Dynamic Exponent Market Maker: Personalized Portfolio Manager and One Pool to Trade Them All

Wittawat Kositwattanarerk

REVIEW 3 major objections 4 minor 30 references

Dynamic Exponent Market Maker: Personalized Portfolio Manager and One Pool to Trade Them All

T0 review · 3 major / 4 minor · reviewed 2026-08-06 · deepseek-v4-flash

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Pith's one-line read Dynamic exponent market maker: a single pool whose invariant exponent adjusts at each deposit so every liquidity provider's token-value ratio is preserved, given no arbitrage.

desk verdict The DEMM math is right, but the flash-loan fix breaks the protocol's own invariant—interesting theory, not yet a deployable design. read the letter →

arxiv 2507.22732 v1 pith:5RWE7WUL submitted 2025-07-30 cs.IT math.IT

Wittawat Kositwattanarerk This is my paper

classification cs.ITmath.IT

keywords automatedmarketmakerdecentralizedexchangeconstantproductformuladynamicexponentone-sidedliquiditytoken-specificLPtokenimpermanentgainflashloanattack

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

This paper proposes a decentralized-exchange protocol, the dynamic exponent market maker (DEMM), in which a single pool holds many tokens and the exponent vector of the constant-product invariant $f(x)=\prod_{t=1}^n x_t^{w_t}$ is updated on every deposit and withdrawal. The central claim, Theorem 3.7, is that each liquidity provider's deposited token-value ratio is preserved over time, given there is no arbitrage opportunity. If the claim holds, liquidity providers could deposit any amount of any token, one-sided or otherwise, and a trader could swap any pair directly in one deep pool instead of hopping between separate pools. The paper also shows that adding liquidity improves the exchange rate for all trades, that partial providers can experience an impermanent gain, and that the protocol is vulnerable to a flash-loan attack unless paired with a preventive measure.

What carries the argument

The central object is the dynamic invariant $f(x)=\prod x_t^{w_t}$, where $w$ doubles as the pool's weight vector and as the count of token-specific LP tokens in circulation. This triple role is what lets the protocol accept arbitrary deposits: a one-sided deposit of token $t$ inflates both $r_t$ and $w_t$ in the same proportion, so the spot price $P_{o/i}=(r_i/w_i)/(r_o/w_o)$ is unchanged by liquidity provision, and each depositor's personal value ratio is frozen into their LP-token holdings.

What would settle it

Simulate the exact four-step sequence of Example 4.10 using the paper's formulas (3), (4), and (5), with a flash loan supplying the initial tokens: if the attacker's final holding exceeds the starting holding in any parameter range without a countermeasure, and if neither the randomized-delay nor the geometric-TWAP rule eliminates the profit, then the deployable-one-pool claim fails. Separately, to test Theorem 3.7, run an arbitrage path where the pool price is re-pegged to external prices after every block and check that a deposit followed by a later withdrawal reproduces the deposited value ratio exactly.

Watch

Extended reading notes

Core claim

The discovery is a self-balancing AMM design built on the identity between three objects: the exponent vector $w$ of the invariant function, the weights of tokens by total value in the pool, and the numbers of token-specific LP tokens in circulation. Liquidity provision updates $w_t$ to $w_t(r_t+\Delta r_t)/r_t$, withdrawal subtracts redeemed LP tokens from $w$, and trading leaves $w$ fixed. Theorem 3.7 proves that, under the no-arbitrage condition, a depositor who later withdraws receives token amounts whose total value ratio equals the ratio at deposit time. The paper further proves (Theorem 4.2) that any liquidity provision improves or leaves unchanged the output of every swap, a violation of the monotone-in-liquidity axiom of the reference axiomatic framework, and it documents a four-step flash-loan attack that exploits one-sided deposits.

Load-bearing premise

The protocol is usable only if a practical defense against the flash-loan attack described in Section 4.6 exists; the paper explicitly says the effectiveness of the two countermeasures it sketches is a topic for future research.

