Knowledge compounds. Painting is good practice for architecture. Stand-up comedy is good training for stage magic.

The knowledge transference between math and physics might be as high as 75%. Unfortunately, the knowledge transference between two random fields tends to be small. Between math and drawing it might be as low as 1%. In my personal experience it's hard to find any pair of broadly applicable subjects that don't have at least 1% overlap. Usually the number will be between 1% and 75%.

Let's suppose mastering a new field of knowledge gives you a 5% discount on average on every subsequent field. Suppose it takes time for someone who knows nothing to master a new field. Then the amount of time it takes to master a new field is a function of how many fields you have already mastered .

How much time does it take to learn fields instead of just the field ?

This is a geometric series.

appears to converge.

converges for every positive transference rate. If we use 1% instead of 5% we just get .

What does this mean?

# The educational phase transition

Obviously, someone who has hit is not going to possess all of human knowledge. No matter how much you know it's still going to take you some minimum time to learn the dialectical quirks of, say, Hejazi Arabic.

What really means is you've hit a certain endgame. The process of learning has undergone a phase transition. All the broad conceptual machinery and widely-applicable facts are there. Picking up anything new is just a matter of plugging new data into preexisting sockets.

More interesting than "what happens at this phase transition" is the idea that "there is a phase transition" and we can reach it in finite time. Perhaps even within a human lifetime.

Much like stream entry, I suspect anyone who achieves this phase transition is better off keeping his/her mouth shut about it in polite society.

# An alternative model

The concept of a singularity in the above model relies on a double-positive feedback loop. It assumes "Each subject you know conveys a compounding 5% discount on learning each subsequent subject." If we tweak this assumption into "Each of study time conveys a compounding 5% discount on learning each subsequent subject" then never converges as .

However, this alternative model still breaks down at high . It just breaks down gradually. For example, at the exponential model predicts an absurd learning rate times that of a beginner. In the human world such a high rate of learning is indistinguishable from infinity. The model has broken down.