Singularity Protocol for Cross Chain AMM without Intermediate Tokens or Bridges

Sumit Vohra

REVIEW 5 major objections 5 minor 6 references

Singularity Protocol for Cross Chain AMM without Intermediate Tokens or Bridges

T0 review · 5 major / 5 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read A proposed AMM invariant splits a cross-chain swap into two one-variable integrals, so each chain computes its own side and only a single value crosses chains, eliminating the need for intermediate tokens or bridges.

desk verdict The core mechanism is a trusted relayer, so the 'bridge-less' claim fails; the math is correct but not new, and the paper is a sketch without a security model. read the letter →

arxiv 2505.24337 v1 pith:ARBFQJDJ submitted 2025-05-30 cs.CE

Sumit Vohra This is my paper

classification cs.CE

keywords cross-chainswapsautomatedmarketmakerAMMinvariantvalueequivalencebridge-freeswaprelayerstablecurveblockchaininteroperability

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 tries to establish that cross-chain token swaps can be performed without a bridge, an intermediate token, or a relay blockchain, by redesigning the AMM invariant so that it splits into two independent one-variable integrals. The paper argues that a price function depending only on each chain's own token balance turns a swap into a scalar value-equivalence message: the source chain computes its integral, sends the number to the destination chain, and the destination chain solves for its output locally. If this works, existing cross-chain AMMs that require double swaps through synthetic tokens like Rune or axlUSDC could be replaced by direct swaps, with lower gas and no bridging risk. The paper also adapts the price curve for stable assets using a Witch of Agnesi flattening, keeping the same split property.

What carries the argument

The engine of the argument is the value-equivalence invariant $\sum_{x=a}^{z}\int_{x_0}^{x_n} P(x)\,dx = 0$, which states that the net change in value of a liquidity pool is always zero. From it the paper derives the swap equivalence equation with $P(x) = W_x/x$; the key property is that $P(x)$ is a function of one asset's balance only, so each side of the swap equation is uni-variate and computable on a single chain. For stable assets, the paper replaces the plain $W_x/x$ curve with a Witch-of-Agnesi-flattened price function that stays integrable while approximating a constant price near the equilibrium quantity $x_{stable}$; the flattening keeps the univariate split intact.

What would settle it

Send a wrong source-chain integral value to a destination chain and see whether it accepts it and mints the output; if it does, the invariant is broken. Alternatively, run two swaps in opposite orders with identical starting pools and check whether the final balances differ, which would show the protocol lacks the atomicity the paper claims not to need.

Watch

Extended reading notes

Core claim

On the paper's own terms, the central discovery is that a cross-chain swap can be written as an equality of two one-variable integrals: $\int_{i_n}^{i_{n+1}} P_x\,dx = -\int_{j_n}^{j_{n+1}} P_x\,dx$, where $P(x) = W_x/x$ is a price function that depends only on the local balance $x$ on a single chain. Because the left side involves only the source chain's state and the right side only the destination chain's state, the source chain can compute its integral and send the result as a scalar message, and the destination chain can solve for the output amount $\Delta j$ on its own. The paper claims this removes the need for an intermediate token, a bridge, or atomic cross-chain state consistency, and that the same split works for a flattened stable-asset price curve built from the Witch of Agnesi bell function.

Load-bearing premise

The design depends on a relayer carrying the source chain's computed value to the destination chain, and on each chain updating its own pool independently preserving the total value-equivalence, with no mechanism to prove the message is authentic or to revert one side if the other does not settle.

Editorial extensions

If this is right

A cross-chain swap can be settled by passing a single scalar (the source-chain integral) through a relayer, instead of executing two AMM swaps through a common token.
Each chain's pool can update independently from a swap, because the invariant change on one chain does not require any state variable from the other chain.
Liquidity can be added or removed on either chain without changing the pool invariant on the other chain, since each pool's value-defining integral stays local.
Bridging risk and wrapped-token volatility disappear because no asset ever leaves its native chain during the swap.
The mechanism works for any mix of L1, L2, and L3 blockchains, since the only cross-chain data is a number, not a state sync.