Editorial extensions

If this is right

Traders would be able to swap any pair of tokens directly in one pool, avoiding multi-hop swaps and their extra fees.
Liquidity providers could deposit any amounts, including a single token, and still have their portfolio value ratio preserved over time, per Theorem 3.7.
Because liquidity provision can only improve quoted swap outputs (Theorem 4.2), the protocol deliberately breaks the monotone-in-liquidity axiom of the AMM axiomatic framework.
Partial providers can gain more of every deposited token when a token they did not supply depreciates, a new 'impermanent gain' effect absent from standard two-token pools.
Deployment requires a working defense against the Section 4.6 flash-loan attack; the paper leaves the effectiveness of its two candidate defenses to future research.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

The paper leaves implicit that the pool's curvature is now a governance variable: by controlling which tokens can be added and removed, a community decides how much value is redistributed between depositors when prices move.
A testable extension not pursued here is to compare the randomized-delay and geometric-TWAP countermeasures under identical price paths to see which one actually blocks the Example 4.10 attack without breaking Theorem 3.7.
The impermanent-gain effect suggests a DEMM pool acts as a transfer between depositors with different rebalancing needs; an explicit calculation of who pays for a partial provider's gain would test whether the effect survives transaction fees.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Start with the punchline: the DEMM math is correct and the idea is genuinely novel, but the flash-loan mitigation section is not just unfinished—the geometric TWAP fix as written contradicts the protocol's own state invariant. So you have a solid theoretical model and a deployability gap the author openly concedes.

What's new: token-specific LP tokens plus an exponent that adapts to deposit proportions. That is distinct from Balancer's fixed weights, Bancor's intermediate token, and the prior generalizations cited. The author proves the key properties cleanly: w equals the LP supply at all times (Prop 3.5), the pool stays exponent-balanced under no arbitrage (Prop 3.6), deposit value ratios are preserved (Thm 3.7), and spot prices are unchanged by liquidity events (Prop 4.1). I checked the algebra; it holds. The 'impermanent gain' phenomenon for partial LPs is a real consequence, and the flash loan attack in Example 4.10 is clearly explained.

Soft spots, proportionate: Theorem 4.2's proof relies on a calculus inequality stated without proof—true, but a referee should ask for a one-line argument. Typos are minor. The serious problem is Section 4.6. The random-delay countermeasure is underspecified: does the deposit execute against the state at submission or at execution? Either choice creates a manipulation window, and the paper doesn't say. The geometric TWAP variant is internally inconsistent. Setting w'_t=(r_t+\Delta r_t)G_t makes the exponent for untouched tokens move whenever G_t differs from the current w_t/r_t—which it will after any trade. Then w no longer equals the LP token supply, so Proposition 3.5 and the withdrawal bookkeeping break. You can patch the minting rule, but as written it doesn't work, and the author's own closing remark that effectiveness 'remains a topic for future research' confirms the gap.

Verdict: this deserves a serious referee, not a desk reject. The mathematical core is worth engaging with, and the author is honest about the vulnerability. The right outcome is a major revision that either fixes the safeguard or scopes the protocol to an idealized no-attack setting. I'd bring it to a reading group for the design discussion, and I'd cite it if I worked on AMM safety, but I wouldn't treat it as deployable yet.

Referee Report

3 major / 4 minor

Summary. The paper proposes a dynamic exponent market maker (DEMM), an AMM whose invariant is f(x)=x^w with w updated on liquidity provision as w'_t=((r_t+\Delta r_t)/r_t)w_t. Token-specific LP tokens are issued and redeemed per asset, and the protocol maintains that w is exactly the total supply of LP tokens (Proposition 3.5). The main formal results are that DEMM is exponent-balanced under no arbitrage (Proposition 3.6), that each liquidity provider's deposit value ratio is preserved under no arbitrage (Theorem 3.7), that relative spot prices are unchanged by liquidity provision and withdrawal (Proposition 4.1), and that liquidity provision can only benefit traders (Theorem 4.2). The paper also discusses transaction fees, impermanent loss and gain, token addition and removal, and a flash-loan attack with two sketch-level countermeasures.

Significance. If the formal claims hold, DEMM addresses a real design gap: one-sided deposits remove the proportional-deposit restriction of CPMM/Balancer pools, and a single composite pool removes multi-hop trading. Credit is due for the precise state-machine formalism, the clean token-specific LP token construction, and the explicit, honest acknowledgment of the flash-loan vulnerability. The core algebra behind Propositions 3.5, 3.6, 4.1 and Theorem 3.7 is sound under the stated no-arbitrage idealization, and those results are clearly definition-theorem style rather than circular. However, the paper's practical claim of a working one-pool protocol is not yet supported: the base protocol is explicitly vulnerable, and the proposed safeguards are either internally inconsistent with the protocol's own invariant or too underspecified to verify. The value of the contribution is therefore conditional on a correct countermeasure that the paper does not currently provide.

major comments (3)

[§4.6 (Flash Loan Attack), TWAP modification of Eq. (4)] The geometric time-weighted safeguard is inconsistent with Proposition 3.5. In the base protocol, w_t is also the total supply of LP token t. A trade (Operation 3.2) changes r_t without changing w_t or the LP supply, so the current ratio w_t/r_t can differ from the historical geometric average G_t used in w'_t=(r_t+\Delta r_t)G_t. If LP tokens are still minted as in Operation 3.3, the new supply is w_t+(\Delta r_t/r_t)w_t, which is generally not equal to w'_t; if the minting rule is changed to force equality, that rule is not stated, and withdrawal pricing under Operation 3.4 (which treats w_t as the LP supply) is undefined. Concretely, in Example 4.10 after Step 1 the pool is ((40,1),(1,1)); taking G_t from the pre-swap state, a one-sided deposit of 1 token t gives w'_t=2·0.1=0.2 while the LP-t supply after the deposit would be 2. Such a mismatch breaks the redemption formula and the exponent-balance guarantee, so the proposed safeguard does not preserve the formal properties proven in Section 3.
[§4.6 (Flash Loan Attack), random-delay countermeasure] The random-delay safeguard is not specified at the semantic level needed to evaluate the protocol. The text does not state whether the deposit's terms are fixed at submission or at execution, how the exponent update (4) and the number of LP tokens minted are computed during the delay, whether the deposited funds are held in escrow, or how traders interact with a pool whose state may be updated asynchronously. Different choices create different manipulation windows or arbitrage opportunities, and the paper's own closing sentence—that the effectiveness of these strategies remains a topic for future research—concedes that no working safeguard is actually established. Because the abstract states that the scheme must be used in conjunction with preventive measures, the paper as it stands does not deliver a deployable protocol, only a vulnerable base design plus untested sketches.
[§4.1, proof of Theorem 4.2] The proof is conditional on an unproved calculus claim: for 0<A and 1<x, (1+A/x)^x>1+A, and for 0<A<1, (1−A/x)^x>1−A. The text says only that the claim can be proved using standard techniques from calculus and gives no proof or reference. Since Theorem 4.2 is one of the paper's substantive economic claims, the lemma should be proved in the manuscript; it is true, but the omission is a completeness gap in a formal result rather than merely a stylistic shortcut.

minor comments (4)

[Section 1] There are several typos, including 'a trade is settled if the there are matching orders'; the manuscript should be carefully proofread.
[Reference [17]] The Balancer whitepaper URL contains '.inance' instead of '.finance'; please correct it.
[Figure 3] The caption uses 'below' where 'bottom' is meant, and the labels in the figure are hard to read; please redraw for clarity.
[Operation 4.8] The requirement that token t and token n+1 are deposited 'of equal value' is not made precise in terms of the current spot price; define the condition formally.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: DEMM is a self-contained mathematical construction; its core theorem follows from explicit definitions, not from fitted inputs or self-citation.

full rationale

The paper is self-contained and non-circular. Operation 3.3 defines the exponent update w'_t = ((r_t+Δr_t)/r_t)w_t, and Operation 3.4 defines withdrawal symmetrically. Proposition 3.5 then proves w equals the LP-token supply by direct induction over these operations. Proposition 3.6 shows exponent-balance is preserved: for deposits the algebra of (4) makes the value ratio (r_s+Δr_s)/r_s w_s : (r_t+Δr_t)/r_t w_t equal to w'_s : w'_t by construction; for withdrawals (5) gives the same identity. Theorem 3.7 is an algebraic transcription of these definitions: using Proposition 3.6 to express unit prices as w_s/r_s : w_t/r_t, the ratio of withdrawn values collapses to w_s/r_s Δr_s : w_t/r_t Δr_t, exactly the deposited value ratio. No parameter is fitted to data, no empirical quantity is later 'predicted' from its own fit, and no load-bearing claim is imported from the author's prior work; the only external result used is Balancer's standard CPMM fact (Proposition 2.6), cited to [17] and used as a known baseline rather than for the proposed construction. The acknowledged flash-loan vulnerability and the sketch of possible mitigations in Section 4.6 are engineering limitations explicitly deferred to future work; they affect deployability, not circularity.

Assumptions & free parameters 0 free parameters · 2 assumptions · 1 invented entities

The central claims rely on the standard CFMM no-arbitrage assumption, frictionless arbitrage, and basic calculus. No free parameters are fitted. The token-specific LP token is a new protocol artifact with no independent evidence outside the paper.

assumptions (2)

domain assumption External market prices exist and arbitrage is frictionless and instantaneous, restoring the exponent-balanced condition after any price change.
Used in Propositions 2.6, 3.6 and Theorem 3.7. The paper states 'given there is no arbitrage opportunity' and relies on arbitrage to keep the pool's value ratios aligned with the exponent.
standard math Standard calculus inequalities, specifically (1 + A/x)^x > 1 + A for x > 1 and A > 0.
Invoked in the proof of Theorem 4.2 but stated without full proof as 'standard techniques from calculus'.

invented entities (1)

Token-specific LP token
purpose: Represents fractional ownership of a single asset in the pool, enabling one-sided deposits and withdrawals.
This is a new protocol artifact introduced by the paper. It has no independent evidence outside the protocol design; its properties are defined and verified only within the paper.

