The Hidden Microstructure of Shared Balance Concentrated Liquidity

Summary

The earlier paper The Geometric Siphon identified a previously unrecognised geometric residual in concentrated liquidity rebalancing. When that dust is retained across rebalances rather than returned to the depositor, it stops being residue and becomes the hidden state of the portfolio. This paper derives its dynamics as a closed-form Master Equation, with a large-portfolio limit governed by propagation of chaos. The result brings statistical mechanics and interacting particle theory into direct contact with AMM microstructure.

Abstract

The geometric residual identified in the prior paper takes a richer form once a concentrated liquidity manager retains it in a depositor-keyed dust ledger shared across multiple pools. The residual is no longer isolated to its own rebalance, but becomes a hidden state coupling the portfolio’s pools. This paper derives the closed-form law governing that state. Theorem 1 (the Master Equation) gives the exact per-event jump law on any pool, induced by the V3 mint geometry at rebalance, and recovers the prior result in the single-pool swap-free corner. Proposition 1 promotes the prior paper’s Connector Rule conjecture to a sign-keyed correlation under explicit V3-mechanical assumptions, while Theorem 2 generalises zero-swap extinction to a per-token mass-conservation identity across multi-pool rebalances, with a donor/absorber partition under heterogeneous arrival. On the homogeneous hub-spoke topology, Proposition 2 establishes a conditional propagation-of-chaos limit under uniform-in-$N$ hub mixing, and Proposition 3 identifies the limiting one-spoke marginal as a binary atom-mixture whose mass and shape are determined by the slippage law of the in-pool swap correction. The half-normal mixture obtained under symmetric Gaussian slippage is a narrow corollary. The empirical anchor spans roughly 34,500 rebalance events over the V6 through V9 evolution of a PM contract family on Aerodrome Slipstream (Base) under one operator, with per-event predictions matching on-chain dust credits at median ratio 1.0000 throughout. Contract-level verification reproduces the closed form against unmodified Aerodrome Slipstream and Uniswap V3 infrastructure to bit-exact tolerance.

The Master Equation

At each rebalance event on a pool $P_k$, the V3 mint geometry fixes the input ratio at the current sqrt-price. With withdrawn amounts $(x_w,y_w)$, standing dust $(x_{\text{dust}},y_{\text{dust}})$ pulled from the depositor-keyed shared ledger, and signed swap correction $({\sigma }_x,{\sigma }_y)$, the mint inputs are

$$\hat{x}=x_w+x_{\text{dust}}-{\sigma }_x,\hat{y}=y_w+y_{\text{dust}}-{\sigma }_y$$

and the LP-binding solve gives the new liquidity

$$L_{\text{new}}=\min \left(\frac{\hat{x}}{g_x},\frac{\hat{y}}{g_y}\right)$$

where $(g_x,g_y)$ are the per-unit-$L$ token amounts at the new range. The dust ledger then jumps deterministically to

$$x_{\text{dust}}^{\prime }=\hat{x}-L_{\text{new}}{g}_x,y_{\text{dust}}^{\prime }=\hat{y}-L_{\text{new}}{g}_y$$

with all other tokens’ ledger entries unchanged. Off a measure-zero locus, exactly one of $(x_{\text{dust}}^{\prime },y_{\text{dust}}^{\prime })$ is zero in the swap-free regime, and the non-zero side holds the entire leftover. This is the per-event jump law on $D_{\text{pool}}(t)$, turning the dust ledger into a piecewise-deterministic Markov process whose increments are governed by the V3 mint geometry rather than a fitted dynamics.

Multi-pool conservation

Under zero-swap rebalances, the geometric extinction of a single position generalises to a per-token mass-conservation identity across multi-pool rebalances. For each portfolio token $T_i$, with $a_{k,i}(\tau )$ position $k$‘s implied amount of $T_i$ at the contemporaneous sqrt-price,

$$\sum _k{a}_{k,i}(\tau )+D_{\text{pool},i}(\tau )=\text{const}$$

is invariant across rebalance events. The single-pool extinction theorem from the prior paper becomes the $m=1$ corner case; for $m\ge 2$ positions sharing a depositor-keyed dust mapping, total token mass is preserved while value redistributes across the portfolio. Under heterogeneous rebalance arrival rates, this conservation law admits a non-trivial donor/absorber partition in which some positions are net donors and others net absorbers over any window, with the sum forced to balance by the identity above.

Propagation of chaos

On the homogeneous hub-spoke topology, the hub token is touched by every rebalance while each spoke is touched only by its own. As the spoke count $N$ grows, the hub mixes faster than any single spoke sees individual updates, and the marginal kernel governing each spoke decouples from the others. Writing $H=D_{\text{pool},T_0}$ for the hub component and $B_k=D_{\text{pool},T_k}$ for spoke $k$, the asymptotic $m$-spoke marginal factorises:

$${\pi }_{(m)}^N⟶{\mu }^{\otimes m}\text{as }N\to \infty$$

with $\mu$ the unique stationary law of the limiting one-spoke chain driven by i.i.d. hub samples. The limit takes the closed form

$$\mu (db)=q{\delta }_0(db)+(1-q){\rho }_{+}(b)db$$

a binary mixture of an atom at zero (the spoke-binding outcomes) and a slack-side density ${\rho }_{+}$, with mass $q={\mathbb{P}}_{{\nu }_{\epsilon }}(\epsilon <0)$ determined by the slippage law of the in-pool swap correction. Under symmetric Gaussian slippage, this specialises to the half-normal mixture

$$\mu (db)=\frac{1}{2}{\delta }_0(db)+\frac{1}{2}\mathrm{HN}(b;\beta )db$$

with the atom mass exactly $1/2$ and scale parameter $\beta =Lw{\sigma }_{\text{slip}}$ in the homogeneous-symmetric leading-order regime.

Cite as

@misc{ryan2026hiddenmicrostructure,
  author       = {Ryan, K. R.},
  title        = {The Hidden Microstructure of Shared Balance Concentrated
                  Liquidity: A Master Equation for the Dust Ledger and
                  Propagation of Chaos},
  year         = {2026},
  month        = may,
  howpublished = {SSRN Preprint},
  url          = {https://papers.ssrn.com/sol3/papers.cfm?abstract_id=6745218},
  doi          = {10.5281/zenodo.20115375},
}

Companion paper to The Geometric Siphon, which establishes the single-pool residual that this paper promotes into a multi-pool dynamical law.