Reading between the lines

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

Because the paper only works through $P(x)=W_x/x$ and the Agnesi-flattened curve, an obvious next probe is to characterize which integrable price functions make the swap equation split; that would define the true size of this 'bridge-free' class.
The relayed integral is assumed genuine, so adding a short zero-knowledge proof of the integral value would likely make the scheme secure against relay tampering; the paper does not address this.
A direct simulation of two chains with adversarial relayer ordering would test the paper's no-atomicity claim: if the value-equivalence invariant depends on the order in which swap messages are processed, then complete independence without atomicity fails.
The stable-asset version requires moving $x_{stable}$ only at liquidity events; comparing its slippage behaviour to Curve's amplification factor on realistic stablecoin pools would be a useful stress test.

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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

Here is my read.

The central claim doesn't survive contact with the mechanism. The paper says a univariate price curve lets you relay the source-chain integral to the destination chain and skip bridges entirely. But the relayer is the bridge. Section IV.C literally says the LHS can be passed as a message using a relayer, with no authentication, no proof of deposit, no timeout, and no revert. A Byzantine relayer can forge a large value and drain the destination pool; a dropped message locks the user's funds. That is the trust assumption of every bridge. So the headline claim of 'no bridging risk' is false as stated.

What the paper does well is small but real: it identifies that standard AMM invariants couple the balances of two chains, and it points out that a price curve of the form P(x)=W/x, which is just Balancer's spot price with a constant virtual asset, makes the swap equation univariate. The integral identity itself is correct and the telescoping argument is fine. The stable-asset curve built on the Witch of Agnesi is at least integrable, and the author honestly says the parameter tuning is future work.

The soft spots are large. Equation (3.8), which governs fees and LP token adjustments, is stated without derivation. The stable section defers the movement of xstable to simulations. There is no implementation and no gas measurement, yet the abstract promises lower gas. The novelty is modest because the core math is a known invariant in a new coat.

This is a sketch, not a demonstration. It might become a useful note after the author adds a proper security model and derives the fee equations. As it stands, a serious referee would send it back with the trust question first. I would desk reject it now, and I would not cite it yet.

Referee Report

5 major / 5 minor

Summary. The paper proposes a cross-chain AMM design called Singularity. It observes that standard AMM invariants couple state variables of the two assets being swapped, which makes cross-chain atomicity hard, and it attempts to define a univariate spot price P(x)=W_x/x by postulating a virtual asset with constant balance. The core swap procedure (Section IV.C) is to compute an integral of P(x) over the source-chain balance change and send that value as a relayer message to the destination chain, which solves for its own balance change and updates independently. The paper also sketches LP/fee adjustments (Section IV.D) and a stable-asset price curve based on a modified Witch of Agnesi (Section IV.E). The abstract and conclusion claim that this eliminates intermediate tokens, bridges, and bridging risk and reduces gas costs.

Significance. If the protocol worked as described, it would address a real pain point in decentralized finance: cross-chain swaps with fragmented liquidity and wrapped-token bridge risk. The paper correctly identifies a desirable property—an invariant that is separable across chains—and the integral manipulation in Section IV.C is elementary but checkable. The authors also state some limitations, such as deferring the xstable movement algorithm to implementation, which is honest. However, the significance is conditional on a security model that the paper never provides. As written, the relayer is a trusted third party, so the central claim of removing bridging risk is not supported. The paper contains no proofs of the fee/LP invariance, no security analysis, no implementation, and no measurements, so the claimed gas savings and risk elimination are unsubstantiated.

major comments (5)

