Local Time Against Deployed Code

Summary

Deployed market code is full of hard edges. A placement rule rounds prices, a trigger line decides when the manager acts, and a range boundary changes how its holdings are valued. The price meets these surfaces again and again, and each meeting can destroy value. One exact law governs what it destroys, separating the cost of holding concentrated liquidity from the cost the operator’s own geometry creates, and both from the effect of watching on a cadence. The operator’s share has a closed form, the mint kerf, the value lost when a position is withdrawn and re-minted at a different composition, named for the material a saw destroys at every cut, fixed by the blade and not by the hand that guides it. Here the blade is the venue’s mint arithmetic, and the operator chooses only how often to cut and how far. The kerf sets a floor no band-keeping policy escapes, and retention deletes it rather than lowering it. Read backwards, the same firing events become an instrument for the price process.

Abstract

Every floor function, trigger line, and threshold in deployed market code is a surface in price space, and automated liquidity management dissipates value where the price meets them. That dissipation obeys one exact law, developed here in three directions and characterised by six theorems. The law is a pathwise identity splitting dissipated value into three terms of a single expansion: a continuous concavity leg, which occupies the slot priced by the loss-versus-rebalancing literature; an event leg, set by the operator’s own geometry; and a monitoring layer. Localised in price, the identity is a measure over the code’s discontinuity set. The event leg is priced exactly. An isolated re-placement pays a mint kerf, the maximum of two log ratios of target share to current share, a potential difference in the position’s composition coordinate computed from the venue’s mint arithmetic rather than assumed. The kerf implies a floor. On a diffusive era every band-maintaining policy in the stated classes dissipates at a rate near twice volatility squared over half-width squared; this tangency constant is sharp in the narrow limit, approached but not attained by two-sided reflection. Corrective swaps replace the constant by the fee scale. The floor is a property of emission, so retention deletes it rather than lowering it. Inverted, the law makes the deployed population a conditional measuring instrument. The naive pooled reading fails, defeated by the operators’ own policy filters; the conditioned small-delay reading identifies integrated jump-tail functionals per pool with explicit error under stated era and monitoring hypotheses, replicating across operator strata within pools. Algebraic, synthetic, production, and bytecode checks support the results: the cost residual at machine precision on 78,908 production rebalances and the exact layer holding against unmodified bytecode on four deployments across three chains. Under the stated landing conditions the jump surcharge closes the loop, hardening the floor at the tails the instrument itself measures.

The dissipation identity

Read from the price process, a venue’s deployed code is a locally finite set of surfaces, and the holding value $f(\cdot ,R)$ at policy state $R$ has kinks on them. Itô–Tanaka between firings, telescoped over the firing times ${\tau }_k$, gives a pathwise identity for the value $J_k$ surrendered at each impulse.

$$\sum _{k:{\tau }_k\le T}{J}_k=\underset{\text{boundary}}{\underbrace{f(X_0,R_0)-f(X_T,R_T)}}+\underset{\text{martingale}}{\underbrace{\sum _k{\int }_{{\tau }_k\wedge T}^{{\tau }_{k+1}\wedge T}{f}_{-}^{\prime }(X_s,R_k)d{X}_s}}+\underset{\text{local time}}{\underbrace{\frac{1}{2}\sum _k{\int }_{\mathbb{R}}{\Delta }_k{\ell }_T^a{f}_{R_k}^{\prime \prime }(da)}}$$

Here ${\ell }_T^a$ is the local time of the price at level $a$, the occupation density measuring how long the path lingers there, and $f_{R_k}^{\prime \prime }$ is the second-derivative measure of the holding value, carried on the code’s own discontinuity set.

The terms belong to one expansion and cannot double-count. The local-time term is the slot the loss-versus-rebalancing literature prices, generated by holding a concave payoff between rebalances. The left side is generated at re-placement instead, by the operator’s own geometry, and it is the term this paper prices exactly.

The mint kerf

A re-placement is not a trade at a price. It is a mint against the venue’s arithmetic, and the mint takes the binding side of two token constraints.

