This is probably not what you wanted, but maybe it can help you somehow. I figured it couldn't hurt.
I googled Arkansas John Pate sawmill, and came up with this military information.

Pate, George

France (Country), 26, m, 5-11, Farmer

Mustered in 3-5-62. Wounded at New Hope Church--gunshot left hand, lost thumb. No 19 United States Army General Hospital Nashville Tennessee. Hospital Louisville Kentucky. Floating United States Army Hospital "OHIO." United States Army General Hospital No 5 & No 8 New Albany Indiana. Transferred to 70th Indiana Infantry. Mustered out Goldsboro North Carolina 4-6-65. Died 2-28-1903

Occupations:

Manual Labor = 95
White Collar = 1
Artisan = 14

Farmer = 95
Clerk = 1
Brick Mason= 2

Laborer = 6

Brick Molder = 1

Sawyer = 3*
Transportation = 0
Shoemaker = 1

Sawmill Operator = 1


The address to the information is;
http://www.geocities.com/Pentagon/Barracks/3627/companyg.htm...

There is also a person who has a photo of 671. John Pate and Wife, Letona?, White County, Arkansas. Here is that address;
http://archives.uca.edu/photograph_collection/p29.html

If it's not your John Pate, maybe it's a relative? I would see if the man that has this photo knows anything about it.

There is also something about a John Pate in Arkansas that has something about a sawmill in the article, but I can't seem to get the thing to open up (my computer is having serious issues today). From what I can tell, the sawmill is called Pate Sawmill.
That address is;
http://www.fpl.fs/fed.us/documnts/fplrp/fplrp594.pdf


Good luck!

bj