The­or­ies 1-4

Guide writ­ten by LE★Baldy & Snaeky. Con­tri­bu­tions from the Amaz­ing Com­munity.

This guide is cur­rently un­der­go­ing change. Keep in mind, strategies may change.

Feel free to use the gloss­ary as needed.

The­ory ba­sics #

Pub­lic­a­tions are equi­val­ent to prestiges for \(f(t)\) so don’t be afraid to use them. However, the best pub­lic­a­tion mul­ti­pli­ers vary from the­ory to the­ory and will change over time. If you are close to a mul­ti­plier you want, turn off auto­buyer and let \(\rho\) in­crease without buy­ing up­grades for a faster short-term in­crease be­fore the pub­lic­a­tion (turn on after you pub­lish). This is known and ref­er­enced as “cruis­ing”. Total \(τ\), found in the equa­tion or at the top of the screen, is a mul­ti­plic­at­ive com­bin­a­tion of all \(τ\) from each the­ory.

Don’t be afraid to skip get­ting all mile­stones to work on the next or a bet­ter the­ory.

Note: If you see # → [# → # → #] → # in the mile­stone route of a the­ory, this is the sec­tion that has an act­ive strategy tied to it.

Gradu­ation rout­ing #

Re­mem­ber to fol­low our rout­ing ad­vice from In­tro­duc­tion to Gradu­ation.

The­ory 1 (20σ / 5k) #

T1 Over­view #

In math­em­at­ics, a re­cur­rence re­la­tion is an equa­tion that re­lies on an ini­tial term and a pre­vi­ous term to change. We start with the cur­rent tick’s term, \(ρ_{n}\), and a con­stant add-on to ob­tain the value of the next tick, \(ρ_{n+1}\). This gives us an equa­tion equi­val­ent to \(ρ=at+con­stant\), with a chan­ging value \(a\) and a con­stant that is the ini­tial value of 1. Later when we add the \(c_{3}ρ_{n-1}^{0.2}\) term, this is now say­ing that we are now adding each tick the value of \(ρ\) from the pre­vi­ous tick ago with a con­stant \(c_{3}\) put to the power of \(0.2\). This is the same with the next term \(c_{4}ρ_{n-2}^{0.3}\), with the value of \(ρ\) two ticks ago and a mul­ti­plier \(c_4\) put to the power \(0.3\). When we mul­tiply the \(c_1c_2\) term by the term \(1+ln(ρ)/​100\) chan­ging the con­stant ad­di­tion to be­ing based on the value of \(ρ\) from the pre­vi­ous tick with the value of \(1+ln(ρ)/​100\). The fi­nal mile­stone up­grade raises the ex­po­nent of \(c_1\) from \(1.00\) to \(1.05\) to \(1.10\) to \(1.15\).

This the­ory also has its ad­jus­ted tick­speed cal­cu­lated by \(q_{1}*q_{2}\). This lengthens the nor­mal tick length of \(0.1/​sec\) to that value which speeds up the the­ory.

T1 for­mula #

Ini­tial

\[ρ_{n+1} = ρ_n + c_1c_2\]

First mile­stone

\[ρ_{n+1} = ρ_n + c_1c_2 + c_3ρ_{n-1}^{0.2}\]

Second mile­stone

\[ρ_{n+1} = ρ_n + c_1c_2 + c_3ρ_{n-1}^{0.2} + c_4ρ_{n-2}^{0.3}\]

Third mile­stone

\[ρ_{n+1} = ρ_n + c_1c_2 \left( 1+\frac{ln(ρ_n)}{100} \right) \\ + c_3ρ_{n-1}^{0.2} + c_4ρ_{n-2}^{0.3}\]

Fourth to Sixth mile­stone

\[ρ_{n+1} = ρ_n + c_1^{1.15}c_2 \left( 1+\frac{ln(ρ_n)}{100} \right) \\ + c_3ρ_{n-1}^{0.2} + c_4ρ_{n-2}^{0.3}\]

T1 strategy #

The pub­lic­a­tion mul­ti­plier has no op­timal fit, as it fluc­tu­ates a lot, but here is known: 4-6 to start; 3-4 between 1e100 and 1e150; the pub­lic­a­tion mul­ti­plier os­cil­lates between 2.5 and 5 past e150. Once you get your first mile­stone, you can turn off \(c_1\) and \(c_2\) un­til e150 act­ive strat.

