A Unified General Formula for Arbitrary Liquidity Operations in Weighted AMMs: Potential Applications to Intelligent Transportation Systems

Giuseppe Guido; Sami Shaffiee Haghshenas; Sina Shaffiee Haghshenas; Vittorio Astarita

arxiv: 2606.22118 · v1 · pith:PLWDY4JOnew · submitted 2026-06-20 · 💱 q-fin.PR

A Unified General Formula for Arbitrary Liquidity Operations in Weighted AMMs: Potential Applications to Intelligent Transportation Systems

Vittorio Astarita , Giuseppe Guido , Sina Shaffiee Haghshenas , Sami Shaffiee Haghshenas This is my paper

Pith reviewed 2026-06-26 11:12 UTC · model grok-4.3

classification 💱 q-fin.PR

keywords weighted AMMliquidity operationsconstant function market makerresource allocationintelligent transportation systemsswap decompositionunified formuladecentralized mechanisms

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The pith

A single closed-form formula unifies all liquidity operations in weighted AMMs by showing the conservation invariant is identical to the general allocation formula.

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

The paper establishes one closed-form expression that covers the four standard liquidity operations in weighted constant-function market makers plus two new cases. It demonstrates this by adapting the weighted invariant to handle arbitrary contributions and withdrawals across multiple resources. A sympathetic reader would care because the same expression now serves for both conservation checks and direct allocation calculations, removing the need for case-by-case derivations. The work further proves that non-proportional operations reduce to a rebalancing swap followed by a proportional one when fees are absent. This unification is presented as directly usable for decentralized allocation of tokenized resources such as freight capacity or charging infrastructure.

Core claim

The weighted invariant supplies a single closed-form formula that computes every liquidity operation, including the two previously undocumented cases of partial-proportional contributions and fully non-proportional operations. The conservation invariant and the allocation formula are structurally the same object; the invariant itself functions as the general allocation formula. Two swap-decomposition theorems establish that, in a fee-less setting, any non-proportional multi-resource operation is equivalent to an internal rebalancing swap combined with a proportional operation, extending earlier single-resource results to the arbitrary case.

What carries the argument

The unified closed-form allocation formula obtained from the weighted constant-product invariant, shown to be identical to the conservation invariant and therefore serving as the direct expression for any operation.

If this is right

All four standard operations plus the two new ones are obtained from the identical expression without separate derivations.
Any non-proportional operation decomposes exactly into one rebalancing swap plus one proportional operation in a fee-less environment.
The same theorems that held for single-resource cases now hold for arbitrary numbers of resources.
The framework supplies a mathematically grounded mechanism for decentralized market-based coordination of transportation resources.

Where Pith is reading between the lines

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

The decomposition suggests that complex allocations in any multi-resource system can be implemented by first adjusting internal balances and then applying a simple proportional change.
If the formula is used in transportation networks, pricing signals for scarce resources could be generated without a central coordinator.
The absence of stability conditions in the base model leaves open whether real-world demand fluctuations would require additional safeguards.
Empirical checks could compare the formula's predictions against observed allocation outcomes in simulated freight or charging markets.

Load-bearing premise

The weighted invariant from Balancer-type AMMs can be directly adapted to model allocation over multiple tokenized resources in intelligent transportation systems without any additional domain-specific constraints or stability conditions.

What would settle it

A concrete multi-resource non-proportional operation whose result computed via the single formula differs from the result obtained by enforcing the original weighted invariant equality would falsify the unification.

read the original abstract

Intelligent transportation systems increasingly rely on decentralized mechanisms to allocate limited resources such as freight capacity, warehouse availability, charging infrastructure, and network bandwidth. Efficient allocation requires pricing mechanisms that adapt dynamically to demand while preserving system stability. This paper investigates weighted constant-function market makers as a decentralized resource allocation mechanism for intelligent transportation systems, adapting the weighted invariant from Balancer-type automated market makers to model a generalized formulation over multiple tokenized resources. The standard literature documents exactly four resource allocation operations: proportional contribution, proportional withdrawal, single-resource contribution, and single-resource withdrawal, each obtained via separate derivations. This paper presents a single closed-form formula that unifies all four cases and extends them to two previously undocumented operations: partial-proportional contributions and fully non-proportional operations. The unified formula reveals that the conservation invariant and the allocation formula are structurally identical; the invariant itself is the general allocation formula. We prove two swap-decomposition theorems showing that, in a fee-less environment, any non-proportional operation is equivalent to an internal rebalancing swap combined with a proportional operation. Both theorems generalize previous propositions from single-resource to arbitrary multi-resource operations. The proposed framework provides a mathematically grounded mechanism for decentralized market-based coordination in transportation networks.

