The problem is that the preprint assumes the "Classical Schrödinger equation" (Eq 6). Is that an usual name? And then after a few steps get the uncertainty principle.

But once you use the Schrödinger equation, the abstract commutation rules for the position and momentum operators are defined. And from these rules it is possible to deduct the uncertainty principle.

The details of the calculations are not important, you will reach the uncertainty principle anyway, because it is hidden in the Schrödinger equation.

Actually, Eq. (9) follows from (7), or (12) follows from (11) using HUP. Of course, we cant get the Schrodinger equation without HUP. But the classical Schrodinger equation uses only the length scales. It is when deriving the quantum Schrodinger equation from the classical Schrodinger equation, that HUP is applied. Also, GR does not use HUP. I derived HUP in an earlier paper from Newtonian gravity without using QM. I showed here if the Riemann tensor in GR was zero, position and momentum would commute, thereby yielding no QM.