Zeros Without Receipts

A note accompanying the July 2026 revision of The Hidden Microstructure¹. The first version of that paper reported an anomaly its own theory could not explain. The revision resolves it, and the fault lay not in the theory but in what an event log is able to say. Self-contained; assumes only passing familiarity with concentrated liquidity.

The dust ledger

A concentrated liquidity position must be rebalanced when the price leaves its range — withdraw, optionally swap, redeposit. The pool’s mint arithmetic demands the two tokens in an exact ratio fixed by the current price, so the deposit almost never fits; one token is consumed exactly (it binds), the other is left slack with a remainder. The system studied in the paper² keeps that remainder in a dust ledger, a standing balance per token per depositor shared across the depositor’s pools, drawn on by later rebalances as deposit material. Retained and recycled in this way, the leftover becomes a state variable of the portfolio, and the paper derives the law governing it.

Setup and notation

Dust ledger $D (t)$ : the manager’s standing-balance record, one entry per token per depositor, written at every rebalance.

DustCredited: the event the manager emits when a rebalance leaves a remainder and credits it to the ledger.

Binding / slack: the side consumed exactly by the new deposit, and the side left with the remainder, respectively.

Spoke: a token appearing in only one or two of a portfolio’s pools. The dataset carries 32 such token-portfolio pairs with enough history to measure (at least 50 own-pool events each).

Atom at zero: probability mass at exactly zero. A distribution “with an atom at zero” is one under which the balance is found at precisely nought some definite fraction of the time.

Walk: an off-chain reconstruction of the ledger’s path through time, assembled from the event history.

The prediction

The paper’s asymptotic theory gives the steady-state law of a spoke’s standing balance:

μ = q δ_{0} + (1 - q) ρ_{+},

an atom at zero with mass $q$ , plus a distribution $ρ_{+}$ of positive amounts. The atom is the theory’s most distinctive fingerprint. A spoke’s balance is zero precisely when its token bound at the last rebalance, so $q$ must equal the token’s binding probability, a quantity measurable per pool from its slippage behaviour. There is no free parameter to tune; the prediction and the measurement arrive by separate routes and are required to agree.

An anomaly, published

They did not agree. The first version of the paper tested the prediction by walking the event history, carrying forward the last amount a DustCredited event had reported for each token. On that walk, 29 of the 32 spokes appeared never to be empty. It reported the numbers and filed the discrepancy as an open problem. That was the honest reading of the data as they stood. It was also wrong.

The event that is not there

Three facts about the manager’s event emissions, each checkable directly against the indexed history, decide where the fault lay.

Among 89,806 DustCredited events there is not a single zero-amount row. The contract never announces a credit of nothing.
85.9% of rebalances credit exactly one of the pool’s two tokens. The slack side gets its receipt; the binding side appears nowhere in the log.
The remaining 13.9% credit both sides by small amounts — the cases where the correcting swap misses slightly, so that neither token binds exactly.

Binding is precisely the transition that empties a ledger entry, and binding emits nothing. A walk that carries forward the last credited amount is therefore not merely inaccurate about zeros; it is structurally incapable of representing one. The instrument was reporting its own blind spot as a property of the system.

Correcting the walk

The correction follows from the event schema, not from the answer one hopes to reach. At any own-pool rebalance a token either received a credit, in which case its balance is the credited amount, or it did not, in which case it bound and its balance is zero. (Sampling must begin at the token’s first own-pool event, else a token that joined its portfolio mid-life carries a spurious stretch of zero balance.) The effect of the correction is total.

Reconstruction	Median $P (balance = 0)$	Spokes reading below 0.05
Credits-only walk (as published)	0.000	32 of 32
Corrected walk	0.382	10 of 32
Binding probability (the prediction)	0.373

The corrected atom masses span nearly the whole unit interval, and they fall on the identity line against each spoke’s measured binding probability — slope 1.026, correlation 0.995, across all 32 spokes. The handful of spokes that genuinely are never empty survive at the tail where the binding probability itself approaches zero.

Why the correction is not fitted

A reconstruction revised until it agrees with the theory would prove nothing. Three checks distinguish this one.

Forced by the schema. The zero-amount count, the one-sided-credit rate, and the both-sided minority were established first; the corrected rule is the only reading of the ledger consistent with them. The slope and correlation of the identity line were never inputs to anything.
Window-independent. Re-derived on a history twelve days longer, the headline statistics reproduce to the third decimal place.
Selective in the right way. The other statistic built on the same walk, the correlation between spokes’ trajectories, barely moves under the correction (median 0.020 as published, 0.0234 corrected). A forward-fill blind spot should corrupt level statistics while leaving co-movement statistics intact, and that is the pattern observed.

Beyond the atom

The positive part of the corrected law resembles the idealised half-normal in bulk (81% of mass below the mean, against 79%) but not in the tails (median excess kurtosis 47.6, against 0.87). The heaviness is inherited from production slippage, which is neither symmetric nor Gaussian. Simulation driven by the empirical slippage law reproduces both the atom and the identity line, and remains stable at ten times the portfolio sizes the paper originally tested.

One loose end survives. Spokes’ balances are not quite independent, where the theory says they very nearly should be. Simulating on the actual shapes of the production portfolios assembles the measured correlation from three sources — wiring denser than the assumed star, several-fold larger leftovers on the minority of rebalances whose correcting swap goes missing, and correlated shocks through shared tokens’ prices, the paper’s own Connector Rule. Four of the five portfolios are reproduced with every input measured rather than fitted; the fifth comes within a factor of two and is recorded as a residual.

What this changes, and what it does not

For the paper, evidential standing rather than content. No theorem moved. Its most distinctive prediction has gone from apparently contradicted by the author’s own data to confirmed on-chain across all 32 spokes, with the per-pool binding probability pinned empirically to the quantity the mechanism requires. The per-spoke question is closed; the joint behaviour of coupled portfolios is the live front.

The exportable lesson is about instruments. Reconstruction from an event log inherits the log’s omissions exactly and silently. A contract that zeroes state without emitting anything is invisible to any history assembled from its emissions, and the error presents not as missing data but as a clean, confident, wrong measurement. The eventual defence was reconciliation against on-chain state, which localised the fault to the one family of statistics the reconciliation had not covered.

The design corollary, for anyone emitting events from stateful contracts, is to emit the consumption, not only the credit. A single zero-amount credit on the binding side would have made the atom directly observable and this note unnecessary. An earlier note, Residual of the Residual, recorded value sitting beneath this same system’s attribution surface; this one records transitions that surface cannot emit. The ledger is the instrument, and the instrument has edges.

References

Ryan, K. R. The Hidden Microstructure of Shared Balance Concentrated Liquidity: A Master Equation for the Dust Ledger and Propagation of Chaos. SSRN Preprint (May 2026, revised July 2026). https://papers.ssrn.com/sol3/papers.cfm?abstract_id=6745218
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

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