On the Middle-Term Mod­el­ing of En­dgame The­or­ies and the Op­timal Pub­lic­a­tion Mul­ti­plier

Guide writ­ten by pa­cowoc. Con­tri­bu­tions from the Amaz­ing Com­munity.

Feel free to use the gloss­ary as needed.

In­tro­duc­tion #

The game fea­tures 8 the­or­ies each with its dis­tinct char­ac­ter­ist­ics and means of be­ha­vior. Al­though they seem wildly dif­fer­ent from each other visu­ally and nu­mer­ic­ally short-term, the gen­eral long-term concept of pur­chas­ing vari­ables of ex­po­nen­tially-higher prices to aid the main cur­rency’s near-ex­po­nen­tial pro­gress re­mains con­stant des­pite the dif­fer­ence of how they ac­tu­ally af­fect the growth of the the­or­ies they be­long to. There­fore, we can use a single model de­duced from the Cost and the Ef­fect of all the vari­ables used in a the­ory to ex­plain the long-term and middle-term be­ha­vior of a the­ory, and make mean­ing­ful con­clu­sions from the ob­served prop­er­ties of the model about the ori­ginal the­or­ies.

Re­quired Back­ground Know­ledge #

1. High school level math­em­at­ics
2. Asymp­totic Nota­tions
3. The Concept of ODEs

How Vari­able Up­grades Work in the Long Term? #

Al­most every up­grade in the game fol­lows a Cost-Level re­la­tion of: \[Cost = ab^{Level}\qquad (1-1)\] In which \(a>0\) and \(b>1\) are pre-de­term­ined con­stants. On the other hand, up­grades have two ways to de­term­ine the value of the vari­able:

Dir­ect Mul­ti­plic­a­tion: #

This type of up­grade has an identical equa­tion to the equa­tion men­tioned above, just with dif­fer­ent con­stants in the places of \(a\) and \(b\). It can be rep­res­en­ted as the ex­pres­sion be­low: \[Value = cd^{Level}\qquad (1-2)\] In which \(c>0\) and \(d>1\) are pre-de­term­ined con­stants.

Lemma 1.1 #

If we al­ways buy the \((L\in\mathbb{N})\)-th and latest up­grade of a “Dir­ect Mul­ti­plic­a­tion” vari­able \(u\), which in­dic­ates \(\rho = Cost(u,L)\). Then, the value of \(u\) and the peak value of its cur­rency \(\rho\) at the pur­chase fol­lows a re­la­tion de­scribed by \(u = K\rho^{r}\), in which \(K\in\mathbb{R}^+\) and \(r\in\mathbb{R}^+\), in which \(r\) only de­pends on the ex­po­nent part of the Cost and Value equa­tions.

Proof #

From (1-1) we have: \[\rho = ab^{L}\] Since \(a \neq 0\), \[\frac{\rho}{a} = b^{L}\] Since \(b > 0\), log­ar­ithm with a base of \(b\) is defined, \[lo­g_b(\frac{\rho}{a}) = L\] Thus, \[lo­g_b(\rho) - lo­g_b(a) = L\qquad (1-3)\] From (1-2) and (1-3) we have: \[u = cd^{lo­g_b(\rho) - lo­g_b(a)}\] \[= \frac{c}{d^{lo­g_b(a)}}d^{lo­g_b(\rho)}\] \[= \frac{c}{d^{\frac{lo­g_d(a)}{lo­g_d(b)}}}d^{\frac{lo­g_d(\rho)}{lo­g_d(b)}}\] \[= \frac{c}{(d^{lo­g_d(a)})^{\frac{1}{lo­g_d(b)}}}(d^{lo­g_d(\rho)})^{\frac{1}{lo­g_d(b)}}\] \[= \frac{c}{a^{lo­g_b(d)}}\rho^{lo­g_b(d)}\] We can see that \(r = lo­g_b(d)\) only de­pends on b and d.

Since \(b>1\) and \(d>1\), \(lo­g_b(d)>0\).

Since \(a>0\) and \(c>0\), \(\frac{c}{a^{lo­g_b(d)}}>0\).

There­fore, there ex­ists a pair of \(K\) and \(r\) that fits the con­di­tions.

