I apparently had a book delivered recently.... the online pbs tracking says that a note was delivered. (?)

But then the post office man delivered something (package) and put it in our front door and put the card about it in our mailbox.

There is NO package in our door.

We've had some thefts recently and apparently we've been had again.

So it "looks" like the book was delivered but I never got it... Do I mark it lost when time runs out  ???

The system will automatically mark the book lost when you do not mark it received.   Have you contacted the PO?  This just had this happen to me, it was a temp carrier, he didn't leave the box, I received an alert from USPS that a notice was left.  No notice and no package.   I went to the PO today, the carrier delivered it today, he didn't realize it should have been left, otherwise it would be at the PO.  I would definitely check with your PO and then notify the sender.  They can open a search with the PO.  It may still turn up.

IMO if you think it was theft, then you should report it to the PO and the police.

It automatically gets marked lost.  As long as you have been logging on to PBS.  (The computer marks it as received if you have not signed on to PBS.)  If the sender used PBS DC, then you should get your credit back.  The sender should also get their credit back.

You can also take the DC# and contact your local post office about the book.   Usually a "note" means that a yellow notice to pick up the book was left with your mail.

I had a book scanned as delivered to me and I never saw the  book. The post office was no help in my case.  So I wouldn't worry about it too much.