The act­ive strat fol­lows but only works when you have all mile­stones past e150. T1 is the only the­ory where the re­cent value of \(ρ\) in­flu­ences the rate of change of \(ρ\) there­fore buy­ing a vari­able as soon as you can af­ford it will slow your pro­gress. Lategame, buy­ing up­grades im­me­di­ately will slow you more than the be­ne­fit of the up­grade be­cause \(c_3\) and \(c_4\) dom­in­ate. If the next level costs \(10ρ\) and you have \(11ρ\), buy­ing that level will re­duce \(ρ_{n+1}\) to \(1\). This re­duces your \(ρ_{n+1}\) by roughly a factor of \(10\). There are \(3\) terms that in­flu­ence the rate of change of \(ρ\), and all are af­fected by the pre­vi­ous state of \(ρ\). The act­ive strategy around this is known as T1Ra­tio. The val­ues in the chart found here are to be only used when you are past \(e150 τ\) and max mile­stones. They rep­res­ent how to pur­chase each vari­able based on the state of the the­ory at the time of pur­chase.

Note: If you are not do­ing the act­ive strat, then simply turn off \(c_1\) and \(c_2\) after mile­stone 1 (e25τ) and auto­buy rest un­til ee6k.

The video be­low is only good for early tau between 1e150 and 1e250.

T1 mile­stone route #

The­ory 2 (25σ / 6k) #

T2 Over­view #

This second the­ory is fo­cus­ing on de­riv­at­ives. De­riv­at­ives in math­em­at­ics are the rate of change of the func­tion they are the de­riv­at­ive of. For the case of \(q_1\) and \(q_2\), \(q_2\) is the de­riv­at­ive of \(q_1\). This fol­lows the power rule for de­riv­at­ives:

In sim­pler terms, it works sim­ilar to how \(x_i\) up­grades work for \(f(t)\) equa­tion with con­tinu­ous ad­di­tion of the pre­vi­ous \(termdt\) to the next \(x_{i+1}\) term, but with con­tinu­ous ad­di­tion of \(q_idt\) to the term above \(q_{i-1}\). These two val­ues of \(r_1\) and \(q_1\) are mul­ti­plied to pro­duce the de­riv­at­ive of \(ρ(t)\), shown by New­ton’s de­riv­at­ive nota­tion \(\dot{ρ}\). This would give the equa­tion of \(ρ\) to be \(ρ(t+dt)=ρ+\dot{ρ}*dt\). The other mile­stones be­sides more \(q\) and \(r\) de­riv­at­ives in­crease the ex­po­nent of \(q\) and \(r\) re­spect­ively. The reason why \(q\) and \(r\) de­riv­at­ives are more power­ful long-term than the ex­po­nents is that they take time to build up and even­tu­ally over­take and keep in­creas­ing \(q_1\) and \(r_1\) while the ex­po­nents have a never-chan­ging boost.

T2 for­mula #

Ini­tial

\[\dot{q_n}=q_{n+1}*dt\] for n=1

\[\dot{r_k}=r_{k+1}*dt\] for k=1

\[\dot{ρ}=q_1r_1\]

First and Second mile­stones

\[\dot{q_n}=q_{n+1}*dt\] for n=1, 2, 3

\[\dot{r_k}=r_{k+1}*dt\] for k=1

\[\dot{ρ}=q_1r_1\]

Third and Fourth mile­stones

\[\dot{q_n}=q_{n+1}*dt\] for n=1, 2, 3

\[\dot{r_k}=r_{k+1}*dt\] for k=1, 2, 3

\[\dot{ρ}=q_1r_1\]

Fifth to Sev­enth mile­stones

\[\dot{q_n}=q_{n+1}*dt\] for n=1, 2, 3

\[\dot{r_k}=r_{k+1}*dt\] for k=1, 2, 3

\[\dot{ρ}=q_1^{1.15}r_1\]

