That tentativity, comparison to reality, reliability and the testing of conclusions/hypotheses are key components of what science is
I don't know how these concepts can be incorporated into any axioms of science that we might eventually decide upon but, it seems to me, that the foundational fact, principle and condition that you specify above are in themselves not wholly adequate in reflecting these criteria.
Or am I pushing the definition of axiom (as I have defined it to be) to breaking point in attempting to include such considerations?
You may be stretching 'axiom' to breaking point, but I think I see where you are going. Certain 'principles' of science are 'unproven' (and instead they rely on strong arguments in favour of). Obviously between Science! and what I've outlined there needs a lot of filling in, but I was trying to keep as close to axioms and avoid as much as possible straying into 'principles' as I could.
The philosophy side of 'knowing', or epistemology is simply about trying to figure out when something can be reasonably called 'knowledge'. Then there needs to be acceptance of certain real-world limitations, followed by figuring out what those limitations are, what compromises need to be made (such as the
problem of induction) and finally some kind of methodology can be constructed.
The methodology (or rather, methodologies) we have ended up with is called 'science', and there is a lot of metaphysical/epistemological groundwork that goes behind it all - not all of it (any of it!?) 'provable'.
So yeah - science shouldn't be considered an epistemology per se, it is a practical pursuit rather than an idealised one. In an ideal world we'd look at the evidence and be able to deduce reliable conclusions from it, just as Rob implied. However - the real world is not ideal and so various compromises and gaffer tape patches are needed.
The philosophy of science is an interesting subject (even if certain science based posters around here would flat out deny that philosophy and science have anything to do with one another), let's hope someone that disagrees with us pops along and that might spur a bit of creative posting.
So science does lean on some other ideas possibly including
PragmatismCoherentismParsimonyFallibilismCorrespondence
And so on and so forth. These kinds of things cannot be proven so are they the kinds of things you are looking for? My favourite is the argument against
verificationism which says that since verificationism itself cannot be verified, verificationism by its own standards is incoherent or meaningless. Such wonderful circles one can run in.
Does that help at all?