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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
48 lines
1.9 KiB
TeX
48 lines
1.9 KiB
TeX
\section{Pipeline Stall Analysis}
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\label{sec:photon-stalls}
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\subsection{Stall Definition}
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\begin{definition}[Pipeline Stall]
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A stall occurs when a pipeline slot $s$ fails to increment its leading confidence
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counter for $\tau$ consecutive rounds, blocking the slot from finalization and preventing
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new transactions from entering.
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\end{definition}
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\begin{theorem}[Stall Duration]
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\label{thm:photon-stall}
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The expected stall duration (number of consecutive non-incrementing rounds) for a
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pipeline slot is:
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\[
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\mathbb{E}[\tau] = \frac{1}{q_1} + O\left(\frac{q_0}{q_1^2}\right)
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\]
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which is $O(1)$ when $q_1 > 1/2$.
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\end{theorem}
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\begin{proof}
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A non-incrementing round occurs when the sample does not produce a supermajority for
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the leading color. After population convergence (which takes $O(\log n)$ rounds), the
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probability of a supermajority round is $q_1 \approx \Phi(p, k, \alpha)$ where $p > 1 - \gamma$.
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The number of rounds until the next supermajority round follows a geometric distribution
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with parameter $q_1$, giving $\mathbb{E}[\tau] = 1/q_1$. For $q_1 = 0.63$ (standard
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parameters with $p = 0.8$, $k = 20$, $\alpha = 0.8$), the expected stall is $1.6$ rounds.
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\end{proof}
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\begin{corollary}[Pipeline Utilization]
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\label{cor:utilization}
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The expected pipeline utilization (fraction of rounds where at least one slot advances)
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is $1 - (1 - q_1)^m$, which exceeds 0.99 for $m \geq 7$ with standard parameters.
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\end{corollary}
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\begin{proof}
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A round is wasted only if none of the $m$ active slots produce a supermajority. Since
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the slots' supermajority events are independent (for independent conflict sets), the
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probability of a fully idle round is $(1 - q_1)^m = (0.37)^m < 0.01$ for $m \geq 7$.
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\end{proof}
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%% -----------------------------------------------------------------------
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%% 6. THROUGHPUT ANALYSIS
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%% -----------------------------------------------------------------------
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