Eight to Tenth mile­stones

\[\dot{q_n}=q_{n+1}*dt\] for n=1, 2, 3

\[\dot{r_k}=r_{k+1}*dt\] for k=1, 2, 3

\[\dot{ρ}=q_1^{1.15}r_1^{1.15}\]

T2 strategy #

The op­timal mul­ti­plier is pretty high and is not known be­fore \(e30\). The the­ory sim will re­com­mend pub­lic­a­tion mul­ti­pli­ers be­low these val­ues, but the sim’s T2MS does not cur­rently have coast­ing. The mul­ti­pli­ers for act­ive play (which do use coast­ing) we know at the mo­ment are:

• \(e25\)-\(e100\) is \(1k\) to \(10k\)
  
• \(e100\)-\(e175\) is \(10k\)-\(100k\)

For both strategies the mile­stones are lis­ted in the or­der X>Y, where X and Y are the mile­stones as nu­mer­ic­ally ordered top to bot­tom in-game, are to be maxed in or­der from left to right.

Idle

For the idle strategy, you want to pri­or­it­ize buy­ing mile­stone levels of 1>2. If you have more than 4 mile­stones, you will pri­or­it­ize mile­stone 1>2>3>4. You will want to pub­lish at about 10-100 mul­ti­plier be­fore \(e75\) and about a \(1000\) mul­ti­plier after \(e75\), but lar­ger mul­ti­pli­ers are fine. If pos­sible, swap to mile­stones 3>4>1>2 at end be­fore pub­lish­ing for an ad­di­tional boost.

Act­ive

The goal of the act­ive strategy is to grow \(q_1\) and \(r_1\) as much as pos­sible while be­ing able to take ad­vant­age of the ex­po­nent mile­stones too, yeild­ing a large boost from that growth. The act­ive for T2 is on a 50-second cycle between two mile­stone sets: 10 seconds for ex­po­nent pri­or­ity (Mile­stones 3 and 4) and 40 seconds for de­riv­at­ive pri­or­ity (Mile­stones 1 and 2) . You will start a pub­lic­a­tion with ex­po­nent pri­or­ity as the cost of the vari­ables gained from mile­stones 1 and 2 are too large for you to get right away. When you can af­ford them, you will start the cycle. The full cycle is lis­ted be­low:

1-3 Mile­stones

3>4 (10s) → 1 (40s) → 3>4 (10s) → 2 (40s) →
re­peat → coast and pub­lish

4+ Mile­stones

3>4>1>2 (10s) → 1>2>3>4 (40s) →
3>4>1>2 (10s) → 2>1>3>4 (40s) →
re­peat → coast and pub­lish

Past \(e175\), the act­ive strat will be­come ex­po­nen­tially less ef­fect­ive. At \(e250\), you would start to idle T2 overnight only. Un­til you have over \(1e350\tau\) from the­ory 2, this is the best the­ory to run idle overnight.

When you get to The­ory 3 at ee7k, move on to push­ing The­ory 3 when act­ive and run­ning T2 overnight. The above is simply an op­tion if you rather not work on T3 now.

T2 mile­stone route #

The­ory 3 (30σ / 7k) #

T3 Over­view #

The basis of this the­ory and un­der­stand­ing how it works is based on mat­rix mul­ti­plic­a­tion. Be­low I have put a color-coded im­age to dis­play how mat­rix mul­ti­plic­a­tion works.

This gives the basis for why cer­tain up­grades are more power­ful than oth­ers. The ex­po­nents on \(b_1\), \(b_2\), and \(b_3\) all dir­ectly af­fect \(ρ_1\) pro­duc­tion which is used for \(\tau\). An ex­tra di­men­sion roughly gives \(50%\) more \(\tau\) pro­duc­tion as it adds an ex­tra term to the \(ρ_1\) pro­duc­tion.

