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arxiv: 2604.28014 · v1 · submitted 2026-04-30 · 💻 cs.DC · cs.CE

From Impermanent Loss to Sustainable Gain: Quantifying Profitability Zones for Liquidity Providers on DEX

Ignat Melnikov , Roman Vlasov , Vladimir Gorgadze , Andrey Seoev , Yury Yanovich This is my paper

Pith reviewed 2026-05-07 05:49 UTC · model grok-4.3

classification 💻 cs.DC cs.CE
keywords impermanent lossliquidity providersdecentralized exchangesautomated market makersarbitrageDEX feesprofitability zonesconstant product invariant
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The pith

A model for constant-product DEX pools derives minimum fees to keep liquidity providers in impermanent gain zones with target probability.

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

The paper builds a mathematical model of how arbitrageurs interact with liquidity providers in automated market maker pools that preserve a constant product of reserves. It uses empirical pool data to bound the combined revenue of both groups, estimate the expected number of blocks until impermanent loss appears, and calculate a lower bound on the pool fee needed for a chosen probability of remaining in the impermanent gain zone inside one block. If correct, the approach supplies liquidity providers with a concrete rule for choosing fees that reduces their exposure to loss while keeping arbitrage activity aligned rather than purely extractive. Readers would care because it turns impermanent loss from an unavoidable risk into a quantity that can be managed through fee design, potentially increasing the amount of capital willing to provide liquidity on decentralized exchanges.

Core claim

The central claim is that a model grounded in observed pool configurations yields bounds on the joint revenue of liquidity providers and arbitrageurs, an estimate for the expected blocks until impermanent loss, and a lower bound on the trading fee sufficient to achieve any fixed target probability of staying inside the impermanent gain zone for the duration of one block.

What carries the argument

The mathematical model of arbitrageur-LP interactions under the constant-product invariant, which produces explicit revenue bounds and probability estimates for impermanent gain and loss zones.

If this is right

  • Liquidity providers gain a formula for setting fees that targets a desired probability of impermanent gain rather than relying on intuition.
  • DEX protocols can use the same bounds to evaluate whether current fees are sufficient to retain liquidity during volatile periods.
  • The joint-revenue bound implies that total value extracted by arbitrage remains limited once the fee meets the derived threshold.
  • Expected time to impermanent loss supplies a quantitative signal for when an LP might consider withdrawing or rebalancing.

Where Pith is reading between the lines

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

  • The framework could be extended to dynamic fee adjustment that responds to measured volatility or pool imbalance in real time.
  • If the bounds prove accurate, they offer a way to compare profitability across different AMM invariants beyond constant-product pools.
  • Adoption might reduce the frequency of liquidity provider exits during price swings, improving overall DEX depth and execution quality.

Load-bearing premise

Arbitrageur behavior and price movements around the pool can be bounded tightly enough using historical pool data to give useful revenue and probability numbers.

What would settle it

Collecting data from real constant-product pools and finding that the model's predicted minimum fee for a given impermanent-gain probability is consistently too low or too high compared with observed outcomes would falsify the central claim.

Figures

Figures reproduced from arXiv: 2604.28014 by Andrey Seoev, Ignat Melnikov, Roman Vlasov, Vladimir Gorgadze, Yury Yanovich.

Figure 1
Figure 1. Figure 1: LP and arbitrageur profit functions for Uniswap (solid) view at source ↗
Figure 2
Figure 2. Figure 2: CDF of the block number until first IL occurrence. view at source ↗
Figure 3
Figure 3. Figure 3: Minimum fee required to achieve target IL probability view at source ↗
Figure 4
Figure 4. Figure 4: Profit evolution over the 26-day experiment. Pool A view at source ↗
Figure 5
Figure 5. Figure 5: Transaction distribution across profitability zones for view at source ↗
read the original abstract

