Thanks for the kind words John :)
In response to your question, I would honestly answer with; I don't know. Given the infinite nature of say, real numbers, it doesn't seem like they are physical. But in saying this, we humans with our mushy brains are finite creatures which dreamed up mathematics, hence why I drew on Landauer's work so much: He would argue mathematics must be bounded by physical constraints for this reason. Therefore, is it possible to bridge these two? I argued that if real Turing machine are an attempt to physically realise mathematics, but because of thermodynamics and the energy requirements to run the thing, you still can't realise all of mathematics i.e the halting problem. Thus, a Turing machine cannot be a viable way to bridge this Platonic divide.
For the wavefunction to evolve via the schrodinger equation, we need some potential energy V(t). Typically, we just add V(t) to our calculation without really asking where it comes from. In reality, it comes via the interaction of your system to an external environment i.e light matter interaction and the dipole moment. Now we can always include this entire interaction ad infinitum in our system, but we are now running with open arms into the church of the higher hilbert space. I'd argue that if your trying to build a Turing machine, eventually, there is going to be some interaction that you just can't as it's lost in the soup. At this point, errors are going to creep in and thermodynamics comes into the fray.
I completely understand the sympathy of your scepticism about hinging my arguments on thermodynamics since it is emergent. But if mathematics is Platonic and only in our heads, then it to must be emergent. I don't believe our brains hit some critical mass which allowed opened up a portal to the Platonic realm: For me, such an argument is necessarily on par with the claim that one can speak to God through prayer.
Again, thanks for taking the time for reading my essay!