T3 strategy #

The op­timal pub­lic­a­tion mul­ti­plier is about 2-3 without cruis­ing and 3-4 with cruis­ing. If you de­cide to play act­ively, there is a form of ex­po­nent swap­ping strat to be aware of. This is a dif­fi­cult strategy be­cause it re­quires you to no­tice when a cer­tain threshold hap­pens. It hap­pens when the fol­low­ing oc­curs:

\[c_{11}*b_{1}^{1.05\text{ or }1.1}<c_{12}*b_{2}^{1.05\text{ or }1.1}\]

When this hap­pens swap your ex­po­nents from \(b_1\) to \(b_2\) and you will get a little up­grade boost. It also al­lows for a slight push of \(ρ_2\) for up­grades to \(b_2\) and \(c_{12}\), but this is a lot less im­pact­ful and less no­tice­able. This strategy also works with \(b_3\) and \(c_{13}\) but is usu­ally not as com­mon.

If you de­cide to buy manu­ally, the fo­cus areas are buy­ing \(b_1\), \(b_2\), and \(b_3\) when their cost is e1 lower than \(c_{11}\), \(c_{12}\), and \(c_{13}\) re­spect­ively. These all dir­ectly boost the pro­duc­tion of \(ρ_1\) which is used for \(\tau\). After this, if you are do­ing the act­ive ex­po­nent swap­ping strategy de­scribed in the pre­vi­ous para­graph, your next fo­cus will be on \(c_{21}\), \(c_{22}\), and \(c_{23}\) as these boost \(b_2\) pro­duc­tion which in­creases the like­li­hood for the ex­po­nent swap to oc­cur. This leaves the \(c_{31}\), \(c_{32}\), and \(c_{33}\) up­grades at the low­est pri­or­ity. If you are not us­ing the ex­po­nent swap­ping strategy from the pre­vi­ous para­graph, then all the re­main­ing up­grades should be bought at equi­val­ent pri­or­ity.

At the end of any pub­lic­a­tion, around a 2-3 mul­ti­plier, you should turn off \(b_1\) and \(c_{31}\) as they cost \(ρ_1\). You will cruise un­til you get to a 3-4 mul­ti­plier. Pub­lish and turn back on \(ρ_1\) cost­ing vari­ables and re­peat.

Com­ment­ary

T3 mile­stone route #

The­ory 4 (35σ / 8k) #

The­ory 4 Over­view #

The­ory 4 is based on Poly­no­mi­als, which con­tain terms of the form \(x^a+x^b+x^c\) etc. In this case, in­stead of ‘x’ it’s ‘q’. The strategies for this the­ory are quite simple com­pared to the pre­vi­ous the­ory, es­pe­cially late game strategies.

The­ory 4 Equa­tion De­scrip­tion #

\(\dot{\rho} = c_1^{1.15}c_2 + c_3q + c_4q^2 + c_5q^3 + c_6q^4\)

\(\dot{q} = 8q_1q_2 / (1 + q)\)

The first line statest that the rate of change of rho is the sum of a bunch of poly­no­mial terms. We have a bunch of ‘c’ vari­ables mul­ti­plied by ‘q’. We can in­crease \(q\) by buy­ing \(q_1\) and \(q_2\) up­grades. Note that this is with all mile­stones. You’ll not have all of these at the be­gin­ning.

The second line is more unique. It says that $\dot{q}$ is pro­por­tional to the in­verse of \(q\) it­self! This means that the more \(q\) we have, the slower \(q\) grows, as $\dot{q}$ de­creases. This means that \(q_1\) and \(q_2\) are not as strong as they first ap­pear. However, we still want to buy them in gen­eral un­less stated oth­er­wise as slow growth is bet­ter than no growth.

For the more math­em­at­ic­ally ob­ser­v­ant reader, we may in­teg­rate the \(\dot{q}\) equa­tion and con­clude that \(q\) is pro­por­tional to the square root of time. This means that even though \(\dot{q}\) grows slower with in­creas­ing \(q\), there is the­or­et­ic­ally no fi­nite limit on the max­imum value of \(q\).