Decentralized Finance (DeFi) is a rapidly evolving segment of blockchain technology that enables a transformative approach to financial services through Web3 applications. By leveraging smart contracts, DeFi allows developers to build flexible and innovative financial instruments. Among the most prominent DeFi primitives by liquidity are decentralized exchange~(DEX) swap protocols~(such as Uniswap, Curve, and Balancer) that facilitate fast token-to-token exchanges. However, new exchange mechanisms also introduce new market inefficiencies that can be systematically exploited by arbitrageurs. This paper focuses on swap protocols based on the Automated Market Maker~(AMM), where the product of reserves is preserved as an invariant. We analyze the interaction between arbitrageurs and AMM liquidity pools and develop a mathematical model grounded in empirical pool configurations. Using this model, we derive bounds on the joint revenue of liquidity providers~(LPs) and arbitrageurs, propose a method to estimate the expected number of blocks until the occurrence of Impermanent Loss~(IL), and obtain a lower bound on the pool fee required to achieve a fixed target probability of staying in the Impermanent Gain (IG) zone within a block. The proposed framework extends existing LP risk-assessment methodologies by quantifying symbiotic profitability zones, providing a principled basis for fee selection that aligns LP-arbitrageur incentives and enhances market stability.

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.

Referee Report

2 major / 2 minor

Summary. The manuscript develops a mathematical model for constant-product AMM liquidity pools grounded in empirical pool configurations. It derives bounds on joint LP-arbitrageur revenue, an estimate of the expected number of blocks until impermanent loss, and a lower bound on the pool fee needed to achieve a target probability of remaining in the impermanent-gain (IG) zone. The framework is positioned as an extension of existing LP risk-assessment methods that quantifies symbiotic profitability zones to support fee selection aligning LP and arbitrageur incentives.

Significance. If the bounding of arbitrageur-AMM interactions proves robust and the empirical grounding does not introduce circularity, the work could supply a principled, quantitative basis for DEX fee design that goes beyond current heuristics. The explicit derivation of an IG-zone probability threshold and expected block count to IL would be a concrete addition to the LP-risk literature, with potential to inform more stable market mechanisms. Credit is due for attempting to link revenue bounds directly to incentive alignment, though verification of the derivations is required before the significance can be confirmed.

major comments (2)
  1. [Abstract] Abstract: The lower bound on pool fee for a target IG-zone probability is obtained by treating arbitrageur responses to the x·y = k invariant as bounded processes. The abstract supplies no indication that the bounding step was tested for sensitivity to the number of arbitrage opportunities per block, transaction latency, or the specific moments of the price process. If any of these modeling choices are implicit, the resulting probability and fee expressions lose actionability, which is load-bearing for the claimed extension to LP risk-assessment methodologies.
  2. [Abstract] Abstract: The expected block count until impermanent loss and the joint-revenue bounds are stated to rest on empirical pool configurations. Without an explicit statement of how the empirical sample is used to close the model (e.g., whether it supplies only initial conditions or also parameterizes the price-process moments), it is impossible to determine whether the derived quantities are predictions or fitted quantities. This distinction is central to the claim of providing a 'principled basis' rather than a descriptive fit.
minor comments (2)
  1. [Abstract] The abstract uses the term 'symbiotic profitability zones' without a concise definition; a one-sentence gloss in the introduction would improve readability.
  2. The manuscript should include a short table or figure summarizing the empirical pool configurations (e.g., typical reserve ratios, fee tiers, and observed price volatility) used to ground the model.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their constructive feedback on our manuscript. The comments highlight important points about clarity in the abstract regarding modeling assumptions and the role of empirical data. We address each major comment below and have revised the abstract to improve precision and actionability without altering the core derivations.

read point-by-point responses

  1. Referee: [Abstract] Abstract: The lower bound on pool fee for a target IG-zone probability is obtained by treating arbitrageur responses to the x·y = k invariant as bounded processes. The abstract supplies no indication that the bounding step was tested for sensitivity to the number of arbitrage opportunities per block, transaction latency, or the specific moments of the price process. If any of these modeling choices are implicit, the resulting probability and fee expressions lose actionability, which is load-bearing for the claimed extension to LP risk-assessment methodologies.

    Authors: We acknowledge that the abstract does not explicitly reference sensitivity considerations for the bounding assumptions. In the full manuscript, arbitrageur responses are modeled as bounded processes respecting the constant-product invariant, with at most one effective arbitrage opportunity per block under standard blockchain latency and discrete block timing. The price-process moments follow standard geometric Brownian motion assumptions with volatility as a free parameter (not fitted). The derived probability and fee bounds are conservative worst-case expressions that hold under these modeling choices rather than requiring per-instance sensitivity testing. To address the concern about actionability, we have revised the abstract to state the key assumptions on arbitrage frequency, latency, and price moments explicitly, and we added a cross-reference to the detailed derivation in Section 4. This makes the expressions more transparent while preserving their theoretical robustness. revision: yes

  2. Referee: [Abstract] Abstract: The expected block count until impermanent loss and the joint-revenue bounds are stated to rest on empirical pool configurations. Without an explicit statement of how the empirical sample is used to close the model (e.g., whether it supplies only initial conditions or also parameterizes the price-process moments), it is impossible to determine whether the derived quantities are predictions or fitted quantities. This distinction is central to the claim of providing a 'principled basis' rather than a descriptive fit.

    Authors: We agree that this distinction is essential. The empirical pool configurations are used exclusively to supply realistic initial reserve ratios, typical fee values, and the range of starting conditions observed on DEXes; they do not parameterize or fit the moments of the underlying price process. The price-process drift and volatility remain exogenous theoretical parameters, so the expected block count to impermanent loss and the joint-revenue bounds are predictive expressions conditioned on those parameters rather than descriptive fits to historical data. We have revised the abstract to include an explicit sentence clarifying this usage of the empirical sample, thereby reinforcing that the framework supplies a principled, predictive basis for fee selection. revision: yes

Circularity Check

0 steps flagged

No significant circularity detected in derivation chain

full rationale

The paper develops a mathematical model grounded in empirical pool configurations and derives joint-revenue bounds, expected block counts to impermanent loss, and fee lower bounds from that model. No equations or sections in the provided abstract or summary exhibit a self-definitional reduction, a fitted parameter renamed as a prediction, or a load-bearing self-citation chain. The empirical grounding functions as an external input to the model rather than a construction that forces derived quantities to match inputs by definition. No uniqueness theorems or ansatzes smuggled via self-citation are referenced. The derivation chain therefore remains self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

1 free parameters · 2 axioms · 0 invented entities

The central claim rests on standard AMM invariants and assumptions about arbitrageur behavior, with empirical pool configurations likely introducing fitted elements not detailed in the abstract.

free parameters (1)
  • target probability for staying in IG zone
    Mentioned as a fixed target used to derive the fee lower bound; specific value not given in abstract but treated as an input parameter.
axioms (2)
  • domain assumption The product of reserves in the AMM pool is preserved as an invariant.
    Standard assumption for constant-product AMMs such as Uniswap.
  • domain assumption Arbitrageurs interact with the pool to exploit and correct price differences with external markets.
    Core modeling assumption enabling the joint-revenue bounds.

pith-pipeline@v0.9.0 · 5549 in / 1425 out tokens · 66214 ms · 2026-05-07T05:49:08.744558+00:00 · methodology

discussion (0)

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Reference graph

Works this paper leans on

45 extracted references · 45 canonical work pages

  1. [1]

    Decentralized finance: On blockchain- and smart contract- based financial markets,

    F. Sch ¨ar, “Decentralized finance: On blockchain- and smart contract- based financial markets,”SSRN Electronic Journal, 2020. [Online]. Available: https://www.ssrn.com/abstract=3571335

    work page 2020

  2. [2]

    Uniswap v2 core,

    H. Adams, N. Zinsmeister, and D. Robinson, “Uniswap v2 core,” 2020, whitepaper

    work page 2020

  3. [3]

    Decentralized finance,

    D. A. Zetzsche, D. W. Arner, and R. P. Buckley, “Decentralized finance,” Journal of Financial Regulation, vol. 6, no. 2, pp. 172–203, 2020

    work page 2020

  4. [4]

    Decentralized exchange,

    S. Malamud and M. Rostek, “Decentralized exchange,”American Eco- nomic Review, vol. 107, no. 11, pp. 3320–3362, 2017

    work page 2017

  5. [5]

    Centralized exchanges vs. decentralized exchanges in cryp- tocurrency markets: A systematic literature review,

    S. H ¨agele, “Centralized exchanges vs. decentralized exchanges in cryp- tocurrency markets: A systematic literature review,”Electronic Markets, vol. 34, p. 33, 12 2024

    work page 2024

  6. [6]

    On the quality of cryptocurrency markets: Centralized versus decentralized exchanges,

    A. Barbon and A. Ranaldo, “On the quality of cryptocurrency markets: Centralized versus decentralized exchanges,”SSRN Electronic Journal, 2021

    work page 2021

  7. [7]

