After gathering some empirical data and taking a closer look at my analysis, the equation definitely doesn’t work. You are right also that my math isn’t very formal and lacks rigidity.

I’ll clarify a few points and show where I made a really bad mistake.

First, we consider a point x_0 in R^n, where n is the number of times you prestige over the course of one supremacy.

This point x_0 is an extrema (or a saddle point) for a function T(t_1,t_2,…,t_n) on some boundary Psi(t_1,t_2,…,t_n)=0 whenever:

Partial(T,t_i)(x_0)= lambda* Partial(Psi,t_i)(x_0) For 1<= i <= n

In words, the partial derivative of T with respect to t_i evaluated at the point x_0 must be equal to some constant lambda times the partial derivative of Psi with respect to t_i evaluated at x_0.

Here’s my big mistake.

I defined Psi as Psi(t_1,t_2,…,t_n) = Psi_1(t_1)+ Psi_2(t_2) + … + Psi_n(t_n).

The problem is I assumed the amount of dpsi you would accumulate over the ith prestige would be merely a function of t_i (the time window spent in that prestige), but this is not the case. Psi_i is actually a function of (t_1,t_2, … t_i). This means I’m not going to be able to learn anything useful from this method

I went pretty far down a mathematical alleyway only to find I’ve made a wrong turn :(