Why is this classical system seemingly non-deterministic?

Question

So, I was solving this problem¹ and I realized that the system given seems to be non-deterministic when analysed with classical laws.

The situation is thus: A rope is wrapped for an angle of $\theta_0$ (symmetrically) around a fixed pole (the circle in the diagram). One end is fixed to a wall, and the other end is pulled with a force/tension $T_0$. The string is massless, but there is friction between the string and the pole of coefficient $\mu$. There is no slipping (the system is static). If necessary, the length of the string and the radius/relative position of the pole can be considered as givens.

Now, if you follow my solution², one gets that the tension in the string along the cylinder, where $\theta=0$ is the part on the right where the string just touches follows the following constraints: $T_0e^{-\mu\theta}\leq T(\theta)\leq T_0e^{\mu\theta}$. Extending this, you get that the tension $T_1$ in the part of the string connected to the wall follows the constraint $T_0e^{-\mu\theta_0}\leq T_1\leq T_0e^{\mu\theta_0}$.

However, I don't see any way to further analyse this system with Newtonian mechanics. Note that the inequality appears even if I consider a string with mass, though the calculation becomes more complicated.

So basically I'm stuck with an inequality. Note that the system, as given, should be completely defined — $\theta_0$, $T$,$\mu$, the radius of pole, and the relative positioning of pole are all that should be required to physically create this system. However, the macroscopic, measurable quantity $T_1$ somehow cannot be determined from Newton's laws.

How does one resolve this apparent paradox? Classical systems are supposed to be able to fully determine measurable quantities. This doesn't seem to be the case here.

^{1. This was a course on Classical (Lagrangian & Hamiltonian) mechanics, and our professor gave us some Newtonian mechanics problems as a first assignment. We're not yet sure why, but most were quite interesting. The assignment due date is over, this is just a residual point that has been bothering me.}

^{2. This is lifted from the document I submitted. I have blurred out the end because it contains my own resolution of the apparent paradox here. I may post it as an answer later, but for now I'd prefer to give everyone a fresh look at this situation.}

Answer 1

The system is not completely determined. For example, if the pole had been rotating very slowly clockwise before being brought to rest by the friction of the string (and then welded to the wall, so as to be a "fixed pole", but with the friction force remaining) you would get one answer. If counterclockwise, you get the opposite.