Do all syllables occur in all tones? For example, ne and nè are words.  But there are no  nē, né, or ně. I tried to Google this but I don't find any references to it.  It's obviously not by final, because kē, ké, and kě all exist.  Is there a rule?
There’s no rule as to what tones can be used with which pronunciations but there is a finite list of words &amp; pronunciations in MSM (Modern Standard Mandarin).

When you start getting into Chinese dialects and topolects any pronunciation is possible.

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Just as a side note né can be found in 哪吒 (Nézhā) the name of a deity.