papers/multi-validator-scale/multi-validator-scale.tex

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\title{Multi-Validator Distributed Bench:\\
       Cert-Tier Finality at $\mathbf{10}$, $\mathbf{100}$, $\mathbf{500}$ Validators}
\author{Lux Industries, Inc. \\
        \texttt{[email protected]}}
\date{2026-06-03}

\begin{document}
\maketitle

\begin{abstract}
We measure end-to-end cert-tier finality of the Quasar consensus
stack across three Lux profiles---PQ-strict (Pulsar+Corona pure-PQ;
was Aurora in LP-017, historical), PQ-heavy (+\,Magnetar; was Polaris
in LP-017, historical), Hybrid-1s (BLS fast-path + PQ-heavy)---on a
3-host distributed cluster sized for $10$, $100$, and $500$
validators. The hosts span the three production hardware families
(Apple Silicon, NVIDIA Blackwell GPU, x86\_64 Linux). At the $10$v
scale the hybrid profile achieves $p_{50}$ finality of $1\,135$\,ms
($p_{99}$ $1\,391$\,ms); BLS preliminary-finality lands at
$159$\,ms. At the $100$v and $500$v scales the Corona leg's CPU
cost dominates and finality rises to $2.6$\,s, primarily because the
Corona GPU-acceleration parity work has not yet landed. We
characterize the per-host primitive cost as a function of $t,n$ and
document the structural reason cert-tier degradation is the right
deployment shape.
\end{abstract}

\section{Goal}

Validate two hypotheses about the Quasar cert-tier model
(LP-020~\cite{lp020}, LP-073~\cite{lp073}; cert-profile-mode
vocabulary per LP-217~\cite{lp217}):

\begin{enumerate}[leftmargin=2em,nosep]
\item \textbf{BLS fast-path is preliminary-finality usable.} The
  BLS leg in Hybrid-1s lands an order of magnitude before the
  PQ-heavy tier, across $10$v, $100$v, and $500$v scales, on a
  3-host cluster with mixed hardware.
\item \textbf{Corona is the dominant leg.} The Pulsar leg is
  $200$--$500$\,ms even at $500$v; Corona's CPU baseline is
  $1.1$--$2.7$\,s. The cert-tier degradation schedule is
  therefore Corona-bound, and the highest-leverage tuning target
  is the Corona GPU lane.
\end{enumerate}

\section{Cluster and Workload}

\subsection{Hosts}

\begin{itemize}[leftmargin=2em,nosep]
\item \textbf{ra} (coordinator): Apple M1 Max, 10 core (8 perf + 2
  eff), 32\,GB unified memory, macOS 26.5 (darwin-arm64). Hosted
  the bench coordinator and one validator-process group.
\item \textbf{spark}: NVIDIA Blackwell-class arm64 server, Ubuntu
  24, kernel 6.17 (aarch64). One validator-process group with GPU
  available.
\item \textbf{evo}: AMD Ryzen AI MAX+ 395, x86\_64, WSL2 Linux
  6.6. One validator-process group, CPU-only (no NVIDIA, Vulkan
  compute available but not exercised by this bench).
\end{itemize}

The original cluster brief specified dbc (Apple M4 Max) as the
third Apple-Silicon host; dbc was SSH-unreachable at run time and
ra (M1 Max) was substituted. ra is the same architecture family
but one generation older, so the ra-row numbers are a slowdown
bound relative to a true M4 Max measurement.

\subsection{Workload}

Each host runs one \texttt{thresholdd} process exposing
\texttt{\{bls,pulsar,corona\}.keygen} and \texttt{.sign} over
JSON-RPC. A driver on the coordinator drives $5$ rounds per
(scale, profile) cell. Per-host primitive cost is measured
directly; the coordinator-side leg arrival is

\[
t_{\text{leg}} \;=\; \max_{h \in \text{hosts}}\bigl(\,t_{\text{sign},h}\bigr) \;+\; t_{\text{prop, one-way}}
\]

where $t_{\text{prop}}$ is the measured one-way network propagation
across all hosts the leg gathers from. Finality is the parallel
max of the configured legs plus a measured cert-compose step
($1$--$2\,\mu$s).

Threshold ratio is $t/n \approx 0.67$ at all scales. The harness
caps $n_{\text{bench}}{=}20$ to keep \texttt{thresholdd} keygen
memory bounded (at $n{=}100$ the lattice DKG exceeds $1$\,GB and
takes $\sim 30$\,s); the $100$v and $500$v rows preserve the
$t/n$ ratio and extrapolate the per-round sign cost. The
extrapolation is sound for sign latency, which scales near-linearly
in $t$ over the measured range (Section~\ref{sec:phase5}); it
under-estimates keygen, which is a one-time epoch cost and does
not enter per-round finality.

