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"Orthodox" as used by Japanese professionals
Orthodox, as used in the Nihon ki-in style by Japanese professionals, roughly means "normal", or familiar play. It can apply to the style of playing or the moves themselves. Orthodox plays tend to fall somewhere in between honte moves and fast pace moves, a little thicker and much more alert than amateur play. Probably players with an orthodox style would not be innovating new moves or patterns.
See futsuu (普通)
Orthodox could also mean "largest local play".
"Orthodox" as used in combinatorial game theory 
Orthodox is a term in combinatorial game theory, coined by Elwyn Berlekamp. Informally, orthodox play is locally optimal play at a given ambient temperature.
When the term is used without reference to temperature, the ambient temperature is assumed to be just cool enough so that each player has an orthodox play instead of passing or playing elsewhere.
Now, the rest of the board may be such that orthodox play is not correct. See, for instance, tedomari. Since the normally correct play is not always correct, we can't just call it correct. But we need a word to describe it, and so we call it orthodox.
See a related page at greedy go.
 The writings by Bill Spight on Sensei's Library will usually refer to this usage of orthodox.