> But the model that correctly predicts all those observations does not have "neutrons" that are countable with integers everywhere and at all times between observations.
Physics equations don't have "if (...) { ... }" conditionals in them. The Universe doesn't seem to run on Boolean algebra!
The rules that govern neutrons either apply everywhere, or nowhere.
If the number of neutrons weren't so thoroughly tied to integer quantities, then deuterium could spontaneously convert to tritium and would decay at some non-zero rate.[1]
A mental model that I like to use (but certainly isn't mainstream) is to think of fermions as topological flaws, much like knots. In this model bosons are like wiggles in the rope.
I like this model because Fermions are very robust to outside interference except when interacting with other Fermions. This is much like knots in a rope. No amount of rope wiggling will ever undo a knot.[2] However, when two knots meet on a rope they obey a kind of "knot algebra" with strange and interesting rules. Preon and Rishon models are an attempt to formalise the mathematics of this, and work surprisingly well.
Continuous transformations of a topological flaw in a 3D volume of the "fabric" of the universe has Spin, which is hard to introduce otherwise. Topological flaws in crystals have even been used to model General Relativity!
Obviously, reality is not a 1-D rope with knots (tell the String Theorists that!). My model of this is a block universe where the parallel universes form a continuum. The knots are point-like only if you take a 3D spatial slice through the higher-dimensional block[3]. However, this point-like behaviour can never 100% manifest physically, because only interactions with other particles can ever be used by in-universe observers.[4] These particles in turn are a continuum across parallel universes. Thus any interaction is essentially the product of two continuous functions with each other, not infinitesimally small points bouncing off of each other.
I know this may seem a bit... "out there", but it's just a minor variation on MWI with elements of Rishon theory sprinkled on top.
[1] I'm obviously pulling this example out of a hat, but you get the idea. I'm not aware of any experiment that can demonstrate neutron number changing in any circumstance other than high energy collisions, or particle exchange between hadrons. A neutron floating about in space will remain a neutron. You won't get two neutrons suddenly turning up where you had one before. More importantly, you can certainly never have fractional neutrons. You can have a fractional expectation of finding a neutron, but you'll find a whole neutron.
[2] This begs the question of how photon-photon pair production can create an electron-positron pair out of nothing! This is like a sufficiently strong wave looping the string back on itself to produce a pair of opposite-handed knots. If separated you can call them two particles if you wish. If you bring them together they "annihilate" to form a rope that's locally highly curved. The curvature flattens, racing outwards as wiggles. These are the gamma rays produced by an positron-electron interaction. Similarly, long-wavelength wiggles move the whole knot around, but very short-wavelength wiggles can loop around the knot and through it. This could explain how some particles interact, but only above certain energy levels.
[3] A common mental model of MWI or parallel universes is something akin to pages of the book with slightly different content printed on each page. But this traps the physicist into thinking that spatial slices can only go parallel to the pages. In my model there's no preferred direction or "grain", so any slice is valid! This means that observers will disagree on essentially all measurements, including particle number. However, all observers will agree that particle numbers are quantised, and all observers agree that the particles follow the same rules. They just disagree about particle histories and particle futures.
[4] The refusal of some physicists to admit that everything obeys QM rules is a disease of the field. Either everything is QM or nothing is. QM experiments don't end at the bench top. The equipment, the physicist, everything is a part of the wave function. This insight is critical, and forms the basis of RQM and MWI, but not mainstream QM as taught in most universities.