Lemma 1.2 #

For a “Dir­ect Mul­ti­plic­a­tion” \(u\), there ex­ists \(r \in \mathbb{R}^+\) so that the value of \(u\) and the peak value of its cur­rency \(\rho\) meet the ex­pres­sion be­low: \[u(\rho) \in {\Theta(\rho^r)}\]

Proof #

Without loss of gen­er­al­ity, let’s sup­pose that we have pur­chased the \((L>0)\)-th up­grade of \(u\) and we are yet to pur­chase the\((L+1)\)-th up­grade. Thus, we have: \[ab^{L} \le \rho < ab^{L+1}\] We can see the up­per bound as an ex­pres­sion of a vir­tual vari­able \(u'\), in which: \[a' = ab\quad c'= c\] \[b' = b \quad d'= d\] Ac­cord­ing to Lemma 1.1, \(u\) and \(u'\) share the same ex­po­nent \(r\), let’s name the \(K\) for the two vari­ables \(K\) and \(K'\) re­spect­ively. Since \(b>1\), \(K'< K\), from Lemma 1.1 we have: \[K'\rho^r \le u \le K\rho^r\] Which is the defin­i­tion of: \[u(\rho) \in \Theta(\rho^r)\]

Step-wise: #

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

1. A cycle of \(v\) spans \(m>0\) levels.
2. On the first cycle of \(v\), it in­creases by \(p>0\) per level.
3. On each cycle, the amount of in­crease per level gets mul­ti­plied by \(q>1\).

Lemma 1.3 #

For each “Step­wise” vari­able \(v\), there ex­ist \((c_1, c_2) \in \mathbb{R}^+\) and \(d \in \mathbb{R}^+\) so that for any level \((L \in \mathbb{N} \ge m)\), the in­equal­ity be­low holds: \[c_1d^L \le v_L \le c_2d^L\]

Proof #

Without loss of gen­er­al­ity, let’s con­sider the \((k+1 \ge 2)\)-th cycle of \(v\), there­fore, the value of \(v\) after the \((0 < l \in \mathbb{N}\ \le m\))-th up­grade in the cycle is \(v_{mk+l}\): From the defin­i­tion of a Step­wise vari­able we have: \[v_{mk+l} = \frac{1}{q-1}p((m+l(q-1))q^k-m)\qquad (1-4)\]

Since \(m\)>0,from (1-4) we have: \[\frac{1}{q-1}p((m+l(q-1))q^k-m) < \frac{1}{q-1}p((m+l(q-1))q^k)=\frac{1}{q-1}mpq^k(1+\frac{l}{m}(q-1))\qquad (1-5)\] Since \(0 \le \frac{l}{m} \le 1\) and \(q>1\), \[\frac{1}{q-1}mpq^k(1+\frac{l}{m}(q-1)) < \frac{1}{q-1}mpq^k(1+(q-1)) = \frac{1}{q-1}mpq^(k+1) < \frac{1}{q-1}mpqq^{\frac{mk+l}{m}}\] \[= \frac{1}{q-1}mpq(q^\frac{1}{m})^{mk+l}\] There­fore, \[v_{mk+l} < mp\frac{q}{q-1}(q^\frac{1}{m})^{mk+l}\qquad (1-6)\]

Since \(k>1\), \(q^{k-1}>1\), from (1-4) we have: \[\frac{1}{q-1}p((m+l(q-1))q^k-m) > \frac{1}{q-1}p((m+l(q-1))q^k-q^{k-1}m)\] \[= \frac{1}{q-1}p(mq^k-mq^{k-1}+l(q-1)q^k)\] \[= \frac{1}{q-1}p(m(q-1)q^{k-1}+l(q-1)q^k)\] \[= mpq^{k-1}(1+\frac{l}{m}q)\qquad (1-7)\] Since \(0 \le \frac{l}{m} \le 1\) and \(q>-1\), we can ap­ply Bernoul­li’s in­equal­ity to get: \[1+\frac{l}{m}q \ge (1+q)^{\frac{l}{m}}\] \[mpq^{k-1}(1+\frac{l}{m}q) \ge mpq^{k-1}(1+q)^{\frac{l}{m}}>mpq^{k-1}q^{\frac{l}{m}}\] \[= \frac{mp}{q}(q^{\frac{1}{m}})^{mk+l}\] There­fore, \[v_{mk+l} > \frac{mp}{q}(q^{\frac{1}{m}})^{mk+l}\qquad (1-8)\]

From (1-6) and (1-8) we have: \[\frac{mp}{q}(q^{\frac{1}{m}})^{mk+l} < v_{mk+l} < mp\frac{q}{q-1}(q^\frac{1}{m})^{mk+l}\] Since \(mk+l = L\) in this case, \[\frac{mp}{q}(q^{\frac{1}{m}})^L < v_L < mp\frac{q}{q-1}(q^\frac{1}{m})^L\] Ap­par­ently, \((\frac{mp}{q},\frac{mpq}{q-1}) \in \mathbb{R}^+\) and \(q^\frac{1}{m} \in \mathbb{R}^+\)

There­fore, there ex­ist \((c_1,c_2)\) and \(d\) that fit the con­di­tions.

