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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
46 lines
1.8 KiB
TeX
46 lines
1.8 KiB
TeX
\section{Partition Recovery}
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\label{sec:nebula-partition}
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\subsection{Partition Model}
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We consider temporary network partitions where the network splits into components
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$P_1, P_2, \ldots, P_m$ for a duration $T_p$, then reconnects.
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\begin{proposition}[Partition Safety]
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\label{prop:partition}
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During a partition, Nebula halts finalization (does not finalize new transactions)
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within each component that contains fewer than $2n/3$ validators. Components with
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$\geq 2n/3$ validators continue to finalize safely.
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\end{proposition}
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\begin{proof}
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Within a component $P_i$ with $|P_i| < 2n/3$, the sampling within $P_i$ cannot
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guarantee honest supermajority (the $|P_i|$ validators include at most $|P_i| - f$
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honest validators, which may be less than $\alpha k$ for small $|P_i|$). The Nebula
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protocol detects insufficient responses and pauses finalization.
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A component with $|P_i| \geq 2n/3$ contains $\geq 2n/3 - f > n/3$ honest validators,
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sufficient for safety.
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\end{proof}
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\subsection{Post-Partition Reconciliation}
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\begin{proposition}[Reconciliation Time]
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\label{prop:reconciliation}
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After partition healing, the Nebula protocol reconciles the network in $O(\log n / \xi + T_p \cdot \xi)$
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rounds, where $T_p$ is the partition duration.
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\end{proposition}
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\begin{proof}
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After reconnection, bridge validators propagate the latest state from the larger
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partition to the smaller ones. The state divergence accumulated during $T_p$ rounds
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creates a ``preference gap'' of at most $T_p$ confidence counts. This gap is resolved
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by the standard convergence dynamics in $O(\log n / \xi)$ rounds, plus the time to
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process the backlog of $T_p$ rounds of unfinalized transactions.
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\end{proof}
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%% -----------------------------------------------------------------------
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%% 7. EVALUATION
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%% -----------------------------------------------------------------------
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