Hi JanSidgwick,
I’m not using Arnold. So, I would go with a texture. This texture can be created either in BodyPaint 3D or with MoGraph.
The problem with this is that in any way, you need a final image resolution that is sufficient for the area with the largest stretch. In spherical projection, that is the “equator” area. If a deformer works on that sphere, it needs to be observed locally more in detail.
The next problem is that you certainly like to have the words not overlapping, while perhaps a semi-automated set up is targeted. I hope I found a good mix in MoGraph. I project single words (numbers in this case) onto the surface via Multi Shader. Randomized the rotation (if needed) and used a Target to keep them centered if you like an easy adjustment. It can work without, just use the Cloner> Transform instead, Target: off!
With a Push apart, I avoided the overlap.
Since this is a light set up, I was able to Bake the lights into a texture that was already using the UV or Projection method. Doesn’t matter so much.
I use this Equirenctangular texture projection a lot in my artwork, so please set the Sampling to something else, other than MIP, to get clean pole-caps.
The MoGraph example is here: Screenshot and files
https://www.amazon.com/clouddrive/share/m16wMnqgxs75Gx4jJqvU3CvDOiemhFfRBuliaxwIJDc
With the texture backed from file …01.c4d, you can see in file …11.c4d how seamless this works. Bake out 1.5 larger than you need, based on how close you get with your camera, etc.
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The BodyPaint 3D option is discussed here (post #4)
https://www.cineversity.com/forums/viewthread/4447/#17958
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My best wishes