Introduction

Cryptarchia consensus leadership is determined by a lottery in which the chances of winning are higher for eligible nodes with a greater stake relative to the total active stake. At the same time, the true total active stake cannot be known by participants due to the privacy properties of Nomos notes. This tension is resolved in Cryptarchia by having the network estimate the total active stake based on the observed activity of the network.

Goals

The Cryptarchia total stake inference algorithm must satisfy the following criteria:

The inference process converges quickly, yielding a mean estimate that closely matches the true total stake. However, mean accuracy alone is not sufficient—if the estimator’s variance remains high at steady state, block production rates may fluctuate significantly. Thus, effective total stake inference requires both rapid, accurate mean convergence and low variance to ensure stable, predictable block production throughout the protocol.
The process can be approximated well enough with the information we have in Cryptarchia.

Overview

This document provides an analysis of the Cryptarchia total stake inference algorithm based on the following criteria:

Accuracy: The closeness of the mean inferred total stake to the true total stake; it measures systematic bias in the estimator.
Precision: The degree to which repeated inferences yield similar results at equilibrium; it is quantified by the variance of the estimator and reflects how tightly values cluster around the mean, independent of accuracy.
Stability Conditions: The range of possible values for the learning rate $\beta$ that result in the stake inference values converging to the true total stake under stable conditions.
Convergence Speed: The bounds under which the total stake inference values converge exponentially to the true total stake under stable conditions. This analysis also includes an optimal value for $\beta$.

Total Stake Inference Process

The inference algorithm is described in Algorithm. In order to analyze the properties of this algorithm, we model it analytically as the following sequence $\{D_\ell\}_{\ell=0}^\infty$. We then verify that this model aligns with the algorithm to ensure that the analysis accurately reflects the actual process.

$$ D_{\ell+1}=D_{\ell}-\frac{\beta}{f}D_\ell\left[f-\frac{\sum_{t=1}^T \mathbf{1}\left[\sum_{i=1}^N s^\ell_i(t)\geq1\right] - n(\ell)}{T}\right] $$

where,

$D_{\ell}$ is the inferred total stake at epoch $\ell$;
$\beta$ is the learning rate which governs how quickly we adjust our estimate to new information;