I've never heard of using UDIMS this way, to have one continuous strip of polygons to go from 1001 to 1002. WEIRD! It looks like texture filtering issues. Trying to wrap my head around this, I'm wondering if you could break texture filtering in a setup like this if you clamped the texture, so texture filtering would be…
UDIMs require padding because of what Thanez says above. If unique painted /baked detail is required and you want to use UDIMs, you need to change the UVs.
You may want to try different wrap modes in the shader: repeat (traditional texture tiling), clamp (edge pixels are stretched out beyond UV 0-1 space), or mirror. More info and a pic: http://wiki.polycount.com/wiki/Texture_Coordinates#UV_Address_Modes