[Section IV.C] Equation (2.4) and the sentence 'The LHS of the equation 2.4 can be passed as a message using a relayer for inter-blockchain exchange' constitute the core transfer mechanism. The manuscript does not state how the relayer message is authenticated, how the source-chain deposit is proven to have occurred, how chain reorgs are handled, or how the destination leg is reverted if the source leg fails. A Byzantine relayer can submit an arbitrary LHS value to the destination contract and receive Δj, and an offline or malicious relayer can drop the message after the source leg commits, locking the user's funds and breaking the invariant ∑∫P dx = 0. This is not an implementation gap: it places trust in the relayer for message validity and delivery, which is exactly the trust model of a bridge that the abstract claims to eliminate.
[Section IV.B] The reduction from Balancer's two-asset spot price SP^{xv}=W_x v/(W_v x) to the univariate P(x)=W_x/x rests on positing a virtual asset v that is never traded and has constant balance, with C_v=v/W_v absorbed into W_x. This is an assumption, not a derivation: no argument is given that such a virtual asset can exist, that its balance remains constant under liquidity additions, fee shifts, or stable-curve updates, or that the resulting P(x) models a realizable pool. Since the univariate property is the central enabling claim of the protocol, this assumption is load-bearing and is not established.
[Section IV.D, Eq. (3.8)] The LP/fee update equation ∫_{x0}^{xp} P(x) dx = ∫_{x0+Δx0}^{xp+Δxp} P(x) dx is asserted without derivation. The text does not specify how Δxp and Δx0 are computed from fees or new liquidity, nor does it prove that the global value-equivalence invariant remains zero after the shift. The minting rule new_sins_minted/burned = (Δxp/xp)·sinpool is likewise asserted without a link to the invariant or to LP value. Because the paper explicitly relies on the invariant not changing, this is a central gap, not a minor omission.
[Section IV.E] The stable-price construction is incomplete. The price function is stated as a modified Witch of Agnesi, but the properties of monotonicity and the limits at x=0 and x→∞ are only asserted as 'straightforward,' and the displayed antiderivative I(x) is given as an unlabeled long expression with no derivation. More importantly, the text states that 'The exact algorithm of movement will be described at the time of implementation and is based on simulations,' so the stable-swap behavior depends on an unspecified algorithm for updating xstable. As written, the stable-asset swap cannot be implemented or tested against the claimed properties.
[Abstract and Section V] The abstract and conclusion claim lower gas costs and no bridging risk, but the paper provides no implementation, no simulations, and no measurements. There is no code, no test suite, no cost model, and no security analysis for the relayed-message protocol. These performance and security claims are therefore unsupported.

minor comments (5)

[Throughout] The equation numbering is inconsistent: the text refers to 'equation 2.4' and 'equation 3.8' but the displayed equations are not numbered. The numbering should be introduced consistently.
[Sections IV.C and IV.D] Notation such as Δi, Δj, i_n, j_n, Δx_p, and Δx_0 is used without precise definitions; in particular, the sign convention for Δj in j_{n+1}=j_n−Δj and the meaning of x_p versus x_0 should be clarified.
[Figures 3 and 4] Figures 3 and 4 are schematic or external and lack adequate labels; Figure 3's caption is incomplete ('Shows swap between asset A and asset C | asset D and asset B') and Figure 4 has no axis labels, which makes the stable versus volatile price curves hard to interpret.
[References] References 4–6 are bare URLs with no access dates or version information; the Thorchain and Axelar references should be cited to specific documents, and the MathWorld reference should include a proper bibliographic entry.
[Throughout] There are typos and formatting issues, including 'htttps://docs.uniswap.org' in Figure 1, 'deﬁllama.com' with a ligature in Figure 2, and inconsistent capitalization such as 'the and Defi'. These should be cleaned up.

Circularity Check

1 steps flagged · score 6.0 of 10

The claimed elimination of bi-state dependency is assumed via a constant virtual asset, not derived; the stable-swap tuning is explicitly deferred to simulations.

self definitional [Section IV.B 'Properties of Price Function Curve' and Section IV.C 'Deriving Swap Value Equivalence Equation']
"A uni-variable price curve can be defined if we assume that there exists a virtual asset ‘v’ which is never traded and has a constant balance. ... The LHS of the equation 2.4 can be passed as a message using a relayer for inter-blockchain exchange and RHS can be calculated to find out Δj, the amount of token to be swapped out. Point to be noted here is that both LHS and RHS equations are uni-variate and only dependent on state variables of their own blockchain."

The paper's central claimed result is an AMM invariant with no bi-state dependency. That property is inserted as an assumption: a virtual asset v with constant balance makes the Balancer spot price P(x)=W_x/x depend only on x. The later statement that the swap-equation LHS/RHS are uni-variate and depend only on local chain state is the same assumption restated, not a consequence derived from independent first principles. Equation (2.4) holds by definition once P is taken to be univariate; nothing forces v to be constant, and no derivation or falsifiable prediction supports it. The claimed elimination of bi-state dependency is therefore equivalent to the chosen ansatz by construction.

full rationale

The only genuine circularity is the self-definitional move in Section IV.B: the claimed derived property (univariate, no bi-state dependency) is introduced by postulating a virtual asset v with constant balance. The swap equation in Section IV.C is then algebraically consistent, but the cross-chain claim rests on this definition rather than on a derivation from more basic principles. I do not count the relayer/atomicity weakness as circularity; that is a correctness and security gap, not a reduction of a prediction to its input. The stable-swap parameters (A, xstable) are explicitly deferred to simulation-based implementation, so no fitted parameter is disguised as a prediction. There are no load-bearing self-citations. The score of 6 reflects partial circularity: one central claim reduces to its ansatz by construction, while the LP/fee equations and the stable-curve construction contain independent mathematical content.

