Excellent, thanks a lot for this!

''Maybe only the part of interacting is not axiomized'' Isn't that basically the only interesting part on the physics side? Only axiomatizing free field theory doesn't say much about whether it can be done for interacting fields.

Do we really know at this point that QFT is ''close to an axiomatic description''? The Wikipedia entry also says that ''Dirac formulated quantum mechanics in a way that is close to an axiomatic system'', but this seems debatable.

Could you provide some more references and details?

The degree to which proofs are carefully checked during peer review can depend on the area.

There is some interesting physics literature based around avoiding the problem that Hodge dualization requires a $\textit{regular}$ metric, whereas in unified theories there could be a ''pregeometric'' regime where the inverse metric does not exist or is singular. journals.aps.org/prd/abstract/10.1103/PhysRevD.105.125010

Is a stochastic modification of the SIR model widely used in applications? (just curious)

Link has now been included.

I'm not sure what you mean. Certain physical phenomena are just better modelled this way eg. if they involve Brownian motion. See for example link.springer.com/article/10.1007/s00220-022-04462-2