If someone learning with fractional aiming estimates a shot to be between a 3/4 and a 5/8 aim, they have a 7mm gap/subdivision to navigate within, which leaves them with three aiming options: Aim in the middle of the gap, aim slightly thicker than a 5/8, or aim slightly thinner than a 3/4. Using the tip/ferrule as a gauge is a good way to pinpoint the thinner or thicker aim lines, because an eighth of a tip is about 1.5mm. So the player must decide which of these options looks or feels right, and that is the aim line they use. Every 1mm off will be about a 1° difference in the shot result. The more the player uses this method the more consistent they become with accurately estimating those in-between shots, shots that don't fall exactly on one of the eighth of a ball references. And it's rediculous just how often a shot actually falls dead in line on one of those references.
With ghostball,...
...Every 1mm that they're off will cause about a 1° difference in the shot result.. This method also requires experience to become consistent with pinpointing that ghostball spot, but after enough experience the player will become more accurate and consistent with it.
With contact points,...
Every 1mm that they're off on the contact point will cause a 2° difference in the shot result.
It's not mathematical error. It's simply errors in estimating the correct aiming references. And that's exactly how shooting goes. We underestimate or overestimate the cut angle sometimes, or we don't compensate accurately for speed ot spin. So many misses are do to simple physical or visual errors. And the acceptable margin for errors can be defined mathematically.
Sounds exactly like mathematical error. Besides, the idea is to attain coherence to the the shot image. This is best done by shooting the shot - note the references - until the results are picture perfect. The work is the work not the discussible details.