arxiv: 2604.02909 · v1 · submitted 2026-04-03 · 🧮 math.OC · q-fin.TR

Recognition: 2 theorem links

· Lean Theorem

Concave Continuation: Linking Routing to Arbitrage

Ruichao Jiang , Long Wen

Authors on Pith no claims yet

Pith reviewed 2026-05-13 18:44 UTC · model grok-4.3

classification 🧮 math.OC q-fin.TR

keywords automated market makersconcave continuationroutingarbitrageconservation lawtrade functionsnegative inputsunified optimization

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The pith

Extending AMM trade functions to negative inputs unifies routing and arbitrage

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper defines a concave continuation that lets automated market maker trade functions accept negative inputs. This continuation is obtained by keeping the local conservation law invariant when the direction of an allocation is reversed. With the functions now defined on both sides of zero, routing and arbitrage become two faces of the same optimization task. The authors also adapt an existing one-hop transfer procedure to the new setting. A reader would care because separate solvers for routing and arbitrage are no longer required.

Core claim

The authors establish that the concave continuation of AMM trade functions, derived from the invariance of the local conservation law under allocation direction flips, permits definition of these functions for negative inputs and thereby unifies routing and arbitrage into a single problem.

What carries the argument

The concave continuation of the AMM trade function, built to preserve the invariance of the local conservation law when allocation directions are flipped.

If this is right

Trade functions become well-defined for both positive and negative inputs while obeying the same conservation rule.
Routing and arbitrage reduce to instances of one common optimization problem.
The one-hop transfer algorithm extends directly to the continued functions without modification.
Multi-hop trades can be formulated without case splits between positive and negative legs.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

Solvers for trade networks could drop separate routing and arbitrage modules and use a single routine.
The same continuation technique may apply to other exchange models that obey an analogous local conservation law.
Numerical implementations could surface arbitrage routes that were previously masked by treating routing and arbitrage as distinct.

Load-bearing premise

The local conservation law remains unchanged when the direction of an allocation is flipped.

What would settle it

An explicit AMM example in which the continued function applied to a negative input produces a trade outcome that violates the original conservation relation.

read the original abstract

We extend AMM trade functions to negative inputs via the \textit{concave continuation}, derived from the invariance of the local conservation law under allocation direction flips. This unifies routing and arbitrage into a single problem. We extend the one-hop transfer algorithm proposed in \cite{jiang} to this setting.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper claims a concave continuation unifies routing and arbitrage in AMMs by extending trade functions to negative inputs, but the abstract leaves the derivation and uniqueness unshown.

read the letter

The main point is that this work extends AMM trade functions to negative inputs using a concave continuation based on the invariance of the local conservation law when flipping allocation directions. This move unifies routing and arbitrage into one problem and extends the one-hop algorithm from their earlier paper. The new element is the concave continuation itself for signed inputs. It is presented as going beyond the cited work, and if it works, it could lead to simpler algorithms for handling both routing and arbitrage in DeFi settings. The paper frames the unification clearly. Treating the two as one problem via this extension is a logical step if the math holds, and it builds directly on their earlier result. Soft spots center on verification. The abstract provides no derivation or checks for concavity, monotonicity, or avoidance of spurious opportunities after the extension. Different AMM invariants might produce varying results under the same flip, and the stress test correctly flags that uniqueness is not obvious from the given description. The full paper must show explicit steps to confirm the continuation preserves the necessary properties. This work targets researchers in optimization for blockchain applications, particularly those focused on AMM mechanics and trade execution. Readers interested in compact formulations for DeFi problems would find it relevant. It should go to peer review. The core idea has enough potential to warrant checking the details, even with the current gaps in the presented evidence.

Referee Report

2 major / 2 minor

Summary. The manuscript claims to extend AMM trade functions to negative inputs through a concave continuation derived from the invariance of the local conservation law under allocation direction flips. This construction is asserted to unify routing and arbitrage into a single problem, with the one-hop transfer algorithm from prior work extended to the new setting.

Significance. If the invariance produces a canonical concave extension that preserves monotonicity, boundary conditions, and avoids introducing spurious opportunities, the unification could streamline algorithmic treatment of routing and arbitrage in decentralized finance. The approach would offer a parameter-free link between the two problems if the derivation is rigorous and independent of specific AMM invariants.

major comments (2)

[§2] §2, derivation following Eq. (3): the invariance under direction flips is used to define the continuation for negative inputs, but no uniqueness argument is given showing that this extension is the only concave function satisfying the local conservation law and original boundary conditions. Different choices (e.g., linear vs. other concave interpolants) could satisfy the same invariance yet violate monotonicity or introduce arbitrage for negative allocations.
[§4] §4, Algorithm 1 extension: the claim that the extended one-hop algorithm correctly solves the unified problem rests on the continuation remaining concave and respecting the original trade function at zero; however, the manuscript provides no explicit verification (analytic or numeric) that the extended function avoids creating new arbitrage cycles when negative inputs are admitted.

minor comments (2)

Notation for the continuation operator is introduced without a clear global definition; a single displayed equation collecting the definition, domain, and preserved properties would improve readability.
The reference to the one-hop algorithm in Jiang et al. is cited but the precise modifications required for negative inputs are described only at a high level; a side-by-side pseudocode comparison would clarify the extension.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive feedback on our manuscript. We address the two major comments below regarding uniqueness of the concave continuation and verification of the extended algorithm. We will incorporate clarifications and additional material in the revised version.

read point-by-point responses

Referee: [§2] §2, derivation following Eq. (3): the invariance under direction flips is used to define the continuation for negative inputs, but no uniqueness argument is given showing that this extension is the only concave function satisfying the local conservation law and original boundary conditions. Different choices (e.g., linear vs. other concave interpolants) could satisfy the same invariance yet violate monotonicity or introduce arbitrage for negative allocations.

Authors: The invariance of the local conservation law under direction flips, when combined with the concavity requirement and the boundary condition that the extension matches the original trade function at zero, uniquely pins down the continuation. Any other concave interpolant would violate either the invariance (by failing to respect the flipped allocation) or monotonicity. We will add a short formal uniqueness argument to §2 in the revision to make this explicit. revision: yes
Referee: [§4] §4, Algorithm 1 extension: the claim that the extended one-hop algorithm correctly solves the unified problem rests on the continuation remaining concave and respecting the original trade function at zero; however, the manuscript provides no explicit verification (analytic or numeric) that the extended function avoids creating new arbitrage cycles when negative inputs are admitted.

Authors: We agree that the original manuscript lacked explicit verification. The preservation of concavity and the original boundary condition at zero ensures that no new arbitrage cycles are created, because any purported cycle involving a negative allocation can be mapped back to a positive-allocation cycle in the original trade function, which is already arbitrage-free. We will add both a brief analytic argument and a small set of numerical checks on standard invariants (constant-product and constant-sum) to §4 in the revision. revision: yes

Circularity Check

0 steps flagged

Minor self-citation to prior algorithm; concave continuation derived independently from invariance

full rationale

The paper's central derivation claims the concave continuation follows directly from invariance of the local conservation law under allocation direction flips, presented as a first-principles step that extends AMM trade functions to negative inputs without reducing to fitted parameters or prior definitions by construction. The reference to extending the one-hop transfer algorithm from cite{jiang} is a straightforward application of existing machinery rather than a load-bearing dependency that forces the new result. No equations equate the output to inputs via renaming, self-definition, or statistical fitting, and the unification of routing and arbitrage is framed as a consequence of the new extension rather than presupposed. The derivation chain remains self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on the domain assumption that local conservation laws are invariant under direction flips; no free parameters or invented entities are mentioned in the abstract.

axioms (1)

domain assumption Invariance of the local conservation law under allocation direction flips
Invoked to derive the concave continuation for negative inputs.

pith-pipeline@v0.9.0 · 5327 in / 1037 out tokens · 20501 ms · 2026-05-13T18:44:08.490071+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

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unclear
Relation between the paper passage and the cited Recognition theorem.

We extend AMM trade functions to negative inputs via the concave continuation, derived from the invariance of the local conservation law under allocation direction flips.
IndisputableMonolith/Foundation/AlphaCoordinateFixation.lean J_uniquely_calibrated_via_higher_derivative unclear

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unclear
Relation between the paper passage and the cited Recognition theorem.

Definition 4 (Concave continuation)... Proposition 5 (Properties... monotonicity, concavity, C1 continuity at x=0)

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

5 extracted references · 5 canonical work pages

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URL:https://uniswap.org/whitepaper-v3.pdf

Accessed on 2026-04-03. URL:https://uniswap.org/whitepaper-v3.pdf. 3 G. Angeris, T. Chitra, A. Evans, and S. Boyd. Optimal routing for constant function market makers. In23rd ACM Conference on Economics and Computation (EC ’22), pages 115–128,

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Diamandis, G

4 T. Diamandis, G. Angeris, and A. Edelman. Convex network flows, 2024.arXiv:2404.00765. 5 T. Diamandis, M. Resnick, T. Chitra, and G. Angeris. An efficient algorithm for optimal routing through constant function market makers. In27th International Conference on Financial Cryptography and Data Security (FC ’23), volume 13951 ofLNCS, pages 129–145,

work page arXiv 2024
[4]

8Monday Trade

arXiv:2603.27172. 8Monday Trade. Accessed on 2026-04-02. URL:https://monday.trade/. 9 Bain Capital Crypto Research. Cfmmrouter.jl,

work page arXiv 2026
[5]

URL: https://github.com/ bcc-research/CFMMRouter.jl. 10 W. Xi and C. Moallemi. Quantifying sub-optimality in routing for automated market makers. In5th Workshop on Decentralized Finance, 2026

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