It is funny that this old post from @azircon still attracts reaction.

I think there may be a misunderstanding of my argument. In particular:

• You mention that my assumptions on the velocities at t₁ and t₂ are incorrect, stating that the motion is vertical at t₁ and horizontal at t₂. However, this is exactly what I explicitly assumed and stated in the proof, so your points (1) and (2) do not contradict my setup.


• Concerning the tension, the sign conventions matter. At t₁, the mass tends to move to the left, i.e. opposite to the positive x-direction, so the corresponding tension is negative. At t₂, the mass tends to move downward, i.e. opposite to the positive z-direction, which again fixes the sign of the tension to be negative. This is why you get -8 instead of +8.

With these conventions applied consistently, I therefore maintain my original conclusion: +8 is the correct answer.