Nick Curtis said a brief bit about this in a lecture once, if I remember rightly it is the accumulation of the dead sole that has not come away naturally, it tends to come over eventually in one massive lump (in a hoof shape) and it can be quite thick.
Both my ponies have had bits of sole come away recently. I thought this was quite normal as I've seen it often not just with them. Same as chestnuts seem to get to a certain size and drop off.
On the other board they seem to think this is a problem and needs a podiatrist (probably spelt wrong).
What you describe sounds perfectly normal to me!
And I wouldn't worry too juch about getting a podiatrist, just routine visits by a qualified farrier will be sufficient!
False sole is just excess sole, like extra skin on our feet. You find it mostly in shod horses (that aren't worked on sand), as the sole isn't contacting the ground. It's quite harmless and most podiatrists will only trim a tiny part of it to see what condition the 'live sole' is in, underneath. False sole is flakier, whilst 'live' sole has a more waxy consistency. When I was studying, we were taught to let false sole wear away naturally and I don't *think* that's changed! You can't put back what you take away! If your ponies aren't lame and able to do the work asked of them, then don't fix what isn't broken is my advice!