What about 'synthetic a priori' statements? I think they're central to science. — Wayfarer

In my view, 'synthetic' means 'empirical' as opposite to 'formal' (content level) and 'a priori' means 'predictive' (formal level) as I said. Hence, 'synthetic a priori' is central to science insofar it fits its "hypothetical" side of the hypothetical-deductive method.

the apriori looks like knowledge, but actually is just the description of a model — gurugeorge

You are assuming that 'a priori' refers to models, that is, formal knowledge. It is 'analytic' in my opinion, but not 'a priori'. Only 'synthetic' knowledge can be 'a priori/posteriori' given that it need to experimentation or experience. Can you gave us other non-mathematical example of 'a priori'?

knowledge can be deduced theoretically without empirical evidence — Kamikaze Butter

In my opinion, "without empirical evidence" only can be understood as referred to certain formal knowledge (logic and mathematics, when the true can be predicted, such as the value of "x" in the equation "2+2=x"). I would accept that this is the "analytic" knowledge. But the temporal dimension of "a priori/posteriori" is neglected in my view.

I think that "prior to experience" means "predictive" insofar it introduces an temporal dimension in the truth. In mi opinion, your example of the degrees of a four sided foundation is not relevant in the a priori/posteriori distinction due to time is not included in the mathematical theories.

Thank for your response @gurugeorge

I think that the "a priori" moves from language to reality, about stating the possibility of a fact; and the "a posterior" moves from reality to language. For example, in empirical science, all hypotheses tested in an experimental research are said to be possible "a priori", but not "a posteriori". If the study is informative or well-designed then results must be discard at least one of the hypotheses.

They are (informal) definitions, so their true is established a priori.
I want to say that a "unnecessary" conclusion can be "invalid" in two ways ("possibly" and "impossibly" true). MP inference is a valid scheme (necessary), AC is invalid but "possible" and MT asymmetric (to infer "A") is "impossible" to be true.
If something is not necessary, then it is contingent; if contingent, it can be possible and impossible. If it possible, also it can be "real" or "empirically" true.

I resume my idea with other words:

"A" is necessarily true iff "not-A -> (B and not-B)"
"A" is possible true iff "A or not-A" is true
"A" is empirically true iff "A" is true

What fails in these definitions?

I think that a non-physical thing is that has not physical properties, so physics is not able to account it. For example, mathematical and logic systems, natural languages, etc.
Regarding conscience, it seems to be the physical thing that make possible the existence of non-physical things.