I see the TM tag with smaller text than it's associated name, does the same apply for the ® symbol.
Should it be the same font size as the name or smaller like exponents are displayed in math?
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The ® symbol doesn't need to be in a smaller font size than the actual text. In Name™ the trademark symbol ™ is not a smaller font size but a special character which has a shape that is smaller than the characters T and M.
Most of the good fonts have a wide range of unicode support which means they have glyphs/letters drawn specifically for these purposes. In the following image you can see how some fonts like Source Sans Pro and Mukta have a small RTM symbol by default.
So you can use it as a superscript if you feel it's attracting too much attention (e.g. Logo design), but when using in a running text it's better to use it the way it's designed.
I could not find anything from the PTO about this. Every time I have used it, it's always been the same size font with a superscript (or exponent, like math symbols). TM, SM R. I always use them as a superscript.