Assumptions & free parameters 3 free parameters · 6 assumptions · 1 invented entities

The protocol depends on two ad hoc mathematical devices (the virtual asset v and the Witch of Agnesi stable curve), on an unspecified xstable tuning rule, and on a relayer assumption that is not security-hardened. The only new entity is the SINS LP token, which has no independent evidence.

free parameters (3)

Pool weights W_x = Chosen by pool creator; no calibration rule given
Set by hand to reflect initial price ratios in P(x)=W_x/x; determines swap curve and is not derived.
Amplification factor A = Undefined
Controls flatness of the stable price curve; no selection rule is provided.
Equilibrium quantity xstable = Undefined; to be set by simulations
Centers the stable price curve; Section IV.E says the exact movement algorithm will be described at the time of implementation and is based on simulations.

assumptions (6)

domain assumption Value equivalence invariant: the total value traded from the pool is zero, i.e. the sum over assets of the integral of P(x) dx over the balance change equals 0.
Stated as the defining property of the pool in Section IV.A; it is a premise, not a derived result.
domain assumption An ideal price curve is decreasing, continuous, differentiable, and has asymptotes at zero and infinity.
Assumed in Section IV.B to guarantee liquidity and monotonic pricing.
ad hoc to paper There exists a virtual asset v that is never traded and has constant balance, allowing Balancer's spot price to be rewritten as P(x)=W_x/x.
Introduced in Section IV.B specifically to make the price curve univariate; this is the source of the claimed no-bi-state-dependency property.
ad hoc to paper The stable price curve uses a modified Witch of Agnesi bell curve to flatten prices around xstable.
Chosen in Section IV.E because it is integrable; no empirical or theoretical justification beyond that.
ad hoc to paper xstable changes only at liquidity addition or removal, following an algorithm to be determined by simulations.
Section IV.E explicitly defers the movement algorithm to implementation time; the protocol behavior depends on this unspecified rule.
standard math Integral additivity and the fundamental theorem of calculus are used to split and evaluate the value-change integrals.
Used in Section IV.C to derive the swap equivalence equation from the price function.

invented entities (1)

SINS token
purpose: LP share token minted and burned on liquidity changes, with initial supply set as a geometric mean of initial balances.
No token economics, issuance rules beyond a sketch, or deployed instance are provided; its behavior is part of the protocol design, not externally verified.

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Cite this review

Pith. "Pith review of Singularity Protocol for Cross Chain AMM without Intermediate Tokens or Bridges." pith.science (2026). https://pith.science/paper/ARBFQJDJ

@misc{pith2026250524337,
  author       = {Pith},
  title        = {Pith review of: Singularity Protocol for Cross Chain AMM without Intermediate Tokens or Bridges},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/ARBFQJDJ}},
  note         = {Machine review of arXiv:2505.24337}
}

read the original abstract

Automated Market Makers (AMMs) are decentralized exchange protocols that provide continuous access to token liquidity without the need for order books or traditional market makers. However, this innovation has failed to scale when it comes to cross-chain swaps. Modern cross-chain swaps employ double-sided AMMs, which are not only inefficient due to liquidity fragmentation but also require an intermediate token. This introduces inherent volatility risk as well as blockchain and bridging risk, especially in the case of wrapped tokens. This paper describes the inefficiencies of existing AMM invariants, particularly their mixed polynomial nature, and derives a new class of AMMs that do not have bi-state dependency between the assets being swapped. We propose a novel method of value transfer swaps using the described invariant that mitigates the need for bi-state dependency and eliminates the need for intermediate tokens or bridging. Furthermore, we show how this mechanism enables efficient cross-chain swaps with lower gas requirements and no bridging risks. The proposed technology is designed to support cross-chain swaps across any permutation of L1, L2, and L3 blockchains.

Figures

Figures reproduced from arXiv: 2505.24337 by the authors.