Lemma 1.4 #

For a “Step-wise” vari­able \(v\), there ex­ists \(r \in \mathbb{R}^+\) so that the value of \(v\) and the peak value of its cur­rency \(\rho\) meet the ex­pres­sion be­low: \[v(\rho) \in {\Theta(\rho^r)}\]

Proof #

From Lemma 1.3 we have: \[c_1d^L \le v_L \le c_2d^L\] We can see the up­per and lower bound as two vir­tual vari­ables, let’s call them \(v_1\) and \(v_2\). \[v_1 \le v \le v_2\qquad (1-9)\] Since \(v_1\) and \(v_2\) are both “Dir­ect Mul­ti­plic­a­tion” vari­ables, we can ap­ply Lemma 1.2 on \(v_1\) and \(v_2\) and get: \[v_1 \in \Theta(\rho^r)\] \[v_2 \in \Theta(\rho^r)\qquad\]
\[(1-10)\] In which \(r \in \mathbb{R}^+\). From (1-9), (1-10) and the defin­i­tion men­tioned in , we have: \[v(\rho) \in \Theta(\rho^r)\]

From Lemma 1.2 and Lemma 1.4 we can ob­tain an im­port­ant con­clu­sion:

Lemma 1.5 #

For any pur­chas­able vari­able \(v\), there ex­ists \(r \in \mathbb{R}^+\) so that the value of \(v\) and the peak value of its cur­rency \(\rho\) meet the ex­pres­sion be­low: \[v(\rho) \in {\Theta(\rho^r)}\]

How Vari­ables Propag­ate to­wards the Main Vari­able #

The­or­ies all have one or more cur­ren­cies, they are used to pur­chase vari­ables or in­ter­act with other cur­ren­cies. One shared fea­ture of the the­or­ies is that the cur­ren­cies they con­tain are either de­rived from a set of other vari­ables and cur­ren­cies via arith­metic op­er­a­tions, or that there is some key fea­tures in a the­ory that makes a few un-pur­chas­able vari­ables prac­tic­ally con­stant in the long term. (For ex­ample, the x,y and z in T8 due to Sol­arswap). There­fore, we can get the re­la­tion of a cer­tain cur­rency to the main cur­rency in the long term by factor­ing in the cur­ren­cies and vari­ables that con­trib­ute in the growth of that spe­cific cur­rency. We can gen­er­al­ize the growth of all but a few de­rived vari­ables or cur­ren­cies into. \[\frac{da_n}{dt} = f(a_1,a_2...,\rho)\] In which \(\rho_i\) are any vari­able, if the right side of the equa­tion in­volves the vari­able it­self, the vari­able is called “self-in­flu­enced”.

Nor­mal Cases #

Lemma 2.1 #

For any De­rived Vari­able \(a\) with uni­form \(t\) ex­po­nents and is neither cyc­lic defined nor self-in­flu­enced, there ex­ists \((x,y) \in \mathbb{R}^+\) so that: \[a(\rho,t) \in \Theta(\rho^xt^y)\]

Proof #

Since all the \(t\) ex­po­nents of the con­trib­ut­ing vari­ables are identical as a pre­requis­ite of this Lemma, we only need to dis­cuss the ex­po­nents of the main cur­rency to ana­lyze the de­rived vari­able asymp­tot­ic­ally. Be­cause \(\frac{∂a}{∂t}\) is the res­ult of a series of Ad­di­tion and Mul­ti­plic­a­tion of the vari­ables, we can use the arithemic laws of poly­no­mial asymp­totic nota­tions.

1. Mul­ti­plic­a­tion: \(\Theta(\rho^{r_1}) \cdot \Theta(\rho^{r_2}) → \Theta(\rho^{(r_1+r_2)})\)
2. Ad­di­tion: \(\Theta(\rho^{r_1}) + \Theta(\rho^{r_2}) → \Theta(\rho^{max(r_1,r_2)})\)

From Lemma 1.5 and the clos­ure un­der ad­di­tion of \(\mathbb{R}^+\), there will al­ways ex­ist a dom­in­ant ex­po­nent \(x \in \mathbb{R}^+\) so that: \[\frac{∂ a(\rho,t)}{∂ t} \in \Theta({\rho^x}t^y)\] In­teg­rate both sides with re­spect to t, \[a = \Theta(\rho^xt^y)t+C\] There­fore, \[a \in \Theta(\rho^xt^{y+1})\] Since \(y \in \mathbb{R}^+\), \((y+1) \in \mathbb{R}^+\).

