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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
59 lines
2.4 KiB
TeX
59 lines
2.4 KiB
TeX
\section{Safety Analysis}
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\label{sec:photon-safety}
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\subsection{Independence of Confidence Counters}
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\begin{theorem}[Pipelined Safety]
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\label{thm:photon-pipeline-safety}
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For $m$ independent transactions (no shared conflict sets) in the Photon pipeline, the
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probability that any transaction experiences a safety violation is at most:
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\[
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\Pr[\text{any violation}] \leq m \cdot (q_0/q_1)^\beta
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\]
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where $q_0, q_1$ are the supermajority probabilities for minority and majority colors.
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\end{theorem}
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\begin{proof}
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Consider two transactions $t_i, t_j$ in the pipeline with independent conflict sets.
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Their confidence counters $D_{i,\cdot}$ and $D_{j,\cdot}$ evolve based on the same
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sample $S$ but independent color queries.
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For transaction $t_i$, the responses from $S$ about $t_i$'s conflict are drawn from
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the population distribution for $t_i$'s conflict set. Similarly for $t_j$. Since the
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conflict sets are disjoint, the responses are independent random variables (the same
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validator may respond differently for $t_i$ and $t_j$).
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Therefore, the confidence counter evolution for $t_i$ is statistically independent of
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$t_j$'s. The safety violation probability for each transaction is $(q_0/q_1)^\beta$
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(from the Wave analysis), and by a union bound over $m$ transactions:
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\[
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\Pr[\text{any violation}] \leq m \cdot (q_0/q_1)^\beta
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\]
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For $m = 100$ and $(q_0/q_1)^\beta < 2^{-128}$: $100 \cdot 2^{-128} < 2^{-121}$,
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still negligible.
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\end{proof}
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\subsection{Conflicting Pipeline Entries}
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\begin{lemma}[Same-Conflict Serialization]
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\label{lem:photon-serialization}
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If two transactions $t_i, t_j$ in the pipeline belong to the same conflict set
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$\mathcal{C}$, the Photon protocol serializes their finalization: $t_j$ cannot finalize
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until $t_i$'s conflict is resolved.
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\end{lemma}
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\begin{proof}
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When $t_i \in \mathcal{C}(t_j)$, the validator's preference for $t_j$ depends on $t_i$'s
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outcome. The confidence counter for $t_j$ cannot reach $\beta$ until the population
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has converged on $t_i$'s resolution, because the supermajority condition for $t_j$
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requires consistency with $t_i$'s resolution. This serialization adds at most $\beta$
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additional rounds for the second transaction, which is the normal single-transaction
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finality time.
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\end{proof}
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%% -----------------------------------------------------------------------
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%% 5. PIPELINE STALL ANALYSIS
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%% -----------------------------------------------------------------------
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