Function Layers

Upon accumulating 1000𝜓 from Supremacies, the first upgrade for function layers (Unlock $$ x_1$$) will be available for purchase.

Purchasing the first function layer unlocks Chapter 12: Honors and redefines the base equation to $$ f(t+dt) = f(t)\times e^{b x_i dt}$$, where $$ x_i$$is the largest function layer unlocked.

Function layers are available from $$ x_1$$to $$ x_8$$, and the cost of each upgrade is driven by an exponential model, which follows the formula $$ b\times 2^{a(i-1)}$$𝜓, where $$ i$$is the subscript of $$ x_i$$. The following values of a and b are used:

Mechanics
The value of each function layer increments each tick based on the following iterative recursive formula:

$$ \begin{align} x_i &\leftarrow x_i + x_{i-1} \cdot dt \\ x_0 &= x \end{align}$$

Using this equation, assuming that $$ x_n$$is the largest functional layer purchased, the iterative formula is given by the following:

$$ x_n \leftarrow \sum_{i=0}^n x_i (dt)^{n-i}$$