Cases with Self-In­ter­fer­ence #

We can de­term­ine the asymp­totic nota­tion of a vari­able of this kind by solv­ing a first or­der ODE, be­low is an ex­ample with the \(q\) vari­able in T4: \[\frac{dq}{dt} = \frac{8q_1q_2}{q+1}\] This is a clas­sic ex­ample of a self-in­flu­enced vari­able, for­tu­nately, the ODE is sep­ar­able: \[(q+1)dq = 8q_1q_2dt\] In­teg­rate both sides, \[\frac{q^2}{2}+q = 8q_1q_2t + C\] As \(\rho\) ap­proaches in­fin­ity, we can see that \(q\) ap­proaches \(\sqrt{16q_1q_2t}\). We can then do the asymp­totic ana­lysis on the en­tire sys­tem nor­mally from here.

More Com­plic­ated Cases #

In more com­plic­ated cases such as the sys­tems that con­tain mul­tiple powers of \(t\) in the growth of a single vari­able, the dif­fer­en­tial equa­tion that cor­res­pounds 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 \(\rho\) of a spe­cific vari­able from the fi­nal De­cay Factor ex­per­i­ment­ally.

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

Lemma 2.2 #

For any the­ory sys­tem in which Lemma 2.1 can ap­ply fully on or can be ana­lyzed asymp­tot­ic­ally by other means, there ex­ists \((x,y) \in \mathbb{R}^+\) so that: \[\frac{d\rho}{dt} \in \Theta(\rho^xt^y)\]

Fi­nally, the Model #

Here, we use the concept from Lemma 2.2, and factor in the mech­anic of pub­lic­a­tion. Since all the­or­ies have the pub­lic­a­tion mul­ti­plier in the form of: \[K\rho^{r}_{pub}\qquad (3-1)\] In which \(K,r \in \mathbb{R}^+\), \(\rho_{pub}\) is the value of \(\rho\) at the last pub­lic­a­tion Since the pub­lic­a­tion mul­ti­plier is dir­ectly mul­ti­plied on the \(\rho\) gain of the the­ory, we can con­struct a model us­ing Lemma 2.2 and (3-1): \[\frac{d\rho}{dt} = K\rho^{x}t^{y}\rho^{z}_{pub}\qquad (3-2)\] In which \(K,x,y,z \in \mathbb{R}^+\) are con­stants.

How to De­duce the Op­timal Pub­lic­a­tion Mul­ti­plier in the En­dgame of a The­ory From the Model #

The­orem 1 #

For any the­ory sys­tem \(T\) that fits the ex­pres­sion be­low asymp­tot­ic­ally \[\frac{d\rho}{dt}=K\rho^{x}t^{y}\rho^{z}_{pub}\] the long-term mean Pub­lic­a­tion Mul­ti­plier of \( T\) when \(\rho → ∞\)is: \[e^{\frac{z(y+1)}{1-x}}\]

Proof #

\[\frac{d\rho}{dt}=K\rho^{x}t^{y}\rho^{z}_{pub}\] This is a sep­ar­able dif­fer­en­tial equa­tion, after some re­arrange­ment we have: \[\rho^{-x}d\rho = Kt^{y}\rho^{z}_{pub}dt\] In­teg­rate both sides, \[\int\rho^{-x}d\rho = \int Kt^{y}\rho^{z}_{pub}dt\] Since \(Kt^{y}\rho^{z}_{pub}\) is a con­stant in a pub­lic­a­tion, \[\int\rho^{-x}d\rho = K\rho^{z}_{pub}\int t^{y}dt\] Eval­u­ate both In­teg­rals, \[\frac{\rho^{1-x}}{1-x} = \frac{K\rho^{z}_{pub}}{z+1}t^{y+1} + C\] In an En­dgame situ­ation, \(t → ∞\), there­fore, \(C\) is ig­nor­able. \[\frac{\rho^{1-x}}{1-x} = K'\rho^{z}_{pub}t^{y+1}\] We solve the equa­tion to get: \[\rho = (K'(1-x)\rho^{z}_{pub}t^{y+1})^{\frac{1}{1-x}}\qquad (3-3)\]

