First, pete, you do yourself a disservice in that opening section. Normally you avoid such sniping and it is a shame you fell into it this time.

Onto more interesting stuff: you talk about adding and removing 'information'. That's rather informal: do you mean information in the precisely defined mathematical sense? If so, I could do it, but it would look like a couple of pages of algebra, and probably still wouldn't capture the essence of what you are really getting at. Is it possible for you to define what you mean by 'information' sufficiently precisely that several readers, including yourself, cannot disagree whether in any particular case it has been added or not?