Given that up to 50 or more discrete aim lines are needed to make all shots on the table (25 for a spot shot into a 4.5" pocket), how does Poolology "mechanically" define that many?
It doesn't define that many. I specifically referenced shots from straight-in to about 38°. Beyond this the player has to estimate where the tip of the cue is pointed, because it's outside the visible edge of the ob. And that does require guesswork or estimation.
In all practicality, we don't really need to define 50+ aim lines, not even close. Nevertheless, for any shot thicker than about a 38° cut (which covers the majority of shots we're normally facing) Poolology defines the aim as simple as this....
Each of these shots is a halfball shot, and they're defined as such based on where the ob's are sitting and where the centerline between cb and ob lands on the rail.
Place an ob anywhere on either of the two thick highlighted lines shown, then place the cb at least a foot away and lined to shoot the ob straight toward the 1st diamond (on the side rail or end rail). This defines a halfball aim will pocket the ball. Of course, not every shot will go in center pocket, but they all go in - some go center pocket, some will go a little left of center and some a little right.
This is just one example of finding an aim line with absolutley no feel or guesswork or rooting around involved. The same method applies for determining a 3/4 aim or a 5/8 aim or whatever.
And if the system defines a shot that's between a 5/8 and a 3/4, then it still isn't guesswork. It either indicates that the aim is dead between those two references or a little thicker or thinner (either a perfect 16th or a little thicker or thinner).
A 16th aim difference is about a quarter of a tip. A 32nd aim difference is about an eight of a tip. Using portions of the tip or ferrule to fine tune 1/8 fractional aiming references is pretty simple and effective.