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: \[Speed ∝ \frac{ln(\rho) - ln(\rho_{pub})}{t}\qquad (3-4)\] We sub­sti­tute (3-3) into (3-4) and get: \[\frac{ln(\rho) - ln(\rho_{pub})}{t} = \frac{ln((K'(1-x)\rho^{z}_{pub}t^{y+1})^{\frac{1}{1-x}}) - ln(\rho_{pub})}{t}\] \[ = \frac{\frac{ln(K'(1-x))+ln(\rho^{z}_{pub})+ln(t^{y+1})}{1-x} - ln(\rho_{pub})}{t}\qquad (3-5)\] In the En­dgame, \(ln(K'(1-x))\) is ig­nor­able: \[\frac{\frac{ln(K'(1-x))+ln(\rho^{z}_{pub})+ln(t^{y+1})}{1-x} - ln(\rho_{pub})}{t} \simeq \frac{\frac{ln(\rho^{z}_{pub})+ln(t^{y+1})}{1-x} - ln(\rho_{pub})}{t}\] \[ = \frac{(\frac{z}{1-x} - 1)ln\rho_{pub} + \frac{y+1}{1-x}ln(t)}{t}\] \[ = \frac{x+z-1}{1-x}ln\rho_{pub}\cdot\frac{1}{t} + \frac{y+1}{1-x}\cdot\frac{ln(t)}{t}\qquad (3-6)\] We use dif­fer­en­ti­ation to find the max­imum of the ex­pres­sion above: \[\frac{d}{dt}(\frac{x+z-1}{1-x}ln\rho_{pub}\cdot\frac{1}{t} + \frac{y+1}{1-x}\cdot\frac{ln(t)}{t}) = 0\] \[\frac{x+z-1}{1-x}ln\rho_{pub}\cdot\frac{d}{dt}\frac{1}{t} + \frac{y+1}{1-x}\cdot\frac{d}{dt}\frac{ln(t)}{t} = 0\] \[\frac{x+z-1}{1-x}ln\rho_{pub}\cdot(-\frac{1}{t^2}) + \frac{y+1}{1-x}\cdot\frac{d}{dt}\frac{\frac{1}{t}\cdot t-ln(t)}{t^2} = 0\] \[\frac{y+1}{1-x}\frac{1-ln(t)}{t^2} = \frac{x+z-1}{1-x}ln\rho_{pub}\frac{1}{t^2}\] Since \(t>0, x \neq 1\), \[(y+1)(1-ln(t)) = (x+z-1)ln\rho_{pub}\] Thus, \[ln(t) = 1-\frac{x+z-1}{y+1}ln\rho_{pub}\qquad (3-7)\] The pub­lic­a­tion mul­ti­plier of the the­ory is: \[r = (\frac{\rho}{\rho_{pub}})^z\] Since \(r>0\), we can take the nat­ural log­ar­ithm on both sides: \[ln(r) = z(ln(\rho)-ln(\rho_{pub}))\] \[ln(r) = zt(\frac{x+z-1}{1-x}ln\rho_{pub}\cdot\frac{1}{t} + \frac{y+1}{1-x}\cdot\frac{ln(t)}{t})\] \[= \frac{x+z-1}{1-x}ln\rho_{pub} + \frac{y+1}{1-x}ln(t)\qquad (3-8)\] Sub­sti­tute (3-7) into (3-8), \[ln(r)=\] \[z(\frac{x+z-1}{1-x}ln\rho_{pub}+\frac{y+1}{1-x}(1-\frac{x+z-1}{y+1}ln\rho_{pub}))\] \[ = z(\frac{(x+z-1)ln\rho_{pub} + (y+1)(1-\frac{x+z-1}{y+1}ln\rho_{pub})}{1-x})\] \[ = z(\frac{(x+z-1)ln\rho_{pub} + (y+1) - (x+z-1)ln\rho_{pub}}{1-x})\] \[ = \frac{z(y+1)}{1-x}\qquad (3-9)\] From (3-9), we get: \[r = e^{\frac{z(y+1)}{1-x}}\] Q.E.D.

