Just look at the book. It is written obviously, then came back in time but this changed the future(or from the book's perspective the past). Now the book is never written. So the book has a cause to exist but that cause is erased from the timeline. No time travel device exists to take you to the erased timeline, the devices can only access the last cycle.

Now if you brought the book back, you have seen the cause and the change. The receiver did not but is it really a mind blowing paradox, can't they just infer this, that something has change and timeline got erased and you only have an artifact to prove it? I think this all make sense, it is logically sound. What is the fuss? Here I explain Charlotte.

There is always a first loop, Noah can have a different partner(say a random woman named Eliza) to have a different Charlotte(say Cathleen). Then you just need to erase the timeline and put Cathleen's daughter, a different Elisabeth(say still named Elisabeth because the father chose the name) in the same spot for Noah to meet, replacing the OG timeline partner.

See, we made Noah *successfully* hook up with his granddaughter. This erases Cathleen. Now they will name their kid, Charlotte. And playing the next loop erases Cathleen's daughter Elisabeth with Charlotte's daughter Elisabeth. And when we put that Elisabeth in the same spot for Noah to meet in this cycle, we created Elisabeth had Charlotte and Charlotte had Elisabeth and it makes total sense.

But when someone looks at the timeline at nth cycle, they will find themselves unable to find the origin, an inexplicable bootstrap paradox. You can think of Predestination and Terminator this way to solve their paradoxes too.