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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
39 lines
1.6 KiB
TeX
39 lines
1.6 KiB
TeX
\section{Stake-Weighted Sampling}
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\label{sec:field-stake}
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In the electromagnetic analogy, particles with larger charge create stronger fields
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and exert more influence on their neighbors. We extend this to stake-weighted consensus.
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\begin{definition}[Field Strength]
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Validator $i$ with stake $w_i$ has field strength $\phi_i = w_i / W$ where $W = \sum_j w_j$.
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The weighted sample mean becomes:
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\[
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\bar{\mathbf{p}}_S = \frac{\sum_{j \in S} w_j \mathbf{p}_j}{\sum_{j \in S} w_j}
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\]
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\end{definition}
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\begin{theorem}[Stake-Weighted Safety]
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\label{thm:field-stake}
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The Field protocol with stake-weighted sampling maintains safety under the condition
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$\sum_{i \in B} w_i < W/3$, i.e., the adversary controls less than $1/3$ of total stake.
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The convergence rate becomes $O(\log(W/w_{\min}))$ where $w_{\min}$ is the minimum
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honest stake.
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\end{theorem}
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\begin{proof}
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The proof follows Theorem~\ref{thm:field-potential} with the replacement of uniform averages
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by stake-weighted averages. The key observation is that the potential function
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$\Phi(\mathbf{P}) = -\frac{1}{W^2}\|\sum_i w_i \mathbf{p}_i\|^2$ remains a Lyapunov
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function under the weighted dynamics. The decrease per round is:
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\[
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\mathbb{E}[\Delta \Phi] \leq -\frac{\eta(2-\eta)}{W} \sum_{i \in H} w_i \|\mathbf{p}_i - \bar{\mathbf{p}}_w\|^2
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\]
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where $\bar{\mathbf{p}}_w$ is the stake-weighted population mean. The convergence time
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is $O(\frac{W}{w_{\min}} \log(W/\epsilon))$, which reduces to $O(\log n)$ for uniform stake.
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\end{proof}
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%% -----------------------------------------------------------------------
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%% 6. PHASE TRANSITIONS
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%% -----------------------------------------------------------------------
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