I watched the Blue Prince blender conference talk, and they showed a shader setup that allows you to scale/tile the UV in a texture atlas, while still staying inside the tile/cell you placed the UV in. The video explains it better than me.
I’ve been trying to recreate it, copied the nodes from the video, but there are a bunch of inputs that are offscreen. I haven’t gotten it to work like in the video. I
It starts from 11:07 and goes on for like a minute. Relevant screenshots from the video below.
(Blue Prince : Beyond The Doors — Blender Conference 2025)
Tried to do this once; do not remember the exact math but Modulo operation was involved. Your math looks more comprehensive. Unfortunately, in my experience this has more academic interest than practical when used in Blender. In game engines, this trick improves performance because it minimizes number of draw calls. In Blender, the main bottleneck for Eevee performance are modifiers (such as Armature), which are CPU intensive. Not rendering pipeline.
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( Some people just ask and never ever do anything else here 
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