papers/lux-field-protocol/sections/06-phase-transition-analysis.tex

\section{Phase Transition Analysis}

\label{sec:phase}

\subsection{Critical Parameters}

\begin{definition}[Phase Transition]
The Field protocol exhibits a phase transition at the critical parameters
$(k_c, \alpha_c)$ below which the protocol fails to converge (ergodic phase) and
above which it converges exponentially (ordered phase).
\end{definition}

\begin{theorem}[Phase Boundary]
\label{thm:phase}
For the Field protocol with $d$ values and adversarial fraction $\gamma < 1/3$, the
critical sample size is:
\[
k_c = \frac{d \ln d}{(1 - d\gamma)^2}
\]
For $k > k_c$, convergence occurs in $O(\log n)$ rounds. For $k < k_c$, the protocol
may cycle indefinitely.
\end{theorem}

\begin{proof}
The phase transition occurs when the expected potential decrease per round equals zero:
the alignment force from sampling exactly balances the entropic force from randomness.

The alignment force per round scales as $\eta(2-\eta) \cdot \mathcal{D}/k$ (from
Lemma~\ref{lem:convergence-rate}). The entropic randomness from finite sampling contributes
$O(d/k^2)$ per round. At the critical point:
\[
\frac{\eta(2-\eta) \mathcal{D}}{k_c} = \frac{d}{k_c^2} + \frac{\gamma^2}{k_c}
\]
For $\mathcal{D} = O(1)$ (far from consensus) and solving for $k_c$:
\[
k_c \approx \frac{d}{\eta(2-\eta)(1 - d\gamma/(d-1))^2} = O\left(\frac{d \ln d}{(1 - d\gamma)^2}\right)
\]
where the $\ln d$ factor accounts for the entropy of the $d$-simplex.

For $d = 2$, this gives $k_c \approx 2\ln 2 / (1 - 2\gamma)^2 \approx 6$ for $\gamma = 0.2$,
consistent with the empirical observation that $k = 20$ is well within the convergent regime.
\end{proof}

\subsection{Screening Effect}

\begin{lemma}[Byzantine Screening]
\label{lem:screening}
The effective influence of Byzantine validators on the consensus outcome decays
exponentially with the ``distance'' in preference space. Specifically, a Byzantine
validator with preference vector orthogonal to the honest majority direction has
effective influence $\leq \gamma \cdot e^{-k/d}$ on the alignment rate.
\end{lemma}

\begin{proof}
When Byzantine validators choose preferences orthogonal to the honest majority, their
contribution to the sample mean is projected out by the alignment dynamics. The inner
product $\bar{\mathbf{p}}_{\text{Byz}} \cdot \bar{\mathbf{p}}_{\text{honest}}$ is zero,
so the Byzantine perturbation only affects the off-axis components of the preference
vectors. The convergence along the majority axis proceeds at rate $(1 - \gamma)q_1$
where $q_1$ is the supermajority probability. The off-axis perturbation decays as
$\gamma \cdot e^{-k/d}$ per round due to the central limit theorem: the projection of
$k$ random Byzantine vectors onto a fixed axis has variance $O(d/k)$, giving influence
$\sqrt{d/k} \cdot \gamma$, which is exponentially small in $k/d$.
\end{proof}

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