This type of up­grade has an identical equa­tion to the equa­tion men­tioned above, just with dif­fer­ent con­stants in the places of and . It can be rep­res­en­ted as the ex­pres­sion be­low:

In which and are pre-de­term­ined con­stants.

▶︎ Lemma 1.1 Permalink: lemma-11#

▶︎ Proof Permalink: proof#

▶︎ Lemma 1.2 Permalink: lemma-12#

▶︎ Proof Permalink: proof-1#

This type of vari­able fol­lows a rather un­ortho­dox path of pro­gres­sion, an ex­plan­a­tion of its mech­anic is ex­plained here. Let’s sup­pose there ex­ist a Step­wise vari­able that has the fol­low­ing prop­er­ties:

1. A cycle of spans levels.
2. On the first cycle of , it in­creases by per level.
3. On each cycle, the amount of in­crease per level gets mul­ti­plied by .

▶︎ Lemma 1.3 Permalink: lemma-13#

▶︎ Proof Permalink: proof-2#

▶︎ Lemma 1.4 Permalink: lemma-14#

▶︎ Proof Permalink: proof-3#

For any pur­chas­able vari­able , there ex­ists so that the value of and the peak value of its cur­rency meet the ex­pres­sion be­low:

▶︎ Lemma 2.1 Permalink: lemma-21#

▶︎ Proof Permalink: proof-4#

We can de­term­ine the asymp­totic nota­tion of a vari­able of this kind by solv­ing a first or­der ODE, be­low is an ex­ample with the vari­able in T4:

This is a clas­sic ex­ample of a self-in­flu­enced vari­able, for­tu­nately, the ODE is sep­ar­able:

In­teg­rate both sides,

As ap­proaches in­fin­ity, we can see that ap­proaches . We can then do the asymp­totic ana­lysis on the en­tire sys­tem nor­mally from here.

In more com­plic­ated cases such as the sys­tems that con­tain mul­tiple powers of in the growth of a single vari­able, the dif­fer­en­tial equa­tion that cor­res­ponds to the main equa­tion of­ten fails to have a closed form solu­tion for us to ana­lyze. Com­mon meth­ods to ana­lyze such a sys­tem are:

1. De­term­ine the strongest term by ex­per­i­ment and use the pre­vi­ous two meth­ods (which is how we ana­lyze T6, which has an in­tric­ate main equa­tion).
2. Use Nu­mer­ical Meth­ods (which the The­ory Sim­u­lator in­cor­por­ates).
3. Re­verse en­gin­eer the ex­po­nent of of a spe­cific vari­able from the fi­nal De­cay Factor ex­per­i­ment­ally.

If we man­age to get everything sor­ted, we will get a power­ful con­clu­sion:

For any the­ory sys­tem in which Lemma 2.1 can ap­ply fully on or can be ana­lyzed asymp­tot­ic­ally by other means, there ex­ists so that:

For any the­ory sys­tem that fits the ex­pres­sion be­low asymp­tot­ic­ally

the long-term mean Pub­lic­a­tion Mul­ti­plier of when is:

This is a sep­ar­able dif­fer­en­tial equa­tion, after some re­arrange­ment we have:

In­teg­rate both sides,

Since is a con­stant in a pub­lic­a­tion,

Eval­u­ate both In­teg­rals,

In an En­dgame situ­ation, , there­fore, is ig­nor­able.

We solve the equa­tion to get:

The goal of ours is to find the Pub­lic­a­tion Mul­ti­plier that gives the max­imum speed meas­ured in a \log­ar­ithmic scale, which can be ex­pressed with the equa­tion be­low:

We sub­sti­tute (3-3) into (3-4) and get:

In the En­dgame, is ig­nor­able:

We use dif­fer­en­ti­ation to find the max­imum of the ex­pres­sion above:

Since ,

Thus,

The pub­lic­a­tion mul­ti­plier of the the­ory is:

Since , we can take the nat­ural \log­ar­ithm on both sides: