Validity Proof
A cryptographic proof that a batch of transactions was executed correctly, checkable by a contract without redoing the work.
The prover runs the computation and produces a succinct proof, and a verifier contract checks it in a fixed small amount of work no matter how large the batch was, which is the property that makes rollup compression possible. Because acceptance depends on mathematics rather than on nobody objecting, a validity proof removes the challenge window that optimistic systems need and gives a clear point at which the base chain considers the layer 2 state settled. The trade-off is asymmetry of cost: proving is far more expensive than verifying, so provers are specialized and often centralized, and proof generation adds latency between execution and settlement. A validity proof is not the same as a fraud proof, which is produced only when someone alleges an error.
In practice
A verifier contract can accept a proof covering thousands of transactions with roughly the same on-chain cost as one covering a handful.
The common misunderstanding
A validity proof shows that the computation followed the rules; it does not show that the rules, or the contract being executed, are free of bugs.