Write ${\omega }_0=x{s}^2/V$ and ${\omega }_1=y/V$ for the value shares the withdrawn position holds at price $s$, and ${\omega }_0^{\prime },{\omega }_1^{\prime }$ for the shares the target range demands there. The per-event log-cost, the mint kerf, is exactly

$$k=\max \left(\ln \frac{{\omega }_0^{\prime }}{{\omega }_0},\ln \frac{{\omega }_1^{\prime }}{{\omega }_1}\right)$$

from any range containing the price to any admissible range containing it, at any width pair. Width, centring and the price level enter only through the shares. Since the shares each sum to one, $k\ge 0$ always, and $k=0$ exactly when target and holdings shares agree.

That equality case is the free locus, and it settles the free-riding question. Composition is the charged variable, not centre motion. A policy may slide its range and widen it without paying a kerf, provided its composition does not move, and every bound below charges composition motion alone.

The floor and its constant

The per-event price implies a rate. Keeping a price inside a band means paying the kerf every time the band is left, and a verification argument on the composition coordinate turns that into a lower bound on the long-run dissipation rate $r$.

The bound is an infimum over designs rather than an optimum for one objective, and it holds for every policy that fixes its width and re-places without a corrective swap.

$$r\ge c_{\min }\frac{{\sigma }_s^2}{h^2},c_{\min }⟶2$$

with $h$ the range half-width in sqrt price and ${\sigma }_s$ the era sqrt-price volatility.

The constant is sharp. A two-sided reflection policy firing at displacement $xh$ and returning infinitesimally pays $c_{\text{refl}}(x)=1/(2x(1-x))$, minimised at $x=1/2$ where it equals exactly $2$, so the floor is approached and never attained inside the class.

Varying the width, or allowing a corrective swap, gives two further classes that enrich this one independently rather than nesting inside it, each with a floor of its own.

Corrective swaps do not remove the floor, they replace its constant with the fee scale. What removes it is retention, since the floor prices the act of emitting value at a re-placement, and an architecture that never emits deletes the leg rather than lowering it.

The population as an instrument

Run the law backwards and the operator population stops being a subject and becomes an apparatus. Every rebalance is a first-passage event stamped on chain, and the overshoot past the trigger records what moved the price there.

A diffusive move cannot manufacture a large overshoot inside a short actuation delay, but a jump can. Under the paper’s stated era and monitoring hypotheses, the rate of large exceedances at small delay reads the jump share ${\pi }_J$ with a controlled error.

$$|\mathcal{E}(\kappa ,\bar{a})-{\pi }_J(\bar{a})|\le \underset{\text{diffusive leakage}}{\underbrace{2Q(\kappa /2)}}+\underset{\text{resolution floor}}{\underbrace{{\pi }_J(\bar{a})\left(1-\frac{\mathcal{J}(2\kappa \sigma \sqrt{\bar{a}})}{\mathcal{J}(0)}\right)}}+\underset{\text{staleness}}{\underbrace{{\epsilon }_{\text{cens}}}}$$

where $\mathcal{E}(\kappa ,\bar{a})$ is the fraction of firings with actuation delay at most $\bar{a}$ whose overshoot exceeds $\kappa$ trailing local scales, ${\pi }_J(\bar{a})$ the jump-crossing share of those firings, $Q$ the normal survival function and $\mathcal{J}$ the integrated jump tail.

Pooling every operator’s events defeats this reading, since operators run their own dead bands and dwell rules, and those filters shape the sample before an analyst sees it.

Conditioned instead on each operator’s recovered geometry and restricted to small delay, it identifies a per-pool integrated jump-tail functional that replicates across operator strata inside the same pool.

What it yields is a bracket with a stated error budget rather than a consistent test, since a deployed trigger is whatever width its operator chose and does not shrink with the sample.

Cite as

@techreport{ryan2026localtime,
  author      = {Ryan, K. R.},
  title       = {Local Time Against Deployed Code: The Exact Cost Law of Concentrated Liquidity Rebalancing, Its Sharp Floor, and Its Inversion},
  institution = {SSRN},
  type        = {SSRN preprint},
  number      = {7259078},
  year        = {2026},
  month       = aug,
  doi         = {10.2139/ssrn.7259078},
  url         = {https://papers.ssrn.com/sol3/papers.cfm?abstract_id=7259078},
}

Fifth in the sequence begun by The Geometric Siphon, whose geometric residual and its vanishing condition this paper upgrades to a cost law, and continued in The Hidden Microstructure, which supplies the mint identity behind the retention result. It consumes the frequency and value channels of Operator & Quantisation Microstructure and reads its instrument on the operator population recovered by Operator Fingerprinting.