> "well, it looks like we are going to have to define friendship." 

Right — the conclusion smuggled in a load-bearing term. If the polis is downstream of the friendships, then everything hangs on what "friendship" means, and Aristotle would be the first to say the English word is hopelessly baggy compared to philia.

His taxonomy in Nicomachean Ethics VIII is the obvious place to start, because it's built exactly for this problem. Three kinds, distinguished by the ground of the affection: friendships of utility (I love what you're useful for), friendships of pleasure (I love how you make me feel), and friendships of virtue or character (I love who you are — your good, for your sake). The first two are accidental in his technical sense: they're really about me, and they dissolve the moment the benefit or the pleasure stops. Only the third is friendship haplōs, without qualification, because only there is the friend loved as the person he is rather than as a delivery mechanism for something else.

And here's why the definition matters for the edumonia argument specifically: only the third kind can do the formative work the conclusion assigned to friendship. A study group held together by utility (networking, credentials by another name) or pleasure (the enjoyment of clever conversation) reproduces exactly the instruction-plus-signaling regime it was supposed to escape. Character friendship is different in kind because it has a shared object — the friends are jointly turned toward the good, and each is, in Aristotle's phrase, "another self" whose perception extends your own. That's why he says friends are necessary even for the self-sufficient happy man: you see your own activity more clearly in a friend than in yourself. The friend is an epistemic instrument for virtue, not just a companion in it.

Which yields the demanding part of the definition: character friendship requires time, shared life (suzēn), and rough equality in virtue — "they must eat the proverbial salt together." It can't be scaled, scheduled, or purchased, which is precisely why it's the scarce resource under the whole argument. Institutions can convene utility and pleasure at will; they can only create conditions for the third kind.

So a working definition for your purposes might be: friendship is the mutual, recognized willing of the other's good for the other's own sake, sustained by shared activity directed at the good. The shared text on the table isn't decoration in that picture — it's the third thing that makes the mutual turning possible.

The pressure point worth deciding on: does the definition require the friends to already be virtuous (Aristotle sometimes talks this way, which makes it circular for education — you need virtue to form the friendships that form virtue), or can there be apprentice versions, friendships of aspiration where what's shared is the pursuit? The second reading is what your small-communities conclusion quietly assumes. It's defensible — arguably it's what NE IX.12 means by friends "becoming better through exercising their friendship" — but it's a choice, and it's where a Thomist would splice in grace doing what nature can't bootstrap.