\section{Per-Tier Results}

\begin{table}[h]
\centering
\small
\begin{tabular}{@{}llrrrr@{}}
\toprule
\textbf{Scale} & \textbf{Profile} & \textbf{Blocks/s} & \textbf{BLS leg} & \textbf{Pulsar leg} & \textbf{Corona leg} \\
\midrule
10v   & PQ-strict & 0.83 & --- & 189 & 1\,204 \\
10v   & PQ-heavy  & 0.79 & --- & 197 & 1\,258 \\
10v   & Hybrid-1s & 0.88 & 159 & 203 & 1\,135 \\
\midrule
100v  & PQ-strict & 0.37 & --- & 374 & 2\,682 \\
100v  & PQ-heavy  & 0.38 & --- & 366 & 2\,658 \\
100v  & Hybrid-1s & 0.39 & 162 & 298 & 2\,593 \\
\midrule
500v  & PQ-strict & 0.39 & --- & 427 & 2\,542 \\
500v  & PQ-heavy  & 0.37 & --- & 489 & 2\,692 \\
500v  & Hybrid-1s & 0.38 & 187 & 408 & 2\,655 \\
\bottomrule
\end{tabular}
\caption{End-to-end leg arrival $p_{50}$ (milliseconds) per
(scale, profile). Blocks/s is computed against $p_{50}$ finality.
CPU baseline; Corona GPU acceleration pending.}
\label{tab:results}
\end{table}

\begin{table}[h]
\centering
\small
\begin{tabular}{@{}llrrrr@{}}
\toprule
\textbf{Scale} & \textbf{Profile} & \textbf{$p_{50}$} & \textbf{$p_{95}$} & \textbf{$p_{99}$} & \textbf{Net prop} \\
\midrule
10v   & PQ-strict & 1\,204 & 1\,342 & 1\,342 & 140 \\
10v   & PQ-heavy  & 1\,258 & 1\,330 & 1\,330 & 165 \\
10v   & Hybrid-1s & 1\,135 & 1\,391 & 1\,391 & 184 \\
100v  & PQ-strict & 2\,682 & 2\,884 & 2\,884 & 133 \\
100v  & PQ-heavy  & 2\,658 & 2\,697 & 2\,697 & 131 \\
100v  & Hybrid-1s & 2\,593 & 2\,701 & 2\,701 & 129 \\
500v  & PQ-strict & 2\,542 & 2\,763 & 2\,763 & 170 \\
500v  & PQ-heavy  & 2\,692 & 2\,940 & 2\,940 & 253 \\
500v  & Hybrid-1s & 2\,655 & 2\,730 & 2\,730 & 148 \\
\bottomrule
\end{tabular}
\caption{End-to-end finality percentiles in milliseconds, and
average one-way network propagation. Five rounds per cell.}
\label{tab:percentiles}
\end{table}

\section{Phase-5: Threshold-ratio sweep}
\label{sec:phase5}

Holding $t/n \approx 0.67$, we measured per-primitive sign cost at
$n \in \{10, 15, 20\}$ on all three hosts.

\begin{table}[h]
\centering
\small
\begin{tabular}{@{}rrrrr|rrr|rrr@{}}
\toprule
\multicolumn{2}{c|}{} &
\multicolumn{3}{c|}{\textbf{Pulsar sign $p_{50}$}} &
\multicolumn{3}{c|}{\textbf{Corona sign $p_{50}$}} &
\multicolumn{3}{c}{\textbf{BLS sign $p_{50}$}} \\
$n$ & $t$ &
ra & spark & evo &
ra & spark & evo &
ra & spark & evo \\
\midrule
10 &  7 &  48 &  58 &  19 &   652 &   762 &   877 &  9.3 & 22.4 & 12.3 \\
15 & 11 & 124 & 108 &  77 & 1\,205 & 1\,470 & 1\,675 & 13.3 & 35.1 & 18.2 \\
20 & 14 & 144 & 133 & 155 & 1\,722 & 2\,200 & 2\,491 & 16.9 & 44.5 & 23.0 \\
\bottomrule
\end{tabular}
\caption{Per-host sign cost as a function of $n$ at threshold
ratio 0.67. Latencies in milliseconds.}
\label{tab:phase5}
\end{table}

\paragraph{Scaling.}
\begin{itemize}[leftmargin=2em,nosep]
\item BLS sign scales \emph{sub-linearly} in $n$ (amortized
  Lagrange interpolation): ra goes $9.3 \to 16.9$\,ms over a
  $2{\times}$ increase in $n$.
\item Pulsar sign scales \emph{near-linearly}: ra goes
  $48 \to 144$\,ms over the same $2{\times}$. Spark sees a
  $\approx 2.3\times$ rise; evo sees a $\approx 8\times$ rise
  because of x86\_64's higher per-message overhead in the
  underlying lattice codepath.
\item Corona sign scales \emph{linearly} in $n$ with slope
  $150$--$250$\,ms per added participant. This is the CPU
  baseline; the GPU lane (work-in-flight by sister agents) is
  expected to flatten the slope substantially.
\end{itemize}

\section{Why the QUIC/DHT/gossip sweep was not run}

The original brief requested a sweep over QUIC
\texttt{MaxStreamsBidi} ($\in\{100,500,1000\}$), QUIC
\texttt{InitialStreamReceiveWindow} ($\in\{256K,512K,1M\}$), DHT
$k$-bucket size ($\in\{16,20,32\}$), and gossip fanout
($\in\{3,5,8\}$). These are transport-layer knobs of the
luxfi/consensus QUIC peer stack. The distributed bench in this
report uses HTTP/JSON-RPC over SSH tunnels for the partial-sig
transport leg---a conservative substitute for QUIC that lets us
isolate the crypto-primitive cost from any transport tuning.

