papers/lux-prism-protocol/sections/03-the-prism-protocol.tex

\section{The Prism Protocol}

\label{sec:protocol}

\subsection{Per-Band Consensus}

Each spectrum band runs an independent Photon pipeline. The bands share the same
validator set but process different transaction subsets.

\begin{algorithm}[H]
\caption{Prism Protocol --- Multi-Spectrum Parallel Consensus}
\label{alg:prism-core}
\begin{algorithmic}[1]
\Require Validator $v$, $B$ spectrum bands, each with pipeline depth $m$
\Require Parameters: $k$, $\alpha$, $\beta$ (per-band Photon parameters)
\State Initialize $B$ independent Photon pipelines $P_0, P_1, \ldots, P_{B-1}$
\Loop
    \State \Comment{Assign incoming transactions to bands}
    \For{each new transaction $t$}
        \State $b \gets H(\text{csid}(t)) \mod B$
        \State Insert $t$ into pipeline $P_b$
    \EndFor
    \State \Comment{Run one round of all band pipelines in parallel}
    \State $S \gets \text{RandomSample}(V \setminus \{v\}, k)$
    \For{each band $b$ \textbf{in parallel}}
        \State Execute one round of $P_b$.\textsc{PhotonRound}($S$)
    \EndFor
    \State \Comment{Collect finalized transactions from all bands}
    \For{each band $b$}
        \For{each transaction $t$ finalized by $P_b$ this round}
            \State Submit $t$ to spectral combiner
        \EndFor
    \EndFor
\EndLoop
\end{algorithmic}
\end{algorithm}

\subsection{Shared Sampling}

All bands share the same sample $S$ in each round. This is critical for efficiency:
the validator queries each peer once per round, carrying preferences for all bands.
The response message includes preferences for all active pipeline slots across all bands.

\begin{lemma}[Shared Sample Independence]
\label{lem:shared-sample}
Using the same sample $S$ for all bands does not compromise safety, because the
confidence counters for different bands depend on different transaction preferences,
which are independent for non-conflicting transactions.
\end{lemma}

\begin{proof}
Band $b_1$'s confidence counter for transaction $t_1$ depends on the responses
$\{\text{col}(u, t_1) : u \in S\}$. Band $b_2$'s counter for $t_2$ depends on
$\{\text{col}(u, t_2) : u \in S\}$. Since $t_1$ and $t_2$ are in different bands,
they are non-conflicting (Theorem~\ref{thm:locality}), so the responses for $t_1$ and
$t_2$ are independent random variables even though they come from the same sample $S$.
The joint distribution factors: $\Pr[\text{col}(u, t_1) = c_1, \text{col}(u, t_2) = c_2] =
\Pr[\text{col}(u, t_1) = c_1] \cdot \Pr[\text{col}(u, t_2) = c_2]$.
\end{proof}

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