Let’s prove that is in­deed the over­push ra­tio. We’ll prove that mul­tiply­ing the pub mult by some factor in­creases the the­ory speed by , which is the same as prov­ing that the time to gain the same amount of is di­vided by when is mul­ti­plied by .

To prove this, we’ll start from equa­tion (3-3) of pa­cowoc’s guide:

We sub­sti­tute by :

Now we solve for :

Now let’s cal­cu­late where we re­place by :

We can see that, when mul­tiply­ing the pub mult by , the time to reach the same is di­vided by .

There­fore, is in­deed the over­push ra­tio of the the­ory.

From pa­cowoc’s model:

We can im­me­di­ately find the over­push ra­tio: .

We can also treat as a con­stant, as it is de­pend­ant on up­grades and does­n’t cor­res­pond to dir­ectly. There­fore, if we in­teg­rate:

Now the OP ra­tio is dir­ectly vis­ible.

For some the­or­ies, we’ll have to go to the log­ar­ithmic space. If we ap­ply lo­g10 to the pre­vi­ous equa­tion:

Now:

Cal­cu­lat­ing EF’s OP factor is com­par­at­ively harder due to it hav­ing mul­tiple cur­ren­cies. In gen­eral, to study the­or­ies with mul­tiple cur­ren­cies, we use lin­ear al­gebra in log­ar­ithmic space.

Late-game, in EF’s equa­tion’s square root, only the term is sig­ni­fic­ant:

Now, let’s ex­press the sys­tem in log­ar­ithmic form, to turn it into a lin­ear sys­tem:

We have:

For the pur­pose of this study, we can ig­nore vari­ables bought with .

To find EF’s OP factor, we need an ex­pres­sion of the form . Then, the OP factor will be given by .

Let’s now com­pute how the vari­ables we need scale with their cur­ren­cies. To do so, we use this model:

• An ex­po­nen­tial vari­able of power and cost scal­ing , bought with cur­rency is ap­prox­im­ated as .
• A step­wise vari­able of power , cycle length and cost scal­ing , bought with cur­rency is ap­prox­im­ated as .

There­fore we can cal­cu­late the fol­low­ing:

• is a step­wise vari­able of power 40, cycle length 10 and cost scal­ing : .
• is an ex­po­nen­tial vari­able of power 2 and cost scal­ing : .
• is a step­wise vari­able of power 2, cycle length 10 and cost scal­ing 200: .
• is an ex­po­nen­tial vari­able of power 1.12 and cost scal­ing 2: .
• is a step­wise vari­able of power 2, cycle length 10 and cost scal­ing 200: .
• is an ex­po­nen­tial vari­able of power 1.125 and cost scal­ing 2: .

Now we can sub­sti­tute in the sys­tem:

These ex­pres­sions will be heavy to ma­nip­u­late so let’s set it to:

Now if we add times equa­tion (2) and times equa­tion (3) to equa­tion (1) we get:

We can fi­nally ex­press EF’s OP factor:

For those who want to see what it looks like if we re­place those :

Fi­nally,

Since RZ also has mul­tiple cur­ren­cies, we’ll fol­low the same steps as with EF. For the pur­pose of this study, and val­ues can be ap­prox­im­ated as con­stants. The black hole has no ef­fect on the over­push factor as the op­timal to ac­tiv­ate the black hole is roughly pro­por­tional to the pub­lic­a­tion time.

Let’s ex­press this sys­tem on its log­ar­ithmic form:

We have:

We can also ig­nore vari­ables bought with .

Let’s now com­pute how the vari­ables we need scale with their cur­ren­cies.

• is a step­wise vari­able of power 2, cycle length 8, fol­low­ing a step­wise cost of power and cycle length 6. Its power is given by: .
• is an ex­po­nen­tial vari­able of power 2 and cost scal­ing 10. Its power is given by .
• is an ex­po­nen­tial vari­able of power 2 and cost scal­ing . Its power is given by .

Now we can sub­sti­tute in the sys­tem: