Bitcoin no longer needs to envy Ethereum: How zkFOL enables BTC to natively support DeFi and privacy

The essence of the problem: Why has Bitcoin been consistently overlooked by the DeFi market?

For over a decade, Bitcoin has adhered to its minimalist design philosophy. The Bitcoin Script language is deliberately restricted—no loops, no recursion, no global mutable state—to ensure that each transaction can be verified within a predictable timeframe. This design guarantees that Bitcoin has never suffered a major vulnerability at the consensus layer.

But at what cost? Bitcoin cannot:

• Store state between transactions
• Execute complex conditional logic
• Native support for automated market makers(AMM), lending protocols, complex vaults
• Handle 64-bit arithmetic or floating-point operations

The result is obvious: with a market cap of nearly $20 trillion, Bitcoin can only watch as Ethereum, Solana, and Avalanche carve up the DeFi pie. Millions of developers move to other chains to build applications, fragmenting the DeFi ecosystem.

Technological breakthrough: Redefining verifiability with mathematical language

The ModulusZK team has broken this deadlock with an elegant mathematical insight—directly converting first-order logic predicates into polynomials.

This idea sounds complex, but the core logic is straightforward. In modern cryptography, polynomials have a decisive advantage over traditional Boolean circuits: they can be verified succinctly. According to the Schwartz-Zippel lemma, verifying whether a polynomial equals zero at a random point is sufficient to prove its identity with a very small error probability.

Recent research by Dr. Murdoch Gabbay (Alonzo Church Award winner) demonstrates that any first-order logic predicate can be directly translated into an equivalent polynomial over a finite field. The specific translation rules are:

• Logical AND(∧) → Addition
• Logical OR(∨) → Multiplication
• Universal quantifier(∀) → Finite summation
• Existential quantifier(∃) → Finite product

What does this mean? A complex logical predicate is compiled into a single polynomial, whose coefficients encode all the constraints of the contract. Verifying whether this polynomial evaluates to zero at a random point is equivalent to verifying the entire contract logic—and this operation can be performed in constant time, regardless of the initial logical complexity.

zkFOL’s two-stage implementation: from Layer-2 to on-chain upgrade

Stage 1: 1:1 anchoring of Layer-2 architecture

zkFOL initially operates as a Layer-2 solution for Bitcoin:

1. Users lock BTC in a multisig vault on the Bitcoin main chain
2. Obtain wBTC-FOL (1:1 mapping) on zkFOL layer
3. All DeFi transactions (swaps, lending, liquidity mining) are executed off-chain, protected by zero-knowledge proofs
4. Proofs are periodically anchored to Bitcoin, ensuring data availability
5. During withdrawal, cryptographic verification unlocks the original BTC

Unlike existing solutions, zkFOL does not rely on centralized validators. Verification is pure mathematics—no third-party trust required.

Stage 2: Mainnet soft fork integration

Once proven secure and efficient on Layer-2, the long-term goal is to incorporate polynomial verification directly into Bitcoin’s base layer via a soft fork (backward-compatible protocol upgrade). This way, all verification occurs on-chain.

Practical example: from logic to proof

A constant-product AMM defined in zkFOL only needs to be written as: