The point I'm making is simply that to be useful, when you close a seal you
have to be able to close it over some data, in particular, another seal. That's
the key thing that makes the idea a useful construct for smart contacts, value
transfer/currency systems, etc.

OK, your second post ("Closed Seal Sets and Truth Lists for Better Privacy and Censorship Resistance") seems to clarify that this data is one of arguments to the condition function.
Frankly this stuff is rather hard to follow. (Or maybe I'm dumb.)

Now I don't get scability properties. Let's consider a simplest scenario where Alice creates some token, sends it to Bob, who sends it to Claire. So now Claire needs to get both a proof that Alice sent it to Bob and that Bob sent it to Claire, right? So Claire needs to verify 2 proofs, and for a chain of N transfers one would need to verify N proofs, right?

And how it works in general:

1. Alice creates a token. To do that she constructs an unique expression which checks her signature and signs a message "This token has such and such meaning and its ownership originally associated with seal <hash of the expression>" with her PGP key.
2. To transfer this token to Bob, she asks Bob for his auth expression and sends a seal oracle a message (Alice_expression (Bob_expression . signature)) where signatures is constructed in such a way that it evaluates as true. Oracle stores this in a map: Alice_expression -> (Bob_expression . signatures)
3. Bob sends token to Claire in a same way: (Bob_expression (Claire_expression . signature))
4. Now Claire asks if Alice_expression->(Bob_expression . _) and Bob_expression->(Claire_expression . _) are in oracle's map. She might trust the oracle to verify signatures, but oracle doesn't understand token semantics. Thus she needs to check if these entries were added.
If I understand correctly, Alice_expression->(Bob_expression . _) record can be communicated in just 3 * size_of_hash_digest bytes.

So this seems to have rather bad scalability even with trusted oracles, am I missing something?