papers/lux-ray-protocol/sections/05-safety-under-adaptive-thresholding.tex

\section{Safety Under Adaptive Thresholding}

\label{sec:safety}

\begin{theorem}[Ray Safety]
\label{thm:ray-safety}
The Ray protocol with adaptive threshold $\beta^*(r)$ achieves safety probability
$\leq 2^{-128} + \delta$ where $\delta$ is the confidence parameter of the adversary
estimator.
\end{theorem}

\begin{proof}
Define event $A$: the adversary estimator correctly bounds the true fraction
($\hat{\gamma}^+ \geq \gamma$). By Lemma~\ref{lem:conservative}, $\Pr[A] \geq 1 - \delta$.

Conditioned on $A$, the threshold $\beta^*$ is sufficient, and by Theorem~\ref{thm:adaptive-safety},
the safety violation probability is $\leq 2^{-128}$.

Conditioned on $\overline{A}$ (estimator underestimates), the threshold may be too low,
but the minimum threshold $\beta_{\min} \geq 3$ still provides base safety of
$(q_0/q_1)^3 \leq 2^{-69}$ for typical parameters (generous upper bound).

By the law of total probability:
\[
\Pr[\text{safety violation}] \leq (1 - \delta) \cdot 2^{-128} + \delta \cdot 2^{-69}
\]
For $\delta = 2^{-60}$, this is $\leq 2^{-128} + 2^{-129} < 2^{-127}$, negligible.
\end{proof}

\subsection{Light-Speed Bound}

\begin{definition}[Light-Speed Confirmation]
The minimum possible confirmation latency for a consensus protocol with round time
$\Delta_r$ is $\beta_{\min} \cdot \Delta_r$, achievable only when the network has no
adversarial interference and all validators agree immediately.
\end{definition}

\begin{proposition}[Ray Achieves Light-Speed]
\label{prop:lightspeed}
Under zero Byzantine faults, the Ray protocol with $\beta_{\min} = 3$ achieves
confirmation in $3 \cdot \Delta_r$ time, matching the light-speed bound.
\end{proposition}

\begin{proof}
With $\gamma = 0$, the confidence velocity $\nu_c = 1$ (supermajority in every round
since all validators agree). The adaptive threshold becomes $\beta^* = \max(3, \lceil \lambda / 1 \rceil) = 3$
since $\lambda = 128 \ln 2 / \ln(q_1/q_0)$ and with $\gamma = 0$, $q_1 \approx 1$ and
$q_0 \approx 0$, making $\lambda$ very small (the ratio $q_1/q_0$ is astronomically large).
Thus finalization occurs after 3 rounds.
\end{proof}

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