Let S be the sphere and consider the chromatic tower:

S--->...--- L_n S--->L_n-1 S --->...--->L_1 S ----> L_0 S = HQ_p

This tower gives rise to a spectral sequence which is at least conditionally convergent. How long can the differentials in this spectral sequence be? I wanna say this ss degenerates really fast but i'm not sure how to show it even converges so...

Honestly my expectations are that this degenerates at E_2 but i learned not to trust my gut by now...

I withdraw my guess. The differentials must be arbitrarily long because the E_1 page spans a half plane but the end result is a first quadrant.

Isn't this known to be a huge mess?

Maybe I'm just discovering this as we speak...

I want to say it is the chromatic SS, but I know this is different (although strongly related)

I thought the beautiful chromatic filtration would give rise to beautiful spectral sequence but im starting to think this is not the case

I thought the chromatic spectral sequence has to do with the E_2 page of ANSS for BP...

I don't know anything about it though...

So I guess I change my question to "how bad is this spectral sequence?"

Cool! mathoverflow.net/a/124668/22810

Haven't read everything yet but its at least relevant.

Actually I was too quick to trash this ss. Its actually upper half plane so the situation is not that bad because there are only finitely many nontrivial entering differentials in each coordinate. If anyone knows anything written about this spectral sequence that would be nice.