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 unifies four liquidity operations in weighted AMMs into one closed-form expression plus two new cases and proves multi-resource swap-decomposition theorems, but the ITS application adds little beyond the math.

read the letter

The main thing here is the unified formula for arbitrary operations in weighted AMMs. The abstract says the four standard cases (proportional contribution and withdrawal, single-resource versions) were previously derived separately, and this work collapses them into one expression that also covers partial-proportional contributions and fully non-proportional operations. It further claims the conservation invariant itself serves as the allocation formula, and it proves two theorems that any non-proportional move equals an internal rebalancing swap plus a proportional one, now generalized from single-resource to arbitrary multi-resource pools.

That unification and the decomposition theorems look like the actual new pieces. If the derivations hold, they clean up some prior work and extend it cleanly in the fee-less setting. The constant-function property is used directly, which is standard in this literature.

The soft spot is the ITS framing. The abstract adapts the Balancer-style weighted invariant to tokenized resources like freight capacity or charging slots, but it does not introduce or test any physical constraints such as location, time windows, or capacity decay. The stress-test note is right on this: without those terms the constant-function property will not survive realistic demand, so the unified formula and theorems do not automatically transfer. The circularity point also lands—the claim that the invariant is the allocation formula reads more like a rephrasing than an independent result.

This is for readers already working on AMM generalizations or decentralized allocation mechanisms. The math section could be worth a referee's time if the derivations are fully spelled out and checked; the transportation application is too thin to carry much weight on its own. I would send it to review for the formal claims but expect the authors to tighten the scope or add domain constraints.

Referee Report

2 major / 1 minor

Summary. The paper claims a single closed-form formula that unifies the four documented liquidity operations in weighted constant-function AMMs (proportional contribution, proportional withdrawal, single-resource contribution, single-resource withdrawal) and extends them to two new operations (partial-proportional contributions and fully non-proportional operations). It states that the conservation invariant is structurally identical to the allocation formula, with the invariant itself serving as the general allocation formula. Two swap-decomposition theorems are proved for fee-less environments, showing non-proportional operations equivalent to an internal rebalancing swap plus a proportional operation, generalizing prior single-resource results to arbitrary multi-resource cases, with applications to decentralized resource allocation in intelligent transportation systems.

Significance. If the formula is independently derived and the theorems hold without circularity or unstated assumptions, the unification could simplify analysis of multi-asset operations in DeFi. However, the direct transfer of the Balancer-style weighted invariant to ITS tokenized resources (freight, charging, bandwidth) lacks justification for ignoring physical constraints such as location, time windows, and capacity decay, limiting practical significance outside pure financial settings. No machine-checked proofs, reproducible code, or falsifiable predictions are provided.

major comments (2)

[Abstract] Abstract: the explicit statement that 'the conservation invariant and the allocation formula are structurally identical; the invariant itself is the general allocation formula' indicates the central result may reduce to a definitional equivalence rather than an independent derivation. This places a high circularity burden on the claim of a 'unified general formula' and requires explicit separation of the invariant definition from the derived allocation expression to support the unification claim.
[Abstract] Abstract (ITS adaptation): the weighted invariant is asserted to model generalized multi-resource operations in ITS without additional domain-specific terms, but the manuscript provides no analysis of how physical constraints (location, time windows, capacity decay) preserve the constant-function property under realistic demand. This assumption is load-bearing for the application claims and risks invalidating both the unified formula and the swap-decomposition theorems outside fee-less financial settings.

minor comments (1)

The abstract would benefit from stating the explicit mathematical form of the unified closed-form formula to allow immediate assessment of its scope.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We appreciate the referee's constructive feedback on our manuscript. We address each of the major comments below and outline the revisions we intend to make.

read point-by-point responses

Referee: [Abstract] Abstract: the explicit statement that 'the conservation invariant and the allocation formula are structurally identical; the invariant itself is the general allocation formula' indicates the central result may reduce to a definitional equivalence rather than an independent derivation. This places a high circularity burden on the claim of a 'unified general formula' and requires explicit separation of the invariant definition from the derived allocation expression to support the unification claim.

