The Geometric Siphon I

Preprint - Dedicated to the GCHQ analyst that has this land on their desk. I hope you find the proofs as elegant as I did when I first realised what I was seeing. Note: This paper has been consolidated into a single revised manuscript.

Summary

This first preprint introduces the Geometric Siphon and proves that autonomously rebalanced concentrated liquidity positions sharing a depositor level token balance can route capital between independent pools. Three theorems characterise the mechanism, establishing an iff vanishing condition for the residual, single-pool convergence in expectation, and geometric extinction without swap correction. A graph-theoretic Connector Rule ties per-pool flow direction to portfolio topology. A 1,380-event dataset on Aerodrome (Base) and a Foundry suite verifying all three theorems against unmodified V3 contracts anchor the framework empirically.

Abstract

This paper identifies and formalises the Geometric Siphon, a previously undocumented mechanism in concentrated liquidity portfolio management. When multiple CL positions share a depositor-level token balance and are autonomously rebalanced, token ratio mismatches between old and new tick ranges produce residuals that flow through the shared balance, transferring capital between positions in independent pools. Three theorems characterise the mechanism. I derive the geometric residual from the Uniswap V3 amount equations. Theorem 1 establishes that the residual is zero if and only if the old and new ranges demand the same token ratio at the current price. Theorem 2 proves that single-pool positions sharing a contract balance converge to equal value under repeated rebalancing. Theorem 3 proves that positions rebalancing without swap correction decay geometrically. A graph-theoretic Connector Rule ties per-pool flow direction to portfolio topology, supported by a token-level decomposition across ten pools and five depositor groups. A Foundry suite verifies all three theorems against unmodified V3 contracts on Base mainnet, returning a 2,061 USDC residual on range changes against an analytical zero on same-range rebalances. A 1,380-event dataset across 39 positions and seven days on Aerodrome (Base) decomposes observed flow into residual and slippage at Spearman correlation 0.976, revealing a ~5%/~95% split between geometric creation and dust redistribution. A controlled value-to-pool-liquidity sweep collapses a 1:3:9 size ratio to 1.37:1:1.20 within 33 events. A USDC/ZARP position loses 99.8% of its capital, falling from $942to$1.59 across 14 zero-swap rebalances over 65 hours, matching the 11-rebalance extinction bound.

The geometric residual

A rebalance withdraws liquidity from range $[s_a,s_b]$ and mints into $[s_a^{\prime },s_b^{\prime }]$ at the same current sqrt price $s$. From the Uniswap V3 amount equations, the residual fraction $\Delta R/V$ vanishes if and only if old and new ranges demand the same token ratio at $s$.

$$\frac{\Delta R}{V}=0⟺R(s,s_a,s_b)=R(s,s_a^{\prime },s_b^{\prime })$$

Here $R(s,s_a,s_b)=(1/s-1/s_b)/(s-s_a)$ is the token ratio a range demands at the current price. Equal ratios redeploy the withdrawn tokens in full; unequal ratios leave a strict surplus of one token that exits the mint as dust.

In a vault-per-pool architecture the surplus stays in the pool that produced it and reads as small slippage. Under a depositor level shared balance it crosses pools, and the shared balance becomes the channel through which the siphon transfers capital across the portfolio.

Cite as

@misc{ryan2026geometric_i,
  author       = {Ryan, K. R.},
  title        = {The Geometric Siphon: Emergent Capital Reallocation in Concentrated Liquidity Portfolios},
  year         = {2026},
  month        = mar,
  howpublished = {SSRN Preprint},
  url          = {https://papers.ssrn.com/sol3/papers.cfm?abstract_id=6374838},
  doi          = {10.5281/zenodo.19526374},
}