The low point near 62,000 surprised me.

[10.029] ‘I am Thorin son of Thrain son of Thror King under the Mountain! I return!’

Exactly there.  I had predicted that paragraph would be a high point of uncommon words since in my heart it is a high point of drama.  Instead, it is a turning point.  The words surrounding it have to do with being soggy and smelling of apples, so I understand the low point.  Remembering that our analysis ignores names, we can see that this phrase uses words so important that they are in common use.  Remembering that it’s a five thousand word window, I’m mightily impressed that it is the exact turning point.

We climb out of the barrels, climb through Laketown, climb the mountain, solve the thrush mystery, climb into the mountain, riddle with Smaug.  Our average of  uncommon words rises with a few local variations (the local peak is [12.021] Thieves! Fire! Murder! Such a thing had not happened since first he came to the Mountain! His rage passes description) to that double-peak surrounding 74,000 words:

[12.093] …Smaug will be coming out at any minute now, …
[13.019] It was the Arkenstone, the Heart of the Mountain.

Is your heart racing?  My heart is racing.