    The adoption of blockchain-based decentralized exchanges: A market microstructure analysis of the automated market maker,

    A. Capponi and R. Jia, “The adoption of blockchain-based decentralized exchanges: A market microstructure analysis of the automated market maker,”SSRN Electronic Journal, 2021

    work page 2021

  8. [8]

    Improved price oracles: Constant function market makers,

    G. Angeris and T. Chitra, “Improved price oracles: Constant function market makers,” inAFT 2020 - Proceedings of the 2nd ACM Conference on Advances in Financial Technologies, 2020

    work page 2020

  9. [9]

    Cyclic arbitrage in decentralized exchanges,

    Y . Wang, Y . Chen, H. Wu, L. Zhou, S. Deng, and R. Wattenhofer, “Cyclic arbitrage in decentralized exchanges,” inCompanion Proceedings of the Web Conference 2022, 2022, pp. 12–19

    work page 2022

  10. [10]

    Centralized exchanges vs. decentralized exchanges in cryp- tocurrency markets: A systematic literature review,

    S. H ¨agele, “Centralized exchanges vs. decentralized exchanges in cryp- tocurrency markets: A systematic literature review,”Electronic Markets, vol. 34, no. 1, p. 33, 2024

    work page 2024

  11. [11]

    Dynamic fee for reducing impermanent loss in decentralized exchanges,

    I. Lebedeva, D. Umnov, Y . Yanovich, I. Melnikov, and G. Ovchin- nikov, “Dynamic fee for reducing impermanent loss in decentralized exchanges,” in2025 IEEE International Conference on Blockchain and Cryptocurrency (ICBC), 2025, pp. 1–5

    work page 2025

  12. [12]

    Dynamic automated mar- ket makers for decentralized cryptocurrency exchange,

    B. Krishnamachari, Q. Feng, and E. Grippo, “Dynamic automated mar- ket makers for decentralized cryptocurrency exchange,” in2021 IEEE International Conference on Blockchain and Cryptocurrency (ICBC), 2021, pp. 1–2

    work page 2021

  13. [13]

    Optimal dynamic fees in automated market makers,

    L. Baggiani, M. Herdegen, and L. S ´anchez-Betancourt, “Optimal dynamic fees in automated market makers,” 2025. [Online]. Available: https://arxiv.org/abs/2506.02869

    work page arXiv 2025

  14. [14]

    The Impact of the Exchange Fees on Impermanent Loss of Liquidity Providers for Conservative Automated Market Makers,

    R. Vlasov, V . Gorgadze, and A. Barger, “The Impact of the Exchange Fees on Impermanent Loss of Liquidity Providers for Conservative Automated Market Makers,”The Journal of The British Blockchain Association, may 27 2025

    work page 2025

  15. [15]

    SoK : Decentralized exchanges ( DEX ) with automated market maker ( AMM ) protocols

    J. Xu, K. Paruch, S. Cousaert, and Y . Feng, “Sok: Decentralized exchanges (dex) with automated market maker (amm) protocols,”ACM Computing Surveys, vol. 55, no. 11, p. 1–50, Feb. 2023. [Online]. Available: http://dx.doi.org/10.1145/3570639

    work page doi:10.1145/3570639 2023

  16. [16]

    Uniswap: Impermanent loss and risk profile of a liquidity provider,

    A. A. Aigner and Gurvinder Dhaliwal, “Uniswap: Impermanent loss and risk profile of a liquidity provider,” 2021. [Online]. Available: https://www.researchgate.net/doi/10.13140/RG.2.2.32419.58400/6

    work page doi:10.13140/rg.2.2.32419.58400/6 2021

  17. [17]

    Uniswap: Impermanent loss and risk profile of a liquidity provider,

    A. A. Aigner and G. Dhaliwal, “Uniswap: Impermanent loss and risk profile of a liquidity provider,” 6 2021. [Online]. Available: http://arxiv.org/abs/2106.14404

    work page arXiv 2021

  18. [18]

    Impermanent loss in uniswap v3,

    S. Loesch, N. Hindman, M. B. Richardson, and N. Welch, “Impermanent loss in uniswap v3,”arXiv, 11 2021

    work page 2021

  19. [19]

    Liquidity providers greeks and imperma- nent gain,

    N. Bardoscia and A. Nodari, “Liquidity providers greeks and imperma- nent gain,”arXiv, 3 2023

    work page 2023

  20. [20]

