The log is an append-only tree. Each committed record is a leaf, and the signed head of the tree is anchored to Bitcoin.
A checkpoint names the log, its size, and the root of the tree at that size, and is signed by the log's key. This is the exact object a witness cosigns and the exact bytes anchored to Bitcoin.
Before a checkpoint is published, it goes to witnesses run by unrelated operators. Each witness checks that the new head extends the history it has already cosigned, then signs it. This is what rules out a second history: the operator cannot show one version of the log to one party and a different version to another, because the witnesses remember what they signed.
A cosignature attests that the log is append-only with a single history. It is not an endorsement of any claim inside the log.
The checkpoint bytes above are timestamped into Bitcoin every hour. A fresh stamp stays pending until a block confirms it.