Authors: We thank the referee for pointing out this potential issue of circularity. Upon review, we recognize that the phrasing in the abstract could be misinterpreted. The central result is an algebraic unification derived by applying the conservation invariant to each operation and showing equivalence. In the revised version, we will add a dedicated section or subsection that first recalls the invariant definition, then derives the general allocation expression step by step for each operation type, explicitly separating the two to demonstrate that the identity is a derived property rather than definitional. This will strengthen the claim of a unified general formula. revision: yes
Referee: [Abstract] Abstract (ITS adaptation): the weighted invariant is asserted to model generalized multi-resource operations in ITS without additional domain-specific terms, but the manuscript provides no analysis of how physical constraints (location, time windows, capacity decay) preserve the constant-function property under realistic demand. This assumption is load-bearing for the application claims and risks invalidating both the unified formula and the swap-decomposition theorems outside fee-less financial settings.

Authors: The core contributions—the unified formula and the swap-decomposition theorems—are derived and proved for fee-less weighted AMMs, independent of the ITS application. The ITS section is presented as a potential application area where tokenized resources could be allocated via such mechanisms. We acknowledge the lack of analysis on physical constraints. In revision, we will modify the abstract and relevant sections to clarify that the model assumes the constant-function property holds in the abstracted setting, and we will add a paragraph discussing the need for future extensions to account for location, time windows, and capacity decay. This does not affect the validity of the mathematical results in their stated domain. revision: partial

Circularity Check

1 steps flagged

Conservation invariant declared structurally identical to allocation formula

specific steps

self definitional [Abstract]
"The unified formula reveals that the conservation invariant and the allocation formula are structurally identical; the invariant itself is the general allocation formula."

The paper claims to derive a single closed-form formula unifying four (plus two new) operations from the weighted invariant, but then asserts the formula and invariant are structurally identical, so the 'unified' result is the invariant restated as the allocation rule by construction.

full rationale

The paper's central unification claim rests on presenting a single closed-form formula for operations and then explicitly stating that this formula is structurally identical to the conservation invariant, with the invariant itself serving as the general allocation formula. This matches the self-definitional pattern: the claimed derivation reduces to re-expressing the input invariant (adapted from Balancer) as the output, rather than an independent derivation. The swap-decomposition theorems and ITS adaptation are presented separately but do not alter the core equivalence. No load-bearing self-citations appear in the provided text. The result has partial circularity because the unification is achieved by noting the definitional identity.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on the standard weighted invariant from Balancer-type AMMs as the foundation for the generalized formulation, with no free parameters or invented entities explicitly introduced in the abstract.

axioms (1)

domain assumption The weighted invariant from Balancer-type automated market makers holds and can be adapted to a generalized formulation over multiple tokenized resources.
Invoked when adapting the mechanism to intelligent transportation systems resource allocation.

pith-pipeline@v0.9.1-grok · 5762 in / 1291 out tokens · 24356 ms · 2026-06-26T11:12:02.350619+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

2 extracted references · 2 canonical work pages · 1 internal anchor

[1]

(Eds.), Handbook on Blockchain

Constant Func- tion Market Makers: Multi-asset Trades via Convex Optimization, in: Tran, D.A., Thai, M.T., Krishnamachari, B. (Eds.), Handbook on Blockchain. Springer International Publishing, Cham, pp. 415–444. https://doi.org/ 10.1007/978-3-031-07535-3_13 Assmann, B., Degenbaev, U.,

work page doi:10.1007/978-3-031-07535-3_13
[2]

From Swap Axioms to Weighted Geometric Means: A Characterization of AMMs

From Swap Axioms to Weighted Geometric Means: A Characterization of AMMs [WWW Document]. https://doi.org/ 10.48550/ARXIV.2604.16898 Martinelli, F., Mushegian, N.,

work page internal anchor Pith review Pith/arXiv arXiv doi:10.48550/arxiv.2604.16898

[1] [1]

(Eds.), Handbook on Blockchain

Constant Func- tion Market Makers: Multi-asset Trades via Convex Optimization, in: Tran, D.A., Thai, M.T., Krishnamachari, B. (Eds.), Handbook on Blockchain. Springer International Publishing, Cham, pp. 415–444. https://doi.org/ 10.1007/978-3-031-07535-3_13 Assmann, B., Degenbaev, U.,

work page doi:10.1007/978-3-031-07535-3_13

[2] [2]

From Swap Axioms to Weighted Geometric Means: A Characterization of AMMs

From Swap Axioms to Weighted Geometric Means: A Characterization of AMMs [WWW Document]. https://doi.org/ 10.48550/ARXIV.2604.16898 Martinelli, F., Mushegian, N.,

work page internal anchor Pith review Pith/arXiv arXiv doi:10.48550/arxiv.2604.16898