Note on consolidation - This article replaces the separate Operator Microstructure and Quantisation Microstructure articles. The two papers were written to complete each other, the first leaving the value channel open and the second closing it, and they are now a single manuscript under the second’s SSRN listing.
Summary
The question The Hidden Microstructure closed on was why the directional asymmetry of dust flows is stronger on volatile–volatile pairs, where the numéraire theorem of The Geometric Siphon does not apply. The answer is the operator itself, and it arrives in two halves. How often a position fires in each direction obeys an exact first-passage law once the barriers are placed on the manager’s trigger lines rather than its range bounds. How much each firing credits comes down to a race inside a single tick, between the rounding bias of a target ratio computed at the floor tick and the corrective swap’s own price impact. Neither half is a strategy. Both are one-line implementation choices, and the paper the study was designed to write would have confirmed the numéraire mechanism where this one bounds it one to two orders below what actually carries the asymmetry.
Abstract
On a production concentrated liquidity position manager, the USD-valued dust flows credited at rebalances are directionally asymmetric. The first companion paper explains the volatile–stable case through a numéraire theorem; the second finds the asymmetry stronger on volatile–volatile pairs, outside that theorem’s domain. This paper decomposes the asymmetry into a frequency channel and a value channel and closes both, on the V9 production dataset (126,339 rebalances, 48 pools, Aerodrome Slipstream on Base). The frequency channel is an exact first-passage law with no free parameters, once the barriers are placed on the operator’s trigger lines rather than the range bounds, which the data disclose as a discontinuity in the firing-position distribution. Measured drift and measured placement enter with slopes consistent with their theoretical value of one wherever the regressor carries signal, and the volatile–volatile excess is asymmetric range placement, itself a grid-quantisation artefact of the manager’s placement rounding. The value channel is first measured, where it is the near-cancellation of two large opposing factors with a direction-conditioned residual, and then derived. Replaying the deployed pipeline per event reproduces the measured credit and the leftover side on 31,340 rebalances with nothing fitted, and the map reduces to a tick race free of range width. The sizing computes its target ratio at the floor tick while executing and minting at the continuous price, so the leftover side is the sign of that floor bias plus the corrective swap’s own tick displacement. Both channels of the directional asymmetry are therefore floor functions, one in range placement and one in the target ratio, and the swap-free numéraire component, derived here in closed form, is bounded one to two orders below them at production range widths. All results are single-operator; the laws’ form and the measurement recipe are portable, and cross-operator replication has since begun.
The barriers are not where you think
The empirical subtlety in the frequency channel is that the first-passage barriers are not the range bounds. The manager fires when price comes within a configured fraction of the range width of either bound, so the effective barriers are the trigger lines inside them. The fraction is recoverable from public chain data alone, as sharp steps at exactly and in the distribution of the price’s position within the range at firing. Fits against the raw bounds produce systematically attenuated slopes; against the trigger lines, the law holds. The volatile–volatile excess that motivated the paper is then placement arithmetic. Those pools sit ranges with the price systematically above the midpoint, because the manager centres each new range by rounding the current tick down to the pool’s tick-spacing grid. The first-passage law converts that rounding bias into the observed firing excess with slope one.
The tick race
Two terms of the same scale decide every leftover in the value channel. The sizing’s target ratio is computed at the floor tick’s price, biasing the mint towards 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 towards 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 earlier measurement could only report. Across range-width quartiles and a temporal split, the inputs drift substantially while the mapping tracks each cell at hit rates near 98%.
Two floors
The unification is the point. The frequency channel’s excess traces to a floor in range placement, the tick rounded down to the spacing grid when a new range is centred. The value channel traces to a second floor, the target ratio evaluated at the floor tick while everything else in the pipeline sees the continuous price. Traditional microstructure learned that apparent anomalies can be artefacts of discreteness in the trading mechanism. Automated liquidity management reproduces the situation with the artefact written in verifiable code, so the attribution can be made exact rather than argued, and the same fact has a use. Quantisation choices are code-determined, stable between deployments, and recoverable from public data alone, so they fingerprint the operator that made them.
Cite as
@techreport{ryan2026operatorquantisation,
author = {Ryan, K. R.},
title = {Operator \& Quantisation Microstructure: The Frequency and Value Channels of Concentrated Liquidity Dust Flows},
institution = {SSRN},
type = {SSRN preprint},
number = {7166739},
year = {2026},
month = jul,
doi = {10.2139/ssrn.7166739},
url = {https://papers.ssrn.com/sol3/papers.cfm?abstract_id=7166739},
}Third in the sequence begun by The Geometric Siphon, whose numéraire theorem this paper bounds at production scale, and continued in The Hidden Microstructure, whose closing observation it explains. Operator Fingerprinting carries the trigger-geometry recovery across operators and measures at population scale the placement quantisation derived here from a single one.