final when its confidence
counter exceeds $\beta_1$, but the counter can theoretically be surpassed by an adversary
(with negligible probability). Flare provides \emph{certificate finality}: once a
certificate is issued, it serves as a transferable, verifiable proof of finality.

\begin{proposition}[Finality Comparison]
\label{prop:finality}
Let $T_W$ and $T_F$ be the expected finalization times for Wave and Flare respectively.
Then:
\[
T_W \leq T_F \leq T_W + O\left(\frac{\beta_2 - \beta_1}{q_1}\right)
\]
where $q_1$ is the supermajority probability for the winning color.
\end{proposition}

\begin{proof}
Wave requires $\beta_1$ total supermajority rounds; Flare requires $\beta_2$ consecutive
rounds. Since consecutive rounds occur at rate $q_1$ per round (after population
convergence), the expected time for $\beta_2$ consecutive is $\sum_{i=1}^{\beta_2} 1/q_1^i
\approx \beta_2/q_1$ when $q_1$ is close to 1. The difference $T_F - T_W$ depends on
the gap between the consecutive requirement and the cumulative requirement, bounded by
$(\beta_2 - \beta_1)/q_1$ when $q_1 > 0.6$.
\end{proof}

\subsection{Latency Analysis}

\begin{table}[h]
\centering
\begin{tabular}{@{}llll@{}}
\toprule
\textbf{Metric} & \textbf{Slush} & \textbf{Wave} & \textbf{Flare} \\
\midrule
Finality type & None & Soft & Certificate \\
Memory & None & Cumulative & Consecutive \\
Expected rounds & $O(\log n)$ & $O(\log n + \beta_1)$ & $O(\log n + \beta_2/q_1)$ \\
Safety probability & $0$ & $1 - (q_0/q_1)^{\beta_1}$ & $1 - e^{-\Omega(k\beta_2)}$ \\
Transferable proof & No & No & Yes \\
\bottomrule
\end{tabular}
\caption{Comparison of metastable consensus protocols}
\label{tab:comparison}
\end{table}

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