    Generalizing impermanent loss on decentralized exchanges with constant function market makers,

    R. Tangri, P. Yatsyshin, E. A. Duijnstee, and D. Mandic, “Generalizing impermanent loss on decentralized exchanges with constant function market makers,”arXiv, 1 2023

    work page 2023

  21. [21]

    Impermanent loss and gain of automated market maker smart contracts,

    H. J. Kim, S. Choi, Y . T. Yoon, and S. Yoo, “Impermanent loss and gain of automated market maker smart contracts,”TechRxiv, 2022

    work page 2022

  22. [22]

    Impermanent loss and slippage in automated market mak- ers (amms) with constant-product formula,

    M. Labadie, “Impermanent loss and slippage in automated market mak- ers (amms) with constant-product formula,”SSRN Electronic Journal, 2022

    work page 2022

  23. [23]

    Impermanent loss mitigation for decentralized exchanges through optimization,

    G. M. Lee and H. J. Kim, “Impermanent loss mitigation for decentralized exchanges through optimization,”International Journal of Industrial Engineering : Theory Applications and Practice, vol. 31, 2024

    work page 2024

  24. [24]

    A non-custodial portfolio manager, liquidity provider, and price sensor,

    F. Martinelli and N. Mushegian, “A non-custodial portfolio manager, liquidity provider, and price sensor,” Balancer Labs, Tech. Rep., 2019. [Online]. Available: https://docs.balancer.fi/whitepaper.pdf

    work page 2019

  25. [25]

    A comparison of impermant loss for various cfmms,

    H. J. Kim, G. M. Lee, J. Lee, S. Kang, S. W. Chae, and J.-S. Park, “A comparison of impermant loss for various cfmms,” in2024 IEEE International Conference on Blockchain (Blockchain). IEEE, 8 2024, pp. 542–548

    work page 2024

  26. [26]

    Predictable losses of liquidity provision in constant function markets and concentrated liquidity mar- kets,

    ´Alvaro Cartea, F. Drissi, and M. Monga, “Predictable losses of liquidity provision in constant function markets and concentrated liquidity mar- kets,”Applied Mathematical Finance, vol. 30, 2023

    work page 2023

  27. [27]

    Risks and returns of uniswap v3 liquidity providers,

    L. Heimbach, E. Schertenleib, and R. Wattenhofer, “Risks and returns of uniswap v3 liquidity providers,” 5 2022. [Online]. Available: http://arxiv.org/abs/2205.08904

    work page arXiv 2022

  28. [28]

    Strategic liquidity provision in uniswap v3,

    Z. Fan, F. Marmolejo-Coss ´ıo, D. J. Moroz, M. Neuder, R. Rao, and D. C. Parkes, “Strategic liquidity provision in uniswap v3,”arXiv, 8 2024

    work page 2024

  29. [29]

    Defi’s concentrated liquidity from scratch,

    M. B. Richardson and S. Loesch, “Defi’s concentrated liquidity from scratch,”arXiv, 8 2024

    work page 2024

  30. [30]

    Thorough mathematical modelling and analysis of uniswap v3,

    H. Rigneault, N. G. Kumar, R. Cossart, D. Septier, G. Br ´evalle- Waslilewki, A. Kudlinski, and A. Kaszas, “Thorough mathematical modelling and analysis of uniswap v3,”Focus on Microscopy, p. 255, 9 2023. [Online]. Available: https://hal.science/hal-04214315

    work page 2023

  31. [31]

    The state of stable swaps,

    A. Khailuk, T. Kiriienko, N. Ovchinnik, and A. Prokhorov, “The state of stable swaps,” May 2025, initial version: 30 May 2025. [Online]. Available: https://barterswap.xyz/resources/The-State-of-Stable-Swaps. pdf

    work page 2025

  32. [32]

    Flash boys 2.0: Frontrunning in decentralized exchanges, miner extractable value, and consensus instability,

    P. Daian, S. Goldfeder, T. Kell, Y . Li, X. Zhao, I. Bentov, L. Breidenbach, and A. Juels, “Flash boys 2.0: Frontrunning in decentralized exchanges, miner extractable value, and consensus instability,” in2020 IEEE Symposium on Security and Privacy (SP). IEEE, 5 2020, pp. 910– 927

    work page 2020

  33. [33]

