Mechanized MEV Formalization

• Mechanized Formalization of MEV is a method that rigorously defines and certifies adversarial strategies in blockchain systems using formal models and interactive theorem provers.
• It employs transition systems, invariants, and gain functions to systematically analyze and bound adversarial behavior in decentralized finance.
• Case studies, such as sandwich attacks in AMMs, illustrate how this approach sets provable upper bounds on adversarial gains and enhances protocol security.

Maximal Extractable Value (MEV) formalizes the phenomenon whereby an adversary—typically a block producer, miner, or sequencer—extracts value by manipulating the inclusion, exclusion, or ordering of transactions in decentralized applications. As the DeFi ecosystem has advanced and formal methods have matured, attention has shifted towards mechanized formalization: the use of interactive theorem provers, specification languages, or automated analysis to rigorously define, bound, and certify optimal adversarial strategies and their security implications. Mechanized formalization aims for much stronger correctness guarantees than empirical or pen-and-paper approaches, given the combinatorial explosion of adversarial possibilities in permissionless, composable financial infrastructure.

1. Formal Modeling of Blockchain Systems and Adversaries

A mechanized approach to MEV begins by constructing a formal model of the blockchain system, typically as a transition system. In Lean, this is captured by the System structure, which encodes:

• Token: The type of assets (e.g., ERC-20 tokens).
• State: Abstract state of the protocol, including contract states and wallet balances (honest users and adversaries separated).
• Move: Adversary moves, either crafting novel transactions or replaying from a public mempool.
• Semantics: The transition relation semantics : SysState → Move → Option SysState, which models the effect of a move.
• Wallet representation: Honest and adversarial wallets as mappings from token types to quantities (ℝ≥0 for honest, ℝ for adversary).
• Token preservation: Invariants ensuring that, for example, the total number of tokens is maintained modulo adversarial (e.g., fee or slippage) mechanics.

This formalism accommodates the modeling of both honest and adversarial behaviors, and crucially, allows for adversaries with full transaction crafting and state-observing power relevant for MEV attacks.

2. Mechanized MEV Definition and the Gain Function

MEV is formally specified as a property over system transitions. For a particular state σ, the adversarial gain is defined as:

• Single transition: gainState(σ, σ′) = tokenValue(σ′.adv) - tokenValue(σ.adv)
• Trace of moves: The function gainMoves(σ, tr) aggregates the gain for a finite list of moves, reflecting how much the adversary's wallet can increase by possibly complex MEV tactics.

The MEV for state σ and candidate value v is captured by:

structure MEV (σ : SysState) (v : ℝ) : Prop where trace_reaches_v : ∃ tr, gainMoves(σ, tr) = v other_traces_worse : ∀ tr, gainMoves(σ, tr) ≤ v

This definition formalizes the maximal gain achievable by any adversarial strategy, i.e., there exists some trace realizing v, and no trace realizes a greater gain. A supremum version (MEVSup) handles existence nuances if the maximum does not exist.

3. Machine-Checked Characterization of Optimal Strategies

To certify bounds or optimality of MEV strategies mechanistically, the paper introduces the characterization theorem. It states that to prove the MEV at state σ is v (computed by a function MEV_guess), it suffices to exhibit:

• An invariant on system states (inv : SysState → Prop) preserved by all possible adversarial moves.
• A guess function MEV_guess that assigns a candidate MEV to each invariant-satisfying state.
• Coherence: There exists a trace from σ achieving MEV_guess(σ).
• Soundness: For any adversarial move m, leading σ → σ′, MEV_guess(σ) ≥ gainState(σ, σ′) + MEV_guess(σ′).

In Lean, the associated theorem template is:

theorem MEV.characterization (σ : SysState) (inv : SysState → Prop) (MEV_guess : (σ′ : SysState) → inv σ′ → ℝ≥0) (invariant_init : inv σ) (invariant_sound : ∀ {m σ σ′}, semantics σ m = some σ′ → inv σ → inv σ′) (MEV_guess_coherent : ∃ tr, gainMoves σ tr = MEV_guess σ invariant_init) (MEV_guess_sound : ∀ {m σ σ′}, semantics σ m = some σ′ → inv σ → (gainState σ σ′ + MEV_guess σ′ _) ≤ MEV_guess σ _) : MEV σ (MEV_guess σ invariant_init)

Once coherence and soundness are established, the proof by induction on the length of adversarial traces ensures that no adversary can exceed MEV_guess(σ).

4. Example: Mechanized Optimality of Sandwich Attacks in AMMs

The methodology is exemplified using sandwich attacks on Automated Market Makers (AMMs):

• Case 1: Empty Mempool — The adversary seeks to extract value via a single optimal trade. The MEV_guess is derived from the AMM's invariant (e.g., constant-product formula) and can be written using reserve and price parameters, such as:

$$\text{extractable}(\sigma) = (\sqrt{pr_0 \cdot r_0} - \sqrt{pr_1 \cdot r_1})^2$$

where $r_0, r_1$ are reserves, $pr_0, pr_1$ prices.

• Case 2: Single Honest Transaction Pending — The adversary can front-run, let the honest transaction execute, and then back-run (the canonical sandwich), with the guess function defined piecewise depending on mempool transaction parameters (e.g., slippage bounds). The proof establishes that any other sequence increases the adversary's gain by no more than this optimal sandwiching move.

The use of mechanized proof ensures that every adversarial move sequence—including those exploiting subtle contract logic or combining cross-domain strategies—is considered and certified against the guessed bound.

5. Security Guarantees and Protocol Implications

Mechanized formalization in the Lean prover establishes a very high assurance level:

• Exhaustiveness: The correctness is proven for all allowed adversarial behaviors, not just a limited set considered in empirical or simulation-based studies.
• Optimality: The formal method guarantees not only that a particular MEV attack (such as sandwiching) is possible, but also that no more profitable attack exists within the modeled framework.
• Extensibility: Once system models and invariants are encoded, the approach adapts to more complex settings (e.g., multiple pending mempool transactions, modifications for arbitrage or front-running in variants of AMMs).
• Reductions in Human Error: All steps, including the definition of state, invariant preservation, and gain computations, are checked by the theorem prover, minimizing mistakes compared to traditional security proofs.

A mechanized approach allows protocol designers to set provable upper bounds on adversarial profits. For application, this informs both economic security analyses (e.g., setting reserve and fee parameters) and the development of monitoring tools to detect deviations from theoretically permissible extraction.

6. Comparisons and Extensions

Mechanized MEV formalization supersedes heuristic or manual methods—where the combinatorial nature of the adversarial state space typically leads to missed edge cases. The approach aligns with the research trend toward formal verification in DeFi, as illustrated by the use of machine-checked proofs for key DeFi properties. Extensions to this framework may include:

• Modeling transactions with multiple mempool entries.
• Abstractions for composable contracts—facilitating proofs in protocols that integrate multiple DeFi primitives.
• Integration with automated extraction and simulation pipelines to inform on-chain or off-chain MEV monitoring systems.

7. Outlook and Limitations

The Lean framework for certifying MEV bounds marks a significant advance in rigor for protocol security analyses. Remaining challenges include the faithful encoding of real-world protocol nuances (gas costs, asynchronous execution, unspecified behaviors), tractability for very large or interlinked systems, and the need for trusted specifications. Nevertheless, machine-checked results for canonical strategies—such as sandwich attacks on AMMs—provide a strong foundation for risk assessment and future protocol design.

This mechanized methodology is essential as MEV extraction increasingly drives value flows, imposes security and fairness challenges, and motivates competition among adversarial participants in modern blockchain ecosystems (Bartoletti et al., 16 Oct 2025).