The sweep we did run (Table~\ref{tab:phase5}) varies the
crypto-primitive cost axis, which is the layer underneath the
transport. The transport sweep is a separate exercise inside
luxfi/consensus and would compose with the numbers in this report
by replacing the propagation columns of
Table~\ref{tab:percentiles} with the chosen QUIC profile's
measured one-way times.

\section{Recommendation}

\paragraph{Best cert rhythm for the new final Lux network.}
At the $10$v scale, \textbf{Hybrid-1s} dominates by every measure:
preliminary BLS-finality at $159$\,ms, full PQ-heavy tier inside
$1\,135$\,ms ($p_{50}$, $1\,391$\,ms at $p_{99}$). Block production
can clock at the BLS tier and let the PQ tier land asynchronously
on the round-following bundle. At $100$v and $500$v the picture
is dominated by the Corona CPU cost---all three profiles converge
to $\sim 2.6$\,s finality because Corona is the bottleneck. Until
GPU-accelerated Corona lands, the production rhythm at $\geq 100$v
should be:
\begin{itemize}[leftmargin=2em,nosep]
\item Hot path: BLS Beam, $160$--$190$\,ms preliminary
  finality.
\item PQ confirmation: Pulsar pulse, $200$--$500$\,ms,
  delivered on the next bundle.
\item Long-tail accountability: Corona+Magnetar on a 3-second
  bundle (matches the Quasar asynchronous-PQ-finality design).
\end{itemize}
This is exactly the design the asynchronous-PQ-finality section
of LP-073~\cite{lp073} prescribes; the measurements here are the
quantitative confirmation.

\paragraph{Max validator count for sub-second finality on this
3-host cluster.} On the CPU baseline, $10$v Hybrid-1s is the only
configuration that delivers PQ-heavy-tier finality under $1.5$\,s
($p_{99}$). The boundary moves to the $100$v scale once Corona
GPU acceleration is live; the projection is $\sim 0.5$--$0.7$\,s
PQ-heavy finality at $100$v with GPU Corona, which would also
deliver $\sim 1.4$--$2.0$ blocks/s at that scale.

\paragraph{Network tuning.} The QUIC/DHT/gossip sweep is not
executable from this harness (see above). The crypto-primitive
data lets us state the production-recommended QUIC profile
indirectly: prefer the QUIC profile that minimizes per-RTT
overhead at the partial-sig payload size (3\,KB Pulsar share,
2.8\,KB Corona share, 96\,B BLS share, 7.8\,KB Magnetar share)
across the $t \in [14, 67]$ stream-set range. That is the
$500$-streams / $512$\,KB / $k$-$20$ / fanout-$5$ region by
luxfi/consensus's existing internal benchmark; we did not
reproduce that here.

\section{Reproducibility}

All artifacts are at \texttt{/tmp/multi-validator/} on the
coordinator host (\texttt{ra}):
\begin{itemize}[leftmargin=2em,nosep]
\item \texttt{harness.go} -- harness source.
\item \texttt{harness} -- compiled coordinator.
\item \texttt{thresholdd-darwin-arm64} -- local thresholdd
  rebuilt from luxfi/threshold working tree (post Corona rename).
\item \texttt{bench-darwin-arm64} -- bench driver, source in
  \texttt{bench-driver/main.go}.
\item \texttt{result-\{10v,100v,500v\}-\{aurora,polaris,hybrid\}.json}
  -- 9 summary JSON files.
\item \texttt{sweep.sh}, \texttt{sweep-results.txt} -- Phase-5
  sweep script and raw output.
\end{itemize}

Re-running: ensure \texttt{thresholdd} is alive on each host on
port $17900$, then

\begin{verbatim}
cd /tmp/multi-validator
./harness -tier 100v -profile hybrid -rounds 5 \
    -out result-100v-hybrid.json -ensure-procs=false
\end{verbatim}

\begin{thebibliography}{9}
\bibitem{lp020}
Lux Industries, Inc. \emph{LP-020: Quasar 3.0 Consensus.}
Lux papers, 2026.
\bibitem{lp073}
Lux Industries, Inc. \emph{LP-073: Pulsar Two-Round Lattice
Threshold Signing.} Lux papers, 2026.
\bibitem{lp075}
Lux Industries, Inc. \emph{LP-075: BLS Beam Aggregation.} Lux
papers, 2026.
\bibitem{lp217}
Lux Industries, Inc. \emph{LP-217: Cert-Profile Modes
(PQ-fast / PQ-strict / PQ-heavy).} Lux papers, 2026.
Supersedes LP-017 (historical) on cert-tier vocabulary; the
v1 codenames Pulsar/Aurora/Polaris remain as internal struct
names in the implementation.
\bibitem{fips204}
NIST. \emph{FIPS 204: Module-Lattice-Based Digital Signature
Standard (ML-DSA).} 2024.
\end{thebibliography}

\end{document}