Recursive covenants differ from regular covenants by extending conditions beyond the immediate next transaction to all future transactions that spend the bitcoins. This perpetuity ensures that conditions, such as requiring a valid signature from a specific public key for spending, are maintained across the entire lifecycle of the bitcoins. Such a mechanism significantly broadens the scope of applications and use cases for Bitcoin, enabling complex transaction logic essential for interoperability with sidechains or Layer 2 solutions.

Bitcoin's UTXO model traditionally supports one-off and stateless smart contracts. However, recursive covenants introduce a way to maintain and update a form of state across multiple transactions. Stateful smart contracts, unlike their traditional counterparts, enable the tracking and modification of state over time. This capability is illustrated through a simple counter contract example, demonstrating how contract states can be incremented and stored in an adjacent output to the contract itself. The state management employs a separate output for state storage to avoid modifying the contract's taproot address, thus maintaining its consistency without the need for complex elliptic curve arithmetic.

The counter contract example further highlights the use of a covenant to access the transaction ID of the previous transaction, allowing the contract to parse the raw transaction and access its outputs. This method enforces a strict relationship between the input and output addresses and the transition of state outputs, ensuring that each state incrementation reflects accurately in subsequent transactions. Such a detailed approach enables a single run to result in a series of transactions, each marked by a unique transaction ID and reflecting an incremental state change.

The practical implementation of such stateful smart contracts on Bitcoin introduces a new dimension to Bitcoin's capabilities, mirroring some functionalities found on other blockchain platforms like Ethereum. With the full code available on GitHub, more sophisticated state machines following this model can be designed, showcasing the versatility and potential for innovation within the Bitcoin ecosystem. The integration with The Mempool Open Source Project® provides real-time insights into transactions, further exemplifying the practical application and monitoring of these stateful contracts within the broader Bitcoin network.