Data Availability
The guarantee that the data behind a block or batch was actually published, so anyone can download it and check the results.
If data is withheld, no one can reconstruct the state, build a fraud proof, or exit a layer 2 without the operator's cooperation, which is why availability is treated as a separate security property from validity. Rollups can publish to the base chain, where every node receives the data, or to an external data availability layer such as Celestia or EigenDA, and validium designs keep data off chain with a committee attesting to its availability, which adds a trust assumption in exchange for lower cost. Data availability sampling lets a light node gain confidence that a large block was fully published by requesting a few random pieces, which is the basis of the danksharding design. Publishing data is usually the largest ongoing cost of operating a rollup.
In practice
A rollup whose operator stops publishing batch data leaves users unable to compute their own balances, even though the assets remain locked in the base-chain contract.
The common misunderstanding
Availability is not permanent storage; the requirement is that data was published long enough for anyone to download it, not that a chain keeps it forever.