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zeekay
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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
35 lines
1.4 KiB
TeX
35 lines
1.4 KiB
TeX
\section{Sparse Network Model}
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\label{sec:nebula-model}
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\begin{definition}[Random Geometric Graph]
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The network is modeled as a random geometric graph $G(n, r)$: $n$ validators are
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placed uniformly at random in $[0, 1]^2$, and validators $v_i, v_j$ can communicate
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if $\|v_i - v_j\| \leq r$. The graph is connected with high probability when
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$r \geq c\sqrt{\log n / n}$ for a constant $c > 0$.
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\end{definition}
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\begin{definition}[Neighborhood]
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The neighborhood of validator $v_i$ is $\mathcal{N}(v_i) = \{v_j : \|v_i - v_j\| \leq r\}$.
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The expected neighborhood size is $|\mathcal{N}(v_i)| \approx n \pi r^2$.
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\end{definition}
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\begin{definition}[Bridge Validator]
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A validator $v$ is a bridge between neighborhoods $\mathcal{N}(v_i)$ and $\mathcal{N}(v_j)$
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if $v \in \mathcal{N}(v_i) \cap \mathcal{N}(v_j)$. Bridges propagate consensus information
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across the network.
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\end{definition}
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\begin{definition}[Expansion Factor]
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The expansion factor $\xi$ of the network is:
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\[
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\xi = \min_{S \subseteq V, |S| \leq n/2} \frac{|\partial S|}{|S|}
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\]
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where $\partial S = \{v \notin S : \exists u \in S, v \in \mathcal{N}(u)\}$ is the
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boundary of $S$. For random geometric graphs with $r = c\sqrt{\log n / n}$, $\xi = \Theta(1)$.
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\end{definition}
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%% -----------------------------------------------------------------------
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%% 3. THE NEBULA PROTOCOL
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%% -----------------------------------------------------------------------
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