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Cite this review

Pith. "Pith review of Dynamic Exponent Market Maker: Personalized Portfolio Manager and One Pool to Trade Them All." pith.science (2026). https://pith.science/paper/5RWE7WUL

@misc{pith2026250722732,
  author       = {Pith},
  title        = {Pith review of: Dynamic Exponent Market Maker: Personalized Portfolio Manager and One Pool to Trade Them All},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/5RWE7WUL}},
  note         = {Machine review of arXiv:2507.22732}
}

read the original abstract

Decentralized exchange platforms such as Uniswap and Balancer operate on several pools where each pool contains two or more cryptocurrencies and constitutes direct trading pairs. The drawbacks here are that liquidity providing requires contribution of tokens in a specific proportion, and trading may require hopping between pools, hence increasing transaction fee and gas fee. We propose an automated market maker (AMM) protocol where liquidity providers can deposit any amount of tokens into the pool. The protocol will preserve the proportion of tokens by total value at the time of deposit and can be seen as a personalized self-balancing portfolio manager. In addition, since the invariant function is dynamic, all exchange pairs are executed from a single composite pool. Nevertheless, the scheme is vulnerable to flash loan attacks and must be used in conjunction with preventive measures.

Figures

Figures reproduced from arXiv: 2507.22732 by the authors.

**Figure 2.** Token t received from trading away token s (left) and token s received from trading away token t (right) from Example 4.3 Example 4.3 Suppose that the unit price of tokens s and t is $2 and $10 respectively. Carol sets up a DEMM pool with 40 token s and 8 token t. This makes χ = ((40, 8),(1, 1)) the initial state of the pool. Later, Dave one-sidedly provides 8 token t, shifting the pool to χ ′ = ((40, 16),(1, 2)) [… view at source ↗

**Figure 3.** Holding options and impermanent loss for Alice (top) and Bob (bottom) from [PITH_FULL_IMAGE:figures/full_fig_p015_3.png] view at source ↗

Figures from the paper (2 more)

**Figure 4.** Figure 4: By net value, the pool’s impermanent loss from Example 4.5 is [PITH_FULL_IMAGE:figures/full_fig_p017_4.png]

**Figure 5.** Figure 5: Relative impermanent loss/gain from Example 4.5 [PITH_FULL_IMAGE:figures/full_fig_p018_5.png]

Discussion (0). Sign in to comment.

Reference graph

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Pith tools

Reviewed August 6, 2026 · model on record in the stance chip above.

[1] [1]

Kamal, E

R. Kamal, E. E.-D. Hemdan, N. El-Fishway, Care4U: Integrated Healthcare Systems Based on Blockchain, Blockchain: Research and Applications (2023), 100151

work page 2023

[2] [2]

Marchesi, M

L. Marchesi, M. Marchesi, R. Tonelli, and M. I. Lunesu, A Blockchain Architecture for Industrial Applications, Blockchain: Research and Applications 3 (2022), no. 4, 100088

work page 2022

[3] [3]

Proelss, S

J. Proelss, S. Sevigny, and D. Schweizer, GameFi: The Perfect Symbiosis of Blockchain, Tokens, DeFi, and NFTs?, International Review of Financial Analysis 90 (2023), 102916. 21

work page 2023

[4] [4]

M. A. Uddin, A. Stranieri, I. Gondal, and V. Balasubramanian, A Survey on the Adoption of Blockchain in IoT: Challenges and Solutions, Blockchain: Research and Applications 2 (2021), no. 2, 100006

work page 2021

[5] [5]

Y. Zou, T. Meng, P. Zhang, W. Zhang and H. Li, Focus on Blockchain: A Compre- hensive Survey on Academic and Application, IEEE Access8 (2020), 187182-187201

work page 2020

[6] [6]

J. Xu, K. Paruch, S. Cousaert and Y. Feng, SoK: Decentralized Exchanges (DEX) with Automated Market Maker (AMM) Protocols, ACM Computing Surveys 55 (2023), no. 11, 1–50

work page 2023

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Xu and Y

J. Xu and Y. Feng, Reap the Harvest on Blockchain: A Survey of Yield Farming Protocols, IEEE Transactions on Network and Service Management 20 (2022), no. 1, 858-869

work page 2022