Technical Summary: Bounding LVR in AMMs via Secant–Tangent Divergence and Collateralized Liquidity Scaling

1. Problem Statement

Automated Market Makers (AMMs), particularly those based on the Constant Product Market Maker (CPMM) model ($x \cdot y = k$), face a fundamental geometric dilemma. To reduce execution slippage for traders, protocols must increase liquidity depth. However, in standard CPMMs, increasing depth requires expanding physical capital exposure, which linearly increases the Liquidity Provider's (LP) exposure to Loss-Versus-Rebalancing (LVR).

LVR represents the permanent capital depletion caused by adverse selection, where arbitrageurs exploit the lag between on-chain AMM prices and external market prices. Unlike "Impermanent Loss" (IL), which is reversible if prices mean-revert, LVR is a path-dependent cost driven by volatility and information asymmetry. The paper argues that the traditional approach of scaling physical liquidity to minimize slippage is structurally inefficient: beyond a certain saturation point, additional capital yields negligible slippage improvements while continuing to expose LPs to toxic arbitrage flows.

2. Methodology and Architecture

2.1 Geometric Analysis: Secant–Tangent Divergence

The authors formalize a critical geometric distinction between two price metrics:

1. Execution Slippage ($S$): The difference between the effective price (secant line) a trader receives and the initial marginal price.
2. Marginal Price Deviation ($\Delta p$): The shift in the pool's marginal price (tangent line) after a trade.

The paper proves that for any trade volume, the marginal price deviation ($\Delta p$) strictly exceeds the execution slippage ($S$). Specifically, for small trades, $\Delta p \approx 2S$. This implies that arbitrageurs can extract profit based on the tangent line deviation while paying only the secant line cost. Consequently, simply increasing passive liquidity ($L \to \infty$) to suppress slippage fails to neutralize LVR, as the ratio between the arbitrageur's cost and their profit remains immutable.

2.2 The Hybrid Liquidity-Collateral Pool (HLCP)

To address this, the paper proposes the HLCP, an architecture that decouples execution depth from continuous capital exposure. The system operates on two components:

• Active Reserves ($L_{active}$): A fraction of total capital ($N \cdot L_{total}$, where $N < 1$) exposed to the constant-product curve.
• Collateral Buffer ($C$): The remaining capital ($(1-N) \cdot L_{total}$) held in a low-risk, yield-generating state (e.g., ERC-4626 vaults).

The N-Scaling Mechanism:
The active pool operates on a virtual invariant: $(x_a/N) \cdot (y_a/N) = L_{total}^2$. This allows the pool to claim the slippage characteristics of a deeper pool ($L_{total}$) while physically exposing only $L_{active}$.

The Kinetic Trigger:
The system employs an endogenous trigger based on marginal price deviation ($\Delta p$).

• Dormant State: When $|\Delta p| \leq \tau$ (where $\tau$ is a tolerance threshold, typically $\geq$ the swap fee), the collateral buffer remains sequestered.
• Activation State: When $|\Delta p| > \tau$, the protocol injects collateral into the active pool.
• Injection Logic: The injection is a price-ratio-preserving homothety. Assets are added in the exact ratio of the current marginal price, expanding the curve outward without shifting the tangent slope. This increases the executable depth ($L_{new} = L_{active} + \Delta C$) to absorb subsequent flow without introducing a new arbitrage opportunity at the moment of injection.

2.3 Control Law and Stability

The paper defines a control rule for collateral deployment $\Delta C$:
$$\Delta C = \frac{N \cdot \alpha \cdot \phi(\Delta p) \cdot C}{1 + N \cdot \alpha \cdot \phi(\Delta p)}$$
where $\phi(\Delta p)$ is the activation function. The authors prove that for any finite shock, $\Delta C < C$, ensuring the buffer is never exhausted in a single step (one-shot depletion is bounded away).

3. Key Contributions

1. Theory of Liquidity Saturation: The paper identifies a saturation region where the marginal utility of physical liquidity for slippage reduction approaches zero, while LVR exposure remains constant. It mathematically establishes that passive scaling is an ineffective defense against adverse selection.
2. HLCP Architecture: It introduces a structural solution that separates execution depth from risk exposure via N-scaling and a collateral buffer.
3. Kinetic Trigger Mechanism: It specifies a deterministic, trigger-based collateral deployment rule driven by marginal price deviations, preserving the marginal price slope during injection.
4. Game-Theoretic Validation: In a stylized duopoly model under hyper-saturated liquidity, the paper demonstrates that adopting HLCP is a Nash equilibrium and a Pareto improvement over standard AMMs, provided there is non-zero volatility or collateral yield.
5. Empirical Validation:
   • Stress Testing (SVJ): Under a Stochastic Volatility with Jumps (SVJ) stress scenario, the HLCP trigger successfully caps cumulative capital loss and avoids one-shot buffer depletion.
   • Historical Backtest: Using 2025 Uniswap V2 data with a conservative assumption of zero collateral yield ($Y_C = 0$), the HLCP ($N=0.5$) achieved a higher net LP return (+7.51%) compared to the standard CPMM (+3.84%) by reducing realized LVR.

4. Results and Findings

• Geometric Divergence: The analysis confirms that $\Delta p > S$ always holds, validating that slippage minimization does not equate to LVR mitigation.
• Buffer Integrity: In the SVJ stress test (12-second resolution), a forced negative price jump and elevated volatility regime caused the standard AMM to incur ~1.93% cumulative loss, whereas the HLCP incurred ~0.51% (a ~73.8% reduction). The trigger deployed ~95.1% of the buffer in a single step but remained bounded away from total depletion.
• Net Yield Improvement: In the 2025 historical backtest, despite the conservative assumption that the collateral buffer earned no yield, the HLCP outperformed the standard CPMM. This improvement was driven solely by the reduction in active capital exposure to LVR, assuming the router tolerance condition ($\Delta S \leq \epsilon_{router}$) was met for fee capture.

5. Significance and Claims

The paper claims that the HLCP offers a structural defense against LVR that does not rely on off-chain auctions, complex derivative layers, or restrictive transaction sequencing. Its significance lies in:

• Decoupling Depth from Risk: It challenges the assumption that deep liquidity must be fully exposed to the market curve.
• Endogenous Resilience: It provides a mechanism to react to acute volatility spikes without requiring external oracles or pre-defined auction rounds.
• Economic Efficiency: It demonstrates that in saturated markets, reducing active exposure can improve LP profitability even without auxiliary yields, provided the execution quality remains within router tolerances.

Limitations and Scope:
The authors explicitly state that their claims are bounded by the model's assumptions:

• The routing logic is stylized (proportional fee routing based on tolerance).
• The backtest uses a uniform CPMM (Uniswap V2) baseline, not concentrated liquidity (V3).
• The stress test is a controlled mechanism evaluation, not a market-complete calibration.
• The analysis assumes the collateral buffer can be deployed in a price-ratio-preserving manner without gas cost prohibitions (a full gas analysis is outside the scope).

The paper concludes that while it does not eliminate the liquidity-depth trade-off, the HLCP significantly improves the trade-off curve, offering a Pareto-improving alternative for LPs in volatile, saturated markets.