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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
34 lines
1.4 KiB
TeX
34 lines
1.4 KiB
TeX
\section{Throughput Scaling}
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\label{sec:prism-throughput}
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\begin{theorem}[Linear Scaling]
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\label{thm:prism-scaling}
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The aggregate throughput of Prism with $B$ spectrum bands is:
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\[
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\Theta_{\text{Prism}} = B \cdot \Theta_{\text{Photon}} \cdot \left(1 - O\left(\frac{\sqrt{\log B}}{B}\right)\right)
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\]
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which is $\Theta(B \cdot \Theta_{\text{Photon}})$ for moderate $B$.
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\end{theorem}
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\begin{proof}
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Each band runs an independent Photon pipeline with throughput $\Theta_{\text{Photon}}$.
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The aggregate throughput is the sum across bands, minus the overhead of the spectral
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combiner and the load imbalance.
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The combiner overhead is $O(1)$ per transaction (one comparison and one append), negligible.
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The load imbalance causes the busiest band to handle $T/B + O(\sqrt{T\log B / B})$
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transactions (Theorem~\ref{thm:prism-balance}). The slowest band determines the overall
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throughput, giving:
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\[
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\Theta_{\text{Prism}} = B \cdot \Theta_{\text{Photon}} \cdot \frac{T/B}{T/B + O(\sqrt{T\log B/B})}
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= B \cdot \Theta_{\text{Photon}} \cdot \left(1 - O\left(\frac{\sqrt{\log B}}{B}\right)\right)
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\]
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For $B = 16$: the correction factor is $1 - O(\sqrt{\ln 16}/16) \approx 1 - 0.12 = 0.88$,
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so scaling is 88\% efficient with 16 bands.
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\end{proof}
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%% -----------------------------------------------------------------------
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%% 7. EVALUATION
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%% -----------------------------------------------------------------------
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