I am still looking for a way to build a bridge between the different senses of complete and incomplete being used in this discussion but while that bridge is under construction it may help to say what I’m saying another way.
Signs do not do anything at all by themselves — except take up space in their media — they do not denote, or mean, or propose anything at all except insofar as they are interpreted to do so.
Of course we all speak of signs denoting this or connoting that, but that is just loose talk, elliptical or informal manners of speaking, which our practice and theory of semiotics has the task of rendering clear.
One way of carrying out the required formalization is to introduce explicit interpreters and to specify exactly what interpretant signs they relate to just what signs in reference to just what objects.
But once we’ve specified that much, it becomes clear we are simply specifying a particular sign relation for specified object, sign, and interpretant domains. It can be a rhetorical convenience to keep the figure of the interpreter as a hypostatic abstraction or personification of the sign relation but all the information about the interpreter’s semiotic conduct is contained in the bare sign relation itself.