mirror of
https://github.com/lux-apps/papers.git
synced 2026-10-30 18:15:21 +00:00
zeekay
1ecc25e9cf
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
69 lines
2.3 KiB
TeX
69 lines
2.3 KiB
TeX
\section{Spectral Decomposition}
|
|
|
|
\label{sec:prism-partition}
|
|
|
|
\subsection{Band Assignment}
|
|
|
|
\begin{definition}[Spectrum Band]
|
|
Given $B$ spectrum bands $\{b_0, b_1, \ldots, b_{B-1}\}$, a transaction $t$ with
|
|
conflict set identifier $\text{csid}(t)$ is assigned to band:
|
|
\[
|
|
\text{band}(t) = H(\text{csid}(t)) \mod B
|
|
\]
|
|
where $H$ is a cryptographic hash function (SHA-256).
|
|
\end{definition}
|
|
|
|
\begin{definition}[Conflict Set Identifier]
|
|
The conflict set identifier $\text{csid}(t)$ is determined by the consumed UTXO(s)
|
|
of transaction $t$. All transactions consuming the same UTXO share the same $\text{csid}$.
|
|
Non-conflicting transactions have distinct $\text{csid}$ values.
|
|
\end{definition}
|
|
|
|
\begin{theorem}[Conflict Locality]
|
|
\label{thm:prism-locality}
|
|
If $t_1 \in \mathcal{C}(t_2)$ (the transactions conflict), then $\text{band}(t_1) = \text{band}(t_2)$.
|
|
Conflicting transactions are always processed in the same spectrum band.
|
|
\end{theorem}
|
|
|
|
\begin{proof}
|
|
By definition, $t_1 \in \mathcal{C}(t_2)$ implies $\text{csid}(t_1) = \text{csid}(t_2)$
|
|
(they consume the same UTXO). Therefore:
|
|
\[
|
|
\text{band}(t_1) = H(\text{csid}(t_1)) \mod B = H(\text{csid}(t_2)) \mod B = \text{band}(t_2)
|
|
\]
|
|
\end{proof}
|
|
|
|
\subsection{Load Balance}
|
|
|
|
\begin{theorem}[Load Balance]
|
|
\label{thm:prism-balance}
|
|
For $T$ transactions with uniformly distributed $\text{csid}$ values and $B$ bands,
|
|
the load on band $b$ satisfies:
|
|
\[
|
|
\Pr\left[|L_b - T/B| > \sqrt{\frac{2T \ln B}{B}}\right] < \frac{2}{B}
|
|
\]
|
|
where $L_b$ is the number of transactions assigned to band $b$.
|
|
\end{theorem}
|
|
|
|
\begin{proof}
|
|
Each transaction is independently assigned to band $b$ with probability $1/B$ (by
|
|
the uniformity of $H$). The load $L_b \sim \text{Binomial}(T, 1/B)$ with mean $T/B$
|
|
and variance $T(B-1)/B^2$.
|
|
|
|
By the Chernoff bound:
|
|
\[
|
|
\Pr[|L_b - T/B| > \delta] \leq 2\exp\left(-\frac{\delta^2 B}{2T}\right)
|
|
\]
|
|
Setting $\delta = \sqrt{2T \ln B / B}$ gives $\Pr[|L_b - T/B| > \delta] < 2/B^2$.
|
|
A union bound over $B$ bands:
|
|
\[
|
|
\Pr[\exists b : |L_b - T/B| > \delta] < B \cdot 2/B^2 = 2/B
|
|
\]
|
|
For $B = 16$: the maximum deviation from mean load is at most $\sqrt{2T \ln 16 / 16} \approx 0.66\sqrt{T}$
|
|
with probability $> 87.5\%$.
|
|
\end{proof}
|
|
|
|
%% -----------------------------------------------------------------------
|
|
%% 3. THE PRISM PROTOCOL
|
|
%% -----------------------------------------------------------------------
|