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
Three directional results extend the Geometric Siphon framework to the asymmetries that arise when a CL position is held through a price move. The absolute residual grows strictly with displacement from the range midpoint. On volatile/stablecoin pairs in V3 ordering, the USD residual is asymmetric between symmetric up and down moves, and below-range exits retain strictly less USD value than above-range exits. Foundry tests verify the three theorems at the integer arithmetic level, and a 36,337-event production scale scan on Aerodrome (Base) records rising market USD flow about 5× larger than falling market flow.
Abstract
In a companion paper, I identified the Geometric Siphon, a mechanism by which concentrated liquidity (CL) positions sharing a depositor-level token balance transfer capital between positions under autonomous rebalancing, with token ratio mismatches between old and new tick ranges producing residuals that flow through the shared balance. This paper establishes three results on the directional properties of that residual. Theorem 4 shows that the absolute residual increases strictly with displacement from the midpoint of the original range, up to the range boundary. Theorems 5 and 6 specialise to pools pairing a volatile asset at token0 with a stablecoin at token1 in Uniswap V3’s pair ordering. Under symmetric in-range sqrt-price displacements, the up-move position retains strictly more USD-denominated value than the down-move; under symmetric displacement past the range boundary, a below-range exit retains strictly less USD value than an above-range exit. Residual magnitudes scale with position value, so the USD-denominated absolute residual inherits these asymmetries by a scaling argument under the same domain assumptions. All three results are verified by Foundry tests against a harness implementing exact Uniswap V3 tick-math and amount-equation constants. Across 36,337 rebalance events on 58 positions over 18 days on Aerodrome (Base), observed dust flows are consistent with the predicted direction; rising markets show USD-denominated flow about 5× larger than falling markets despite a near-identical frequency of positive dust events. The dataset is unsegmented by pool composition, so these figures support the prediction at production scale without strictly confirming Theorems 5 and 6 in their volatile/stablecoin domain.
Directional asymmetry
On a volatile/stablecoin pair in V3 ordering, the up-move position retains strictly more USD value than the symmetric down-move position. Direct expansion of the value function gives the identity
with the factor strictly positive for any current sqrt price in the interior of the range. The up-move therefore retains strictly more USD value at every admissible displacement. The asymmetry is numéraire dependent; in token0 units the inequality reverses. Empirically, the 36,337-event production-scale scan on Aerodrome (Base) records rising-market USD flow about 5× larger than falling market flow at near identical event frequency, consistent with the theorem at the dataset level.
Cite as
@misc{ryan2026geometric_ii,
author = {Ryan, K. R.},
title = {The Geometric Siphon II: Directional Properties},
year = {2026},
month = mar,
howpublished = {SSRN Preprint},
url = {https://papers.ssrn.com/sol3/papers.cfm?abstract_id=6481498},
doi = {10.5281/zenodo.19526374},
}