The Ob­ser­va­tions We Can Get from The­orem 1 #

1. The more time-sens­it­ive the the­ory is, which is re­flec­ted on a higher \(y\),the higher the pub­lic­a­tion mul­ti­plier will be.
2. The lower the in­tra-pub­lic­a­tion de­cay, which is re­flec­ted on a higher \(x\), the longer the pub­lic­a­tion will be. If the in­tra-pub­lic­a­tion de­cay is neg­at­ive, which means the the­ory is di­ver­ging without any aid from pub­lic­a­tion and \(x>1\), the pub­lic­a­tion mul­ti­plier will fall be­low 1, mak­ing pub­lic­a­tion un-eco­nom­ical.
3. In a bal­anced the­ory that is not sens­it­ive to time (All main the­or­ies ex­cept T2,4,6), the pub­lic­a­tion mul­ti­plier will be slightly lower than \(e\), the more severe the de­cay is, the lower it goes.

Ex­amples of an Ana­lysis #

This sec­tion will use the first two the­or­ies to ex­plain how you com­pute the op­timal pub­lic­a­tion mul­ti­plier of a the­ory solely from the in­form­a­tion about a the­ory sys­tem.

The­ory 1 #

The main equa­tion in the En­dgame is: \[\frac{d\rho}{dt} = q_1q_2(c_1^{1.15}c_2 + c_3\rho^{0.2} + c_4\rho^{0.3})\] Us­ing Lemma 1.5, we can get: \[q_1, c_1 \in \Theta(\rho^{0.1})\] \[q_2, c_2 \in \Theta(\rho^{log2})\] \[c_3 \in \Theta(\rho^{\frac{2}{9}})\] \[c_4 \in \Theta(\rho^{\frac{1}{8}})\] Us­ing Lemma 2.2, we can see that the dom­in­ant term in the par­en­theses on the right is \(c_4\rho^{0.3}\), and the pub­lic­a­tion mul­ti­plier equa­tion of it is \(K\rho^{0.164}_{pub}\) \[\frac{d\rho}{dt} \in \Theta(\rho^{0.3+0.125+0.1+log2}t^0\rho^{0.164}_{pub})\] Ther­fore, \(x = 0.525+log2, y = 0, z = 0.164\)

Thus, from The­orem 1 we get: \[r = e^{\frac{0.164}{0.475-log2}} \simeq 2.5669\] This is very close to the geo­met­ric mean of the res­ults of The­ory Sim­u­lator from e1000 to e1500.

The­ory 2 #

The main equa­tion is: \[\frac{d\rho}{dt} = (q_1r_1)^{1.15}\] This the­ory fea­tures four lay­ers of non-main equa­tions, all in the form of: \[q_{n-1} = c_n­q_n\] or \[r_{n-1} = d_n­r_n\] In which \(c_n\) and \(d_n\) are pur­chas­able vari­ables. Us­ing Lemma 1.5, we can get: \[c_1, d_1 \in \Theta(\rho^{0.1})\] \[c_2, d_2 \in \Theta(\rho^{0.1})\] \[c_3, d_3 \in \Theta(\rho^{\frac{log2}{10­log3}})\] \[c_4, d_4 \in \Theta(\rho^{0.05})\] Us­ing Lemma 2.2, we can ana­lyze all the lay­ers and factor in the pub­lic­a­tion mul­ti­plier equa­tion of \(K\rho^{0.198}\)and get: \[\frac{d\rho}{dt} \in \Theta(\rho^{1.15*2*(0.1+0.1+\frac{log2}{10­log3}+0.05)}t^{9.2}\rho^{0.198}_{pub})\] There­fore, \(x \simeq 0.72011 , y = 9.2, z = 0.198\) Thus, from The­orem 1 we get: \[r \simeq e^{\frac{0.198*10.2}{0.27989}} \simeq 1360.61\] Which is close to the res­ults of The­ory Sim­u­lator without T2MC from e1000 to e1500. This be­ha­vior is ex­pec­ted since Coast­ing – The main concept of T2MC is not con­sidered at all in this ana­lysis.

Cred­its #

(The people be­low are not sor­ted)

Thanks the fol­low­ing people for their re­search con­tri­bu­tion! #

1. Ghost (Playsprout)
2. Spiral (a_spir­al­ist)
3. Megam­inx

Thanks the fol­low­ing people for their feed­backs! #

1. Mr. Op­timal (NGZ)
2. Speedrun­ner (Jooo)
3. scide (scbose)
4. Ghost (Playsprout)
5. Helper (Axiss)
6. nubest

Thanks the en­tire com­munity for sup­port­ing my on the way of re­search­ing and cre­at­ing such a won­der­ful en­vir­on­ment!