LAWYER!
The county will not install a tinhorn on a private road, or a privately maintained road.
Sounds to me like you need to buy the tinhorn, and install it across his driveway.
If not that, then buy a tinhorn and install it to cross the road where it's being washed out.
Money out of your pocket, but cheaper than constantly re-graveling.
I've had to rebuild my neighbor's driveway twice, to keep water from running down his driveway into mine. I finally moved my driveway.
nothing is listed but I am looking at a document showing private roadway easementPut in your address or your neighbors to see if there is an easement.
https://www.okc.gov/departments/city-clerk/easement-deed-land-document-locator
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