Constructive Gauges

Working Paper - This paper is currently undergoing work and is subject to revision.

Summary

Constructive gauges redesign the standard reward accumulator so that emissions track tradeable depth rather than nominal stake at the active tick. The framework multiplies each position’s liquidity by a concentration weight, a freshness weight, and an in-range indicator, then distributes emissions in proportion to the resulting score. It closes the parasitic liquidity vector formalised in the companion paper, with formal bounds on residual extraction, a minimality theorem on the three factors, and calibration against operational data from a live concentrated liquidity operator.

Abstract

This paper identifies and corrects a measurement gap in ve(3,3) concentrated liquidity gauges. The Synthetix reward accumulator distributes voter directed emissions in proportion to instantaneous staked liquidity, while the protocol intends to incentivise sustained tradeable depth. That gap admits the parasitic extraction vector of Ryan (2026), in which a position satisfying gauge staking conditions can collect emissions without serving a swap. Constructive gauges close the gap by attaching three multiplicative factors to each position’s nominal liquidity: a concentration weight non-increasing in range width, a freshness weight non-increasing in time-since-rebalance, and an in-range indicator. Three theorems characterise the framework. Theorem 1 bounds the ratio of corrective to standard gauge parasitic revenue by a freshness ratio times a concentration ratio, under a Brownian log-price in the large population mean-field regime. Theorem 2 gives the analogous equality for legitimate LP yield. Theorem 3 shows that any proper subset of the three factors reopens at least one of the short-cycle, out-of-range, and width-arbitrage pathways, without invoking the price model or population statistics. A Foundry fork test suite exercises the accumulator against live Aerodrome Slipstream state on Base, with cross-chain replications on PancakeSwap V3, Thena FUSION, and Ramses V2. The Gaussian in-range probability is checked against 24-hour tick traces from two Aerodrome CL100 pools; empirical departures at multi-hour horizons leave the parasitic bound conservative. Calibration against 94,469 rebalance events across 48 pools (a single-operator dataset, 7 March to 13 April 2026) sets a warmup window of 10 to 30 seconds.

The corrective scoring function

A constructive gauge replaces $L_i$ in the standard accumulator with a score

$$S_i(t)=\underset{\text{liquidity}}{\underbrace{L_i}}\cdot \underset{\text{concentration}}{\underbrace{c(u_i-{\ell }_i)}}\cdot \underset{\text{freshness}}{\underbrace{f(t-t_i)}}\cdot \underset{\text{in-range}}{\underbrace{1[\tau \in [{\ell }_i,u_i]]}}$$

Each factor closes one of the three gaps that Parasitic Liquidity showed exploitable.

The concentration weight $c$ ties the reward to range width. The freshness weight $f$ ties it to time since the position’s last rebalance. The in-range indicator zeroes positions whose range no longer covers the current tick.

Emissions accrue in proportion to $S_i/{\sum }_j{S}_j$ rather than $L_i/{\sum }_j{L}_j$, the only algebraic change to the Synthetix accumulator. Theorem 3 of the paper shows the three factors are jointly necessary, since any proper subset reopens at least one of the short cycle, out-of-range and width arbitrage pathways.

Cite as

@misc{ryan2026constructive,
  author       = {Ryan, K. R.},
  title        = {Constructive Gauges: Remediating Parasitic Liquidity in Concentrated Liquidity Emissions},
  year         = {2026},
  month        = apr,
  howpublished = {SSRN Working Paper},
  url          = {https://papers.ssrn.com/sol3/papers.cfm?abstract_id=6625980},
  doi          = {10.5281/zenodo.19690075},
}

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