Summary
The successor to Operator Microstructure closes the front that paper left open, the per-event law of the direction-conditioned execution residual. The answer is that there is no execution economics in it at all. The deployed sizing routine rounds at the floor tick but executes at the continuous price, and the leftover side comes down to a race between that rounding bias and the corrective swap’s own impact. Range width drops out, because the local share-per-tick slope cancels in the sign. Replayed against production data, the mapping is exact, and the binding probabilities that the companion papers could only report as measurements now follow from geometry, with nothing fitted.
Abstract
This paper identifies quantisation microstructure, the structured, directional, heavy-tailed flows that rounding choices in automated liquidity management generate and that imitate execution economics. The instance is the law that Operator Microstructure left open. The execution residual measured there is direction-conditioned, since after up-moves the leftover lands on the bought token in more than nine events in ten, while after down-moves the split is nearly even, and no recorded mechanism explained both facts. Replaying the deployed pipeline per event reproduces the measured dust credit at a median predicted-to-measured ratio of 1.000 and the leftover side in 98.4 to 99.9% of 31,340 production rebalances (Aerodrome Slipstream on Base), with no fitted parameters. The map then reduces to a tick race free of range width. The deployed sizing computes its target ratio at the floor tick while executing and minting at the continuous price. The floor bias pushes every mint toward the token0 side by the price’s within-tick offset, the corrective swap’s impact shifts the mint requirement by its tick displacement, and the leftover side is decided by the sign of their sum. One proposition states the race. With the offset empirically uniform, the race reproduces the measured binding probabilities of 0.078 and 0.524 to within a point and a half, and the displacement has an ex-ante law in a footprint ratio, position value over active liquidity. With the companion paper’s placement-quantisation result, both channels of the directional dust asymmetry are floor functions interacting with the sign of price impact. All results are single-operator, but quantisation signatures are code-determined and recoverable from public data alone, and cross-operator replication of the recipe turns the artefact into a taxonomy of automated liquidity management.
The tick race
Two terms of the same scale decide every leftover. The sizing’s target ratio is computed at the floor tick’s price, biasing the mint toward token0 by the within-tick offset; the corrective swap’s own impact moves the requirement by its realised tick displacement. Both scale with the local share-per-tick slope, which cancels in the sign, so the side law is identical across pools whose ranges span two orders of magnitude in width. After up-moves both terms push toward token0 and only a reversed impact flips the side; after down-moves a uniform offset races a median impact of about a third of a tick, the coin flip the companion papers measured. Across range-width quartiles and a temporal split, the inputs drift substantially while the mapping tracks each cell at hit rates near 98%.
Cite as
@techreport{ryan2026quantisationmicrostructure,
author = {Ryan, K. R.},
title = {Quantisation Microstructure: The Tick-Race Law of Rebalancing Dust},
institution = {SSRN},
type = {SSRN preprint},
number = {7166739},
year = {2026},
month = jul,
url = {https://papers.ssrn.com/sol3/papers.cfm?abstract_id=7166739},
}Fourth paper in the sequence begun by The Geometric Siphon, continued in The Hidden Microstructure, and preceded by Operator Microstructure, whose open analytical front this paper closes. Operator Fingerprinting carries the recovery recipe across operators and measures the placement quantisation this paper derives from a single operator.