starting with the surprising result from over twenty years ago that all programs can be converted into interactive proofs,

The idea I'm calling CoinWitness is this:

You write down a small program which verifies the faithfulness of one of these transcripts for your chosen verifiable off-chain system. The program requires that the last transaction in the transcript is special in that it pays to a Bitcoin scrippubkey/p2sh. The same address must also be provided as a public input to the program. We call this program a "witness" because it will witness the transcript and accept if and only if the transcript is valid.

You then use the SCIP proof system to convert the program into a verifying key.  When someone wants to create a Bitcoin in an off-chain system, they pay that coin to the hash of that verifying key. People then transact in the off-chain system as they wish. To be confident that the system works faithfully they could repeat the computationally-expensive verifying key generation process to confirm that it corresponds to the transaction rules they are expecting.

When a user of one of these coins wants to exit the system (to compact its history, to move to another system, to spend plain Bitcoins, or for any other reason), they form a final transaction paying to a Bitcoin address, and run the witness on their transcript under SCIP and produce a proof. They create a Bitcoin transaction redeeming the coin providing the proof in their script (but not the transcript, thats kept private), and the Bitcoin network validates the proof and the transaction output. The public learns nothing about the intermediate transactions, improving fungibility, but unlike other ideas which improve fungibility this idea has the potential to both improve Bitcoin's scalability and securely integrate new and innovative alternative transaction methods and expand Bitcoin's zero-trust nature to more types of transactions.