Residual of the Residual

A short empirical note on a smaller remainder sitting beneath the attributed dust of the Geometric Siphon¹. The original paper identified the geometric residual at rebalance level and measured it through a per-portfolio dust ledger. What is recorded here is a narrower object; the portion of that residual which escapes portfolio attribution altogether and accumulates instead in the contract’s physical token inventory.

The prediction

The Geometric Siphon paper identifies a closed-form residual at every concentrated liquidity rebalance. For a position rebalanced from range $[s_a,s_b]$ to $[s_a^{\prime },s_b^{\prime }]$ at sqrt price $s$, Theorem 1 establishes that the signed residual value satisfies

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

with $R$ the position’s token ratio. Whenever the old and new ranges disagree on the ratio at the current price, the rebalance produces a strictly non-zero residual; the binding token redeploys in full and the surplus exits the mint as dust.¹

That paper measured the residual through a position manager exposing a dustBalance[wallet][portfolioId][token] mapping, written on each rebalance from the amounts returned by the NFPM. This made the residual empirically visible position by position rather than leaving it submerged in a single opaque contract balance.

The distinction matters. The original paper did not argue that the shared balance architecture creates the residual. The residual is a consequence of concentrated liquidity rebalance geometry itself. What the shared balance architecture adds is observability across positions and, more importantly, a transfer channel through which residuals from one pool may later be absorbed by another. The present note pushes on layer lower. It records evidence that, even beneath the attribution layer of the discovery architecture, a smaller remainder survives at the seam between V3 integer arithmetic and the manager’s subsequent ledger write.

Setup and notation

• Geometric residual $\Delta R$: the per-rebalance signed token leftover predicted by Theorem 1 of the Geometric Siphon, zero iff the old and new ranges require the same token ratio at the current price.¹
• Attributed dust $\Delta R^{\text{att}}$: the portion of $\Delta R$ that a wei-precise per-portfolio ledger writes during the rebalance.
• Sub-attribution remainder $\epsilon$: the portion that escapes that ledger, accumulating in the contract’s physical inventory.
• PM V9: a custodial position manager that hosts $N$ concentrated liquidity positions and writes a dustBalance[wallet][portfolioId][token] entry on every rebalance.
• NFPM: the unmodified Slipstream NonfungiblePositionManager, the upstream side of the seam at which $\epsilon$ accrues.
• Q64.96: the 160-bit fixed-point sqrt-price representation used throughout V3-lineage CL math.

The attributed-dust ledger

The ledger catches the macroscopic flow. On the system under observation, with 177,353 lifetime rebalances across 43–53 concentrated liquidity positions, the total dust attributed by pm.dustBalance summed across every wallet, portfolio, and token is $2,487.62. Restricted to the three hub tokens that hold the bulk of the dust inventory, USDC, EURC, and WETH, the attributed sum is $676.87. The contract’s physical token balance at the same instant, read directly from each ERC-20’s balanceOf(PM_V9, t), sums to $679.70 across those three tokens.

The two views describe the same physical tokens through different descriptions. The difference between them is the portion of the contract inventory which exists physically but is not named by any per-portfolio attribution entry.

A direct observation

The gap was surfaced during an unrelated TVL audit. A double count between contractDust.totalUsd and the sum of portfolioDust.totalUsd forced a side-by-side comparison on precisely those tokens where rebalance traffic is concentrated. After de-duplication, the arithmetic leaves a $2.83 residue unattributable to any portfolio.

Token	Decimals	Contract balance	Attributed dust	Unattributed amount	Unattributed USD	Per-rebalance equivalent
WETH	18	0.0212839	0.0204190	0.0008649	$1.75	4.88 gwei/rebal
USDC	6	32.5464	31.59	0.9564	$0.96	5,396 wei/rebal
EURC	6	520.602	520.50	0.1021	$0.06	576 wei/rebal
Total					$2.83

The table does three things at once. It shows that the gap exists, gives its per-token structure, and places the per-event magnitudes in the band, single gwei for WETH, single thousands of wei for six-decimal stables, in which an integer-arithmetic floor remainder would naturally be expected to sit.

The seam between arithmetic and attribution

Decompose the per-event residual as $\Delta R_i=\Delta R_i^{\text{att}}+{\epsilon }_i$, where $\Delta R_i^{\text{att}}$ is the amount the manager writes into dustBalance at event $i$ and ${\epsilon }_i$ is the floor remainder accumulating in the contract’s physical balance. Summed over 177,353 events,

$${\sum }_i\Delta R_i^{\text{att}}=\$676.87(\text{three hub tokens}),{\sum }_i{\epsilon }_i=\$2.83.$$

NFPM’s mint and burn callbacks return amounts computed in Q64.96 fixed-point sqrt-price arithmetic. The manager’s subsequent dustBalance write is bounded above by what the NFPM declares it received or returned. Any rational quantity exceeding that declared amount remains as physical token at the contract address, without any corresponding per-portfolio ledger entry. By construction ${\epsilon }_i\ge 0$ per token per event, so ${\sum }_i{\epsilon }_i$ accumulates monotonically.

Taken together, the two terms are the full geometric-residual flow at this rebalance volume. The attributed term is the part the original empirical apparatus was built to surface. The remainder is the part the instrument cannot reach, precisely because of where it sits in the call stack.

Why the magnitudes matter

Three observations support the integer-floor reading over the obvious alternatives.

First, the per-token magnitudes scale with decimal place. The per-rebalance figures are of order ~5 gwei for WETH (5 × 10⁻⁹ tokens), ~5,400 wei for USDC (5 × 10⁻⁶ tokens), and ~580 wei for EURC (6 × 10⁻⁷ tokens). Those magnitudes sit three orders of magnitude apart, in line with the atomic units of the underlying tokens.

Secondly, the USDC figure sits where a single sqrt-price round trip would suggest. Q64.96 divisions in converting between sqrt price and token amounts carry a remainder bounded by tick spacing and a small constant reflecting discarded integer precision.² For a CL100 pool, the resulting floor remainder on a six-decimal token lies in the single-thousands-of-wei band. The observed 5,396-wei figure for USDC sits squarely in that range.

Thirdly, the magnitudes are uniform per token across the lifetime. Direct transfers would arrive as one-off lumps sums of irregular size. What is observed here is instead a per-token constant of approximately “a few atomic units per operation”. That is the shape integer-floor leakage would produce.

Per-event corroboration

The argument above is inferred from the cumulative gap. A second measurement is available at event level from the system’s diffusion log, which records $\text{dustPnl}=V_{\text{pre}}-V_{\text{post}}$ at float64 precision for each rebalance. The display-layer rounding discussed below affects only the aggregate API payload and does not propagate into the on-disk log, so the per-event reading resolves residuals well beneath the wei-equivalent magnitudes of interest.

The cleanest subset consists of those rebalances in which neither a range change nor an operator-requested swap occurred, that is, those satisfying oldRange == newRange and swap.amountIn == 0. These are exactly the events for which Theorem 1’s vanishing condition predicts zero geometric residual under any account that does not admit an integer-arithmetic remainder. Across the tail 5,000 events of the diffusion log, spanning 56.6 hours from 21 to 23 May 2026, this same-range zero-swap subset accounts for 2,561 events (51.2% of the tail).

A non-trivial fraction of those events lands at the float64-precise zero floor.

Bucket of |dustPnl|	Count	% of same-range
= 0 (float-precise)	686	26.8
> 0 to 10⁻⁶	20	0.8
> 10⁻⁶ to 10⁻⁴	946	36.9
> 10⁻⁴ to 10⁻²	457	17.8
> 10⁻² to 1	213	8.3
> 1 to 10	98	3.8
> 10 to 100	75	2.9
> 100	34	1.3

The first three buckets together account for 64.5% of same-range rebalances. That is the natural “paper territory” floor population; the exactly-zero column the per-event analogue of Theorem 1’s iff vanishing condition, while the sub-cent tail sits at the scale one would expect from the Q64.96 remainder.

Floor-band statistics, paper-territory subset

Restricted to the 2,141 same-range zero-swap events with |dustPnl| < $0.01:

Statistic	Value
Exactly zero	686/2,141 (32.0%)
Nonzero median	$5.56 × 10⁻⁶
Nonzero $q_{75}$	$4.87 × 10⁻⁴
Nonzero $q_{90}$	$2.22 × 10⁻³

At the $2,030 WETH spot price over the sample window, the median corresponds to 2.74 gwei WETH per event.

The aggregate gap implies ~4.88 gwei WETH per rebalance averaged across the full 177,353 event lifetime. The per-event median over the cleanest same-range subset is 2.74 gwei. The two measurements therefore land in the same gwei-scale band, with the aggregate somewhat higher because it includes the heavier range-change tail, and the per-event median somewhat lower because it isolates the floor. Two independent measurements on a production system, neither requiring fork instrumentation and agreeing on the same order of magnitude is the sort of corroboration one would want.

A per-pair breakdown sharpens the point. The well-behaved pairs sit cleanly at the arithmetic floor, with WETH/EURC ($n$ = 1,231) at $q_{50}$ = $5.0 × 10⁻⁶ and WETH/USDC ($n$ = 76) at $q_{50}$ = $1.3 × 10⁻⁵, both with and without zeros included. The VVV pairs are bimodal, mostly exactly zero with a tail at 10⁻³ to 10⁻² USD range, consistent with higher volatility introducing genuine inter-block friction on a subset of operations.

The shape of the distribution is itself informative. Roughly 64% of events at the wei-floor; roughly 36% sit in a dollar-or-greater friction regime; little mass lie inbetween. The separation between those two regimes is exactly what the framework would lead one to expect. Events are either at the arithmetic floor or at an externally driven friction scale. A single diffuse friction distribution would not naturally produce that bimodality.

Two scaling laws

The attributed dust and the sub-attribution remainder obey different scaling laws, and that difference explains why the latter has not previously been reported.

Term	What it is	Scales with	Visible at typical ALM volume?
$\Delta R^{\text{att}}$	per-portfolio leftover written to dustBalance	position size × geometric mismatch	yes
$\epsilon$	floor remainder at the NFPM / dustBalance seam	operation count × constant per token	no

A normal ALM running on the order of 100 rebalances per month, roughly three orders of magnitude below this rig, would accumulate only about $0.003 of sub-attribution remainder across several years. That sits comfortably inside the noise from spam ERC-20s, fee residue, dashboard rounding, and other operational debris. Without both high volume and a reason to look for this pattern, it would be invisible.

The 177,353-rebalance count is what carries the highest decimal leg above the dollar threshold. That volume is the reason the remainder surfaces as a measurement rather than remaining theoretical dust. In that sense the observation depends on scale, but the mechanism does not.

Why ledgered ALMs can see this, and unledgered ALMs cannot

A position manager that does not write a per-portfolio dustBalance entry at rebalance cannot isolate the seam at all. The full flow, both $\Delta R^{\text{att}}$ and $\epsilon$, sits at the contract address as a single opaque inventory and is typically described as operator fees, residue, or simply ignored. The seam becomes visible only through the subtraction balanceOf - Σ dustBalance, and that subtraction is possible only because the ledger exists.

Caveats on the exact figure

Two reasons counsel caution about the headline $2.83 figure. Neither affect the structural reading, and neither alters the per-event corroboration above.

Display-layer rounding inflates noise on the aggregate alone. The state payload surfacing portfolioDust.entries[].amount rounds the displayed amount to a few decimal places per entry. Across the thirteen entries used in the attribution sum, this introduces up to about $\sim$ $0.85 of measurement noise on the USDC and EURC totals. The wei-precise gap is therefore likely in the $1.95–$2.20 range, dominated by the $1.75 WETH leg. WETH is displayed at a higher precision, so its contribution to that rounding error is sub-cent. The on-disk diffusion log used for the per-event analysis is unaffected by this display layer, and the precise aggregate can in any case be recovered directly on chain through paired balanceOf and dustBalance calls.

Non-rebalance sources cannot be excluded with absolute certainty from the aggregate alone. Direct ERC-20 transfers are admissible in principle. Against those possibilities, however, the observed magnitudes are uniform per token and lie within an order of magnitude of “a few atomic units per operation”, which the shape integer-arithmetic leakage would produce and not the shape one-off deposits would be expected to produce. The per-event distribution sharpens the point further; one-off deposits would not place 26.8% of same-range rebalances at float64-precise zero and leave the rest in a clean two-regime bimodal distribution. On both signals, the sub-attribution reading remains the more natural one.

A final caution is that the same-range filter does not isolate events with literally no token movement. The rebalance path performs internal token movement even with swap.amountIn == 0, because reminting at the same range with a different liquidity amount, for example after dust-compounder absorption, still requires transfers. The paper-territory median is therefore slightly inflated by such implicit swap events. Tightening the filter to require zero raw amounts on both legs of the swap would move the median still closer toward the pure arithmetic floor. The 2.74 gwei figure should be read as an upper bound on that floor on this rig.

What this changes, and what it does not

The Geometric Siphon framework predicts a closed-form per-rebalance residual, and the original existence claim was tested at the per-portfolio level against 35,910 production events. The observation reported here sits one level deeper, and in two independent forms. A cumulative inventory gap of roughly $2.83 implies a WETH-equivalent remainder of about ~4.88 gwei per rebalance over the system lifetime. A per-event distribution over 2,561-event same-range zero-swap subset places the median at 2.74 gwei WETH, with 26.8% of events at float64-precise zero and the remainder split between a floor band and a separate external friction shoulder. The two measurements land in the same gwei-scale band, and the per-event distribution carries the bimodal shape the closed form predicts at the seam between integer arithmetic and ledger attribution.

The conceptual point is narrower than the original paper, but sharper in one respect. The shared-balance architecture remains the precondition for cross-pool circulation and for the full siphon mechanism described there. Nothing in this note weakens that claim. What the present observation does suggest is that the residual generating geometry is more primitive than the architecture that first exposed it. The architecture did not create the residual; it made it visible, attributable, and reusable across positions. What is now observed is a smaller remainder surviving even beneath that attribution layer.

The two terms in $\Delta R=\Delta R^{\text{att}}+\epsilon$ obey different scaling laws. The attributed term scales with geometry and position size, and is therefore operationally meaningful even at moderate volume. The sub-attribution remainder scales with operation count and token-level constants, and is operationally negligible at ordinary volume, though mechanism-confirming where rebalance counts are sufficiently high.

The original framework named the residual and showed how a particular architecture makes it economically active across positions. This note records a smaller remainder surviving below that same architecture’s attribution surface. It is a lower level trace of the same one.

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References

1. Ryan, K. R. The Geometric Siphon: Existence, Equilibrium, and Directional Properties of the Residual in Concentrated Liquidity Portfolios. SSRN Preprint (April 2026). https://dx.doi.org/10.2139/ssrn.6686798
2. Adams, H., Zinsmeister, N., Salem, M., Keefer, R., Robinson, D. Uniswap v3 Core. Technical report (March 2021). https://uniswap.org/whitepaper-v3.pdf