Is there a rigorous definition for what a "complete" theory without "hidden variables" is? I find only vague characterizations of what a theory that is complete and without hidden variables is supposed to be. Such as a theory that is complete means that there is no lack of knowledge about an underlying physical reality and hidden variables being unknown physical quantities. Is there a more rigorous way to define the notion of "completeness" of a theory? How exactly are "hidden variables" defined? If there are no such definitions, how would you describe it in more precise terms?
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define the notion of "completeness" of a theory? How exactly are "hidden variables" defined? If there are no such definitions, how would you describe it in more precise terms?

First let us be clear of what a "theory" means in physics. It is a mathematical model that fits existing data and observations and , very important in order to be validated, makes correct predictions for new data and observations.
Mathematics has complete theories starting from axioms and developing into theorems rigorously. The mathematics of calculus and differential equations allow an infinity of possibilities for rigorous solutions. Physics theories impose extra axiomatic statements, called laws, postulates, and tables of constants which pick up from the infinity of solutions those solutions that fit the data and observations, relating abstract numbers to physical units. Look at the postulates of quantum mechanics, how involved and complicated they are . The elementary particle table is also axiomatic for the current main stream physics standard model.
So "completeness", the final mathematical model, is complicated by all the assumptions that enter  in the theoretical physics model, it is
more a philosophical search, like the platonic ideal forms.
That is why you find "only vague characterizations of what a theory that is complete and without hidden variables is supposed to be." For such a physics theory the postulates should be simple, connecting the physical units to the mathematical model ( example the geometrical algebraic  model) and all the present  complexity of the physics theory should emerge through mathematical calculations. It is  too complex to define.