**Figure 2.** Figure2 : Total DEX transaction volume : https://de [PITH_FULL_IMAGE:figures/full_fig_p001_2.png] view at source ↗

**Figure 4.** Price curve for volatile assets vs price curve for stable assets [PITH_FULL_IMAGE:figures/full_fig_p004_4.png] view at source ↗

Figures from the paper (1 more)

**Figure 3.** Figure 3: AMM which enables swap to be executed cross chain. Shows swap between asset A and asset C | asset D and asset B. [PITH_FULL_IMAGE:figures/full_fig_p004_3.png]

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Reference graph

Works this paper leans on

6 extracted references · 6 canonical work pages

[1]

DEXs align perfectly with this goal by allowing users to trade directly with each other, eliminating the need for a centralized exchange operator

Eliminating Intermediaries: One of the primary tenets of DeFi is the removal of intermediaries such as banks and financial institutions. DEXs align perfectly with this goal by allowing users to trade directly with each other, eliminating the need for a centralized exchange operator

work page
[2]

DEXs provide users with greater control over their funds since trades occur directly from their wallets

Enhanced Security and Control: DeFi emphasizes user control and ownership of assets. DEXs provide users with greater control over their funds since trades occur directly from their wallets. Users retain ownership of their private keys, reducing the risk of hacks and unauthorized access

work page
[3]

DEXs leverage blockchain technology to record all transactions on a public ledger, enabling anyone to audit and verify trades

Transparency and Audit-ability: Transparency is a key feature of DeFi applications. DEXs leverage blockchain technology to record all transactions on a public ledger, enabling anyone to audit and verify trades. This transparency enhances trust within the DeFi ecosystem

work page
[4]

As long as users have a compatible cryptocurrency wallet, they can access the DEX and start trading immediately

Permissionless Access: DEXs do not impose restrictions on who can participate in trading. As long as users have a compatible cryptocurrency wallet, they can access the DEX and start trading immediately. This permissionless nature aligns with DeFi's inclusive philosophy

work page
[5]

Programmable and Composable Finance: D e F i aims to democratize financial services by allowing users to program their financial interactions. DEXs integrate with DeFi protocols and smart contracts, enabling users to create sophisticated trading strategies, conduct automated trades, and execute complex financial operations

work page
[6]

virtual balances

Liquidity Provision: Liquidity is crucial for both DEXs and DeFi platforms. DEXs use liquidity pools to facilitate trading, and many DeFi protocols rely on these pools for operations like lending, borrowing, and yield farming. DEXs provide a marketplace for users to contribute liquidity and earn rewards. In summary, DEXs are an essential component of the ...

work page

Pith tools

Reviewed August 7, 2026 · model on record in the stance chip above.

[1] [1]

DEXs align perfectly with this goal by allowing users to trade directly with each other, eliminating the need for a centralized exchange operator

Eliminating Intermediaries: One of the primary tenets of DeFi is the removal of intermediaries such as banks and financial institutions. DEXs align perfectly with this goal by allowing users to trade directly with each other, eliminating the need for a centralized exchange operator

work page

[2] [2]

DEXs provide users with greater control over their funds since trades occur directly from their wallets

Enhanced Security and Control: DeFi emphasizes user control and ownership of assets. DEXs provide users with greater control over their funds since trades occur directly from their wallets. Users retain ownership of their private keys, reducing the risk of hacks and unauthorized access

work page

[3] [3]

DEXs leverage blockchain technology to record all transactions on a public ledger, enabling anyone to audit and verify trades

Transparency and Audit-ability: Transparency is a key feature of DeFi applications. DEXs leverage blockchain technology to record all transactions on a public ledger, enabling anyone to audit and verify trades. This transparency enhances trust within the DeFi ecosystem

work page

[4] [4]

As long as users have a compatible cryptocurrency wallet, they can access the DEX and start trading immediately

Permissionless Access: DEXs do not impose restrictions on who can participate in trading. As long as users have a compatible cryptocurrency wallet, they can access the DEX and start trading immediately. This permissionless nature aligns with DeFi's inclusive philosophy

work page

[5] [5]

Programmable and Composable Finance: D e F i aims to democratize financial services by allowing users to program their financial interactions. DEXs integrate with DeFi protocols and smart contracts, enabling users to create sophisticated trading strategies, conduct automated trades, and execute complex financial operations

work page

[6] [6]

virtual balances

Liquidity Provision: Liquidity is crucial for both DEXs and DeFi platforms. DEXs use liquidity pools to facilitate trading, and many DeFi protocols rely on these pools for operations like lending, borrowing, and yield farming. DEXs provide a marketplace for users to contribute liquidity and earn rewards. In summary, DEXs are an essential component of the ...

work page