Elastic promises "open"—delivers proprietary

It *can't* say what is a derived work. Derived work (or more commonly "derivative work") is a term of art in copyright law. Judges decide what it is; licenses don't.

The FSF has taken the position that dynamic linking creates a derivative work; they may even be right about that. But legally, it would be up to a judge to decide on a case-by-case basis, or up to a national legislature to codify explicitly. If dynamic linking does not create a derivative work, then there's no substantive difference between the GPL and the LGPL, never mind the AGPL.

> if you provide a web mail service and want to use an AGPL-ed PDF renderer in an iframe, then will AGPL propagate to all of the software?

There is nothing in AGPL which suggests that the answer should be "yes." You would not need to modify the renderer in order to do this, and the extra AGPL obligations (in section 13) attach at time of modification. If the software is unmodified, the AGPL is materially identical to the GPL.

Even if the software *is* modified, however, the normal GPL limitations in section 0 still apply, particularly this:

> To "propagate" a work means to do anything with it that, without permission, would make you directly or secondarily liable for infringement under applicable copyright law, except executing it on a computer or modifying a private copy. Propagation includes copying, distribution (with or without modification), making available to the public, and in some countries other activities as well.
>
> To "convey" a work means any kind of propagation that enables other parties to make or receive copies. Mere interaction with a user through a computer network, with no transfer of a copy, is not conveying.

In other words, the AGPL does not have a different "test" for what counts as "modification" than the GPL; they both use exactly the same set of legal definitions, right down to limiting "conveying" to *not* include "mere interaction [...] through a computer network." The AGPL is almost entirely identical to the GPL, in fact. Section 13 is pretty much the only difference, so if your hypothetical does not implicate section 13, then you can analyze most of the legal consequences in the same way you would analyze the GPL. If section 13 is implicated, it still has to be analyzed in the context of the rest of the GPL, because that's what the AGPL is - the GPL with an extra section.

In short, the AGPL probably doesn't cover your hypothetical.

> SSPL on the other hand does limit it

"It" in this context does not refer to "derivative work." It refers to "making the functionality of the Program or modified version available to third parties as a service," which is a contractual term, not a copyright term. You can define and limit contractual terms in whatever way you like.