To expand a bit with some examples: the energy a photon in a standing wave in a specific cavity can have is quantised. But a photon out in space can have any old energy it wants to. (Of course, a given energy level will correspond to a specific wave length etc.)
Similar, an electron in a single isolated atom has specific quantised energy levels. But if you look at the electrons in a hunk of copper, they are essentially free to absorb and emit energy in almost arbitrary amounts.
An even stronger example is time: as far as I am aware, time is not quantised in any of our accepted theories.
There's some reasonable speculation that ultimately everything is quantised at the Planck scale, including time. But that's just a very reasonable hunch, not something that 'physics has shown'. (And you already point out that trying to marry quantum mechanics with general relativity is a hot mess.)
I agree that 'quantisation all the way down' is the way to bet. But that's just speculation.
(But I strongly disagree with your claim that physics is build on integers. Yes, it might be discretised, but there are plenty of discrete structures that are not integers. Look at a Rubik's cube for a simple example. On top of that: almost any real world measurement is better described by a probability distribution than by single number, be that an integer or otherwise.)
Similar, an electron in a single isolated atom has specific quantised energy levels. But if you look at the electrons in a hunk of copper, they are essentially free to absorb and emit energy in almost arbitrary amounts.
An even stronger example is time: as far as I am aware, time is not quantised in any of our accepted theories.
There's some reasonable speculation that ultimately everything is quantised at the Planck scale, including time. But that's just a very reasonable hunch, not something that 'physics has shown'. (And you already point out that trying to marry quantum mechanics with general relativity is a hot mess.)
I agree that 'quantisation all the way down' is the way to bet. But that's just speculation.
(But I strongly disagree with your claim that physics is build on integers. Yes, it might be discretised, but there are plenty of discrete structures that are not integers. Look at a Rubik's cube for a simple example. On top of that: almost any real world measurement is better described by a probability distribution than by single number, be that an integer or otherwise.)