$$ \newcommand{\R}{\mathbb{R}} \newcommand{\N}{\mathbb{N}} \newcommand{\E}{\mathbb{E}} \newcommand{\Csmooth}{\bar{C}} \newcommand{\Craw}{C} \newcommand{\tasktype}{\tau} \newcommand{\bps}{\mathrm{BPS}} $$

Protocol Design

We implement BPE on Ethereum (Base L2) using Superfluid’s General Distribution Agreement (GDA) (Superfluid, 2024). The architecture comprises four contracts.

Architecture Overview

BPE protocol architecture. Solid arrows denote read dependencies; the dashed arrow represents the auto-slash feedback path from CompletionTracker to StakeManager.

StakeManager

Sinks deposit ERC-20 tokens to stake. The capacity cap is computed as \(\text{cap}(k) = \sqrt{\text{stake}(k) / u}\) where \(u\) is the stake unit parameter (set to \(10^{18}\)). This concave function implements Sybil resistance: splitting stake \(S\) into \(n\) sinks yields total capacity \(n \cdot \sqrt{S/(nu)} = \sqrt{n} \cdot \sqrt{S/u} < n \cdot \sqrt{S/u}\) for \(n > 1\)—strictly less than the capacity from a single identity. A minimum per-sink stake \(S_{\min}\) further penalizes splitting.

CapacityRegistry

The registry manages task type creation, sink registration, and capacity signaling. Capacity updates use a two-phase commit-reveal protocol for MEV resistance:

1. Commit: Sink submits \(h = \texttt{keccak256}(C, \text{nonce})\).

2. Reveal: Within 20 blocks, sink reveals \((C, \text{nonce})\). The contract verifies the hash, applies the stake cap, and computes the EWMA update: \(\Csmooth_{\text{new}} = 0.3 \cdot C_{\text{capped}} + 0.7 \cdot \Csmooth_{\text{old}}\).

The registry maintains an aggregate totalCapacity per task type, updated atomically on each reveal.

BackpressurePool

For each task type, the pool creates a Superfluid GDA pool and acts as its admin. On rebalance, it reads all sink capacities from the registry and sets each sink’s pool units proportional to their smoothed capacity: \(\text{units}(k) = \lfloor C_{\text{smooth}}(k) \cdot 10^9 / C_{\text{total}} \rfloor\).

Sources create Constant Flow Agreement (CFA) streams to the pool address. The GDA automatically distributes incoming flows to members proportional to their units—achieving capacity-proportional payment routing without centralized scheduling.

Rebalance is permissionless: any address may call it. A 5% threshold (measured in basis points of total capacity change) prevents unnecessary gas expenditure.

EscrowBuffer

When all sinks in a task type are at capacity, excess payments are deposited into the buffer. The drain function distributes buffered funds proportionally to current sink capacities. A configurable bufferMax caps the buffer size; when full, sources must pause.

Pipeline Composition

For multi-stage agent pipelines, the Pipeline contract chains BackpressurePools. The key mechanism is upstream propagation: each stage’s effective capacity is constrained by its downstream stage: $$ \begin{equation} C_{\text{eff}}(\text{stage}i) = \min\bigl(C}}(\text{stagei),\; C)\bigr) \end{equation} $$ Rebalance proceeds back-to-front, ensuring downstream congestion propagates upstream before any stage is rebalanced.}}(\text{stage}_{i+1