***H: There are hundreds and thousands of deadends that are mathematically valid A: Since you acknowledge that the problem exists, how do you know that the "mathematically valid" solutions of today won't turn out to be invalid as we gather more empirical knowledge?***
My view on this is that science is a discipline focused on predictions of approximate behavior of nature. The mathematically valid theories are successful at making these kind of suitable predictions (which is measured against a statistical accuracy rating (e.g., sigma 6)). Phrasing the experimentally confirmed theory in a mathematical language provides justification that the theory is consistent with the manner in which we observe nature to flow. In addition, the theory should be mathematically and logically consistent with other theories having these features (i.e., theory coherence).
The actual ontological nature of the theory is never set it stone. In fact, a number of well-known philosophers of science outright reject scientific theories as being true.
So, to answer your question, we never know if a theory is true or not, but we are confident right away if the theory is mathematically valid. If a theory (i.e., an established hypothesis having a great deal of scrutiny by others) were not mathematically valid, then there would be errors in the math and someone would catch those errors prior to it being elevated to a scientific theory status. There might always exist some exotic philosophical problems with a theory (e.g., duality in QM, time travel in GR, etc), but this by itself cannot be used to thwart the acceptance of a theory since our experience has shown that later knowledge can often resolve exotic philosophical dilemmas.
***If you're smart enough you can only answer the above with "only time will tell". That's acceptable but now you have a different problem: how is it that some of our mathematically valid solutions can't be proven wrong?***
The theory in question as long as it does a great job at approximation and is mathematically solid (which is known fairly quickly after being taken seriously as a hypothesis) will serve as a valid model for quite some time to come. That doesn't mean that a better and more fundamental theory won't emerge (thus relegating our old theory to the history of science or some introductory course where the classical theory is taught, etc). Progress in science almost requires older theories to be seen as less fundamental (and even invalid in certain special cases - such as when approaching constant mass is no longer constant when approaching the speed of light, etc). If scientists are doing their job correctly, then previous theories should be approximately valid in a number of specific cases where they should always serve as approximately valid models (e.g.., classical physics). If more simplified and more general models come available (e.g., quantum field-theoretic methods that shared many of the results of S-matrix theory within particle physics), then the former theory may still be taught (e.g., S-matrix), but the focus may be toward the more general and simplified theory (e.g., field theory).
***Come on, you're smart enough to understand the situation. You can find the answer if you: a) avoid thinking this is a problem of ontology or epistemology b) refrain from searching for a ready made answer on lanl c) realize it's an abstract problem which can only be solved with logic***
I can't agree to any of your choices in (a)-(c). Firstly (a), this is a problem between epistemology and ontology. Science is mainly advocating an epistemological attitude toward theory while ontology remains outside science as being metaphysical (i.e., something that philosophers of science discuss); (b) is too limiting in that a number of scientists have inputted their perspectives on these issues and their opinion counts; and (c), these problems are probably unsolvable even by logic and therefore there will always be a degree of trepidation when talking in terms of a theory's ontological status.
Warm regards, Harv