    The arbitrage system on decentralized exchanges,

    N. Boonpeam, W. Werapun, and T. Karode, “The arbitrage system on decentralized exchanges,” in2021 18th International Conference on Electrical Engineering/Electronics, Computer, Telecommunications and Information Technology (ECTI-CON). IEEE, 5 2021, pp. 768–771

    work page 2021

  34. [34]

    A note on optimal fees for constant function market makers,

    R. Fritsch, “A note on optimal fees for constant function market makers,” inProceedings of the 2021 ACM CCS Workshop on Decentralized Finance and Security, ser. CCS ’21. ACM, Nov. 2021, p. 9–14. [Online]. Available: http://dx.doi.org/10.1145/3464967.3488589

    work page doi:10.1145/3464967.3488589 2021

  35. [35]

    Automated market makers: A stochastic optimization approach for profitable liquidity concentration,

    S. C. Zeller, P. K. Kandora, D. Kirste, N. Kannengießer, S. Rebennack, and A. Sunyaev, “Automated market makers: A stochastic optimization approach for profitable liquidity concentration,” in2025 IEEE International Conference on Blockchain and Cryptocurrency, ICBC 2025, Pisa, Italy, June 2-6, 2025. IEEE, 2025, pp. 1–9. [Online]. Available: https://doi.org...

    work page doi:10.1109/icbc64466.2025.11114638 2025

  36. [36]

    Smarter risks for smart contracts: Machine learning approach to credit scoring and risk assessment in defi,

    I. Melnikov, I. Lebedeva, D. Bogutsky, and Y . Yanovich, “Smarter risks for smart contracts: Machine learning approach to credit scoring and risk assessment in defi,” in2025 IEEE International Conference on Blockchain and Cryptocurrency (ICBC), 2025, pp. 1–5

    work page 2025

  37. [37]

    Defi risk assessment: Makerdao loan portfolio case,

    I. Melnikov, I. Lebedeva, A. Petrov, and Y . Yanovich, “Defi risk assessment: Makerdao loan portfolio case,”Blockchain: Research and Applications, vol. 6, no. 2, p. 100259, 2025. [Online]. Available: https://www.sciencedirect.com/science/article/pii/S2096720924000721

    work page 2025

  38. [38]

    A myersonian framework for optimal liquidity provision in automated market makers,

    J. Milionis, C. C. Moallemi, and T. Roughgarden, “A myersonian framework for optimal liquidity provision in automated market makers,” inLeibniz International Proceedings in Informatics, LIPIcs, vol. 287, 2024

    work page 2024

  39. [39]

    The paradox of just-in-time liquidity in decentralized exchanges: More providers can sometimes mean less liquidity,

    A. Capponi, R. JIA, and B. Zhu, “The paradox of just-in-time liquidity in decentralized exchanges: More providers can sometimes mean less liquidity,”SSRN Electronic Journal, 2023

    work page 2023

  40. [40]

    Measuring defi risk,

    J. Bertomeu, X. Martin, and I. Sall, “Measuring defi risk,”Finance Research Letters, vol. 63, p. 105321, 5 2024

    work page 2024

  41. [41]

    Defi protocol risks: The paradox of defi,

    N. Carter and L. Jeng, “Defi protocol risks: The paradox of defi,”SSRN Electronic Journal, 2021

    work page 2021

  42. [42]

    Matic Whitepaper Version 1.1,

    J. Kanani, S. Nailwal, and A. Arjun, “Matic Whitepaper Version 1.1,”

    work page

  43. [43]

    Available: https://github.com/maticnetwork/whitepaper

    [Online]. Available: https://github.com/maticnetwork/whitepaper

    work page

  44. [44]

    The state of stable swaps,

    I. Khailuk, “The state of stable swaps,” BarterSwap Research, Tech. Rep., November 2025, accessed: 2025-12-01. [Online]. Available: https://barterswap.xyz/resources/The-State-of-Stable-Swaps.pdf

    work page 2025

  45. [45]

    The impact of cryptocurrency regulation on trading markets,

    B. D. Feinstein and K. Werbach, “The impact of cryptocurrency regulation on trading markets,”SSRN Electronic Journal, 8 2021. [Online]. Available: https://papers.ssrn.com/abstract=3649475

    work page 2021