Never assume that your problem is self-evident or easy for the developer to reproduce.
I think it is pretty easy to reproduce, as long as the initial setup has the problem.
I need to find some time to work with this. I need to do another restore to "factory default", and then do another eCS install, tracking what happens along the way.
I did find the reason why LVM is all worried about the partition table. The Lenovo recovery partition extended past what LVM seems to think is the last valid cylinder on the disk. When I used DFSEE to shrink that partition, by the smallest possible amount, the LVM problem went away. Now, however, Win7 (PRO, if that matters) refuses to use that partition. It is clean, and it can be read. Win7 simply refuses to use it (it probably assumes that it has been corrupted). Not an ideal solution, and it is deadly if the user failed to create the recovery media (DVDs).
I am pretty sure, that the initial "problem" that was fixed by the Integrity checker, during install, was that the Extended partition didn't start on a cylinder boundary. That seems to have been fixed properly.
As I said, LVM needs to be fixed, or replaced. DFSEE does the partitioning, without complaining about a non-issue, and as long as I leave the recovery partition alone, Win7 seems to be happy with it. Only LVM has a problem.
If eCS was to change to using DFSEE to do the partitioning (which seems to be the obvious answer), I suppose that miniLVM would also need to suppress the complaints, when a user wants to do other things with (mini)LVM.