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can they be considered to have compressed the entirety of their training data into their weights?

If you consider them lossy compression, then yes.

The goal would be to find the minimum model that, with a fixed seed, would exactly reproduce your text.

So I could hide information in a model basically

I think this would be the exact opposite. This model would be optimized to only output your information.

thats what I meant. For a given input give me the output I want without anyone knowing why that is there