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The seven engine specifications (lux-{photon,wave,nova,nebula,prism,ray,field}-protocol)
were fragmented short notes documenting layers of the same Quasar consensus stack.
This commit merges them into a single canonical reference, lux-consensus-engines.tex,
that uses one preamble, one set of theorems, one bibliography, and one composition
chapter showing how the engines layer into the LP-020 finality pipeline.
The seven originals are reduced to single-page breadcrumb stubs pointing at the
consolidated paper; their sections/ content is preserved on disk for git history.
- new: lux-consensus-engines/ (50 pages, 646KB PDF, ~2900 lines tex)
- breadcrumbs: each original keeps cover + title + stub paragraph naming its chapter
- composition: chapter 9 records the layered contract Photon < Wave < Nova < ...
- bibliography: consolidated from the seven original thebibliography blocks
55 lines
2.2 KiB
TeX
55 lines
2.2 KiB
TeX
\section{Safety Under Adaptive Thresholding}
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\label{sec:ray-safety}
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\begin{theorem}[Ray Safety]
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\label{thm:ray-ray-safety}
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The Ray protocol with adaptive threshold $\beta^*(r)$ achieves safety probability
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$\leq 2^{-128} + \delta$ where $\delta$ is the confidence parameter of the adversary
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estimator.
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\end{theorem}
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\begin{proof}
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Define event $A$: the adversary estimator correctly bounds the true fraction
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($\hat{\gamma}^+ \geq \gamma$). By Lemma~\ref{lem:ray-conservative}, $\Pr[A] \geq 1 - \delta$.
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Conditioned on $A$, the threshold $\beta^*$ is sufficient, and by Theorem~\ref{thm:ray-adaptive-safety},
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the safety violation probability is $\leq 2^{-128}$.
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Conditioned on $\overline{A}$ (estimator underestimates), the threshold may be too low,
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but the minimum threshold $\beta_{\min} \geq 3$ still provides base safety of
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$(q_0/q_1)^3 \leq 2^{-69}$ for typical parameters (generous upper bound).
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By the law of total probability:
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\[
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\Pr[\text{safety violation}] \leq (1 - \delta) \cdot 2^{-128} + \delta \cdot 2^{-69}
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\]
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For $\delta = 2^{-60}$, this is $\leq 2^{-128} + 2^{-129} < 2^{-127}$, negligible.
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\end{proof}
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\subsection{Light-Speed Bound}
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\begin{definition}[Light-Speed Confirmation]
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The minimum possible confirmation latency for a consensus protocol with round time
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$\Delta_r$ is $\beta_{\min} \cdot \Delta_r$, achievable only when the network has no
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adversarial interference and all validators agree immediately.
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\end{definition}
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\begin{proposition}[Ray Achieves Light-Speed]
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\label{prop:lightspeed}
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Under zero Byzantine faults, the Ray protocol with $\beta_{\min} = 3$ achieves
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confirmation in $3 \cdot \Delta_r$ time, matching the light-speed bound.
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\end{proposition}
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\begin{proof}
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With $\gamma = 0$, the confidence velocity $\nu_c = 1$ (supermajority in every round
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since all validators agree). The adaptive threshold becomes $\beta^* = \max(3, \lceil \lambda / 1 \rceil) = 3$
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since $\lambda = 128 \ln 2 / \ln(q_1/q_0)$ and with $\gamma = 0$, $q_1 \approx 1$ and
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$q_0 \approx 0$, making $\lambda$ very small (the ratio $q_1/q_0$ is astronomically large).
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Thus finalization occurs after 3 rounds.
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\end{proof}
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%% -----------------------------------------------------------------------
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%% 6. EVALUATION
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%% -----------------------------------------------------------------------
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