I'll admit to having no idea what those two images show :)
You mention using checkerboard in bump and displacement. Based on that, I added the C4D native Checkerboard surface to the bump channel of one CoronaMtl, and to the Displacement of another, and these are the quick and dirty results, bump on left, displace on right (used displacement World Size of 0.02 to give it fine detail and no artifacts).
Let me know more specifically what you were doing, if this was not what you meant.
Thanks!
Tom