It depends on several factors, the main ones being tractor weight, soil type, and drainage. A CUT with unloaded R1's will do little to no damage to a well drained, sandy-soil lawn. My FIL cuts a lawn like that with a 4500 lb tractor with R1's and I have never noticed any damage. My own lawn is not as well drained, with dark, rich topsoil and, except for mid-summer, I need turfs to avoid lawn damage. When you are off the lawn, in wet conditions, nothing provides traction like R1's.
I have never liked R4's in non-industrial applications and consider them to be "the worst of both worlds", providing traction in the mud, little if any better than turfs, and doing nearly as much damage to soft lawns as R1's.