The­ory Vari­able De­scrip­tion #

Ap­prox­im­ate vari­able strengths on \(\dot\rho\) with all mile­stones are as fol­lows:

The­ory 4 strategy #

The strengths of each vari­able are as fol­lows:

Early game (be­fore 14k ft)

\(c_6\) > \(c_5\) > \(c_4\) > \(q_2\) > \(c_2\) > \(q_1\) > \(c_3\) > \(c_1\)

From 14k ft to mid-late game (about e350+ T4)

\(c_2\) > \(c_3\) > \(q_2\) > \(c_1\) > \(q_1\) > everything else

From e350+ T4 to end game

\(c_3\) > \(q_2\) > \(q_1\) > everything else

Idle

T4 is quite idle friendly com­pared to T3 and T1. Here are some simple idle strategies for T4:

Start to e25

Auto­buy $c_1$, $c_2$. DON’T buy $c_3$, $q_1$, $q_2$! The \(c_3q\) term is bad early on. Pub­lish at about 2.5-3 if pos­sible.

e25 to e175

Get the ‘Add the term’ mile­stones. Pri­or­it­ise these ones first un­til max­imum. Now we auto­buy $c_4$, $q_1$, $q_2$ ONLY. Best pub­lic­a­tion mul­ti­plier is about 6-7.

When we un­lock $c_5$ and $c_6$, we can add these to the auto­buy vari­ables. DON’T auto­buy $c_3$, $c_2$, $c_1$! Pri­or­it­ise the $\dot{q}$ mile­stones over the $c_1$ ex­po­nents. Try to pub­lish between 12-20. See the idle sec­tion of mile­stone or­der be­low.

e175 to en­dgame

Simply auto­buy $c_3$, $q_1$, $q_2$ ONLY. Buy 1 level of $c_1$ to start the the­ory. Pub­lish at about 4-5.

Semi-Idle

There’s no stra­tegic dif­fer­ence between semi-idle and idle for this the­ory. The main dif­fer­ence is with semi-idle, we would pub­lish more of­ten since we check the game more of­ten. We would­n’t over­shoot the op­timal mul­ti­plier as much.

Act­ive

T4 act­ive is more in­volved. However it is not as de­mand­ing as T3 or T1 act­ive.

Start to e75

Auto­buy $c_2$. DON’T buy $c_3$, $q_1$, $q_2$! The \(c_3q\) term is bad early on. Buy $c_1$ un­til its cost ex­ceed about 15% of $c_2$ cost. Pub­lish at about 2.5-3 if pos­sible. When we reach e25 rho, we get the $c_1$ ex­po­nent mile­stone (note the dif­fer­ence between this strategy and the idle strategy). With the $c_1$ ex­po­nent mile­stone, the $c_1{c}_2$ term re­mains the strongest term IF we can babysit and pub­lish of­ten (at about 2.5-3). The strategy re­mains the same oth­er­wise. Note that since we’re only buy­ing $c_1$ and $c_2$ (NO $c_3$, $c_4$, $c_5$, $c_6$, $q_1$, $q_2$!), all the ‘q’ re­lated mile­stones are use­less for now.

e75 to e175 OR 14k ft

Now here is where we can ap­ply some more ad­vanced strategies. Con­sider that the $c_1{c}_2$ term is strong early on, but falls off as the value of $q$ in­creases. Then we can con­clude that we can start with the same strategy as be­fore. But once we reach our pre­vi­ous pub­lic­a­tion point, we can switch to the fol­low­ing strategy:

1. Do the same strategy as be­fore un­til we reach our pre­vi­ous pub­lic­a­tion point.
2. Take point(s) out of the $c_1$ ex­po­nent mile­stones and un­lock all the terms (the first mile­stone). We should now have ac­cess to $c_6$.
3. Auto­buy $c_4$, $c_5$, $c_6$, $q_2$.
4. If you want to op­tim­ise a bit more, you can buy $q_1$ un­til its cost ex­ceed about 15% of $q_2$. Oth­er­wise it’s ok to also auto­buy $q_1$.
5. DO NOT auto­buy $c_1$, $c_2$, $c_3$.
6. Pub­lish at about 10-20. Once pub­lished, re­mem­ber to take out the mile­stone point and put it back into the $c_1$ ex­po­nent to re­peat step 1.

If done right, this strategy is sig­ni­fic­antly faster than the idle strategies above. The lo­gic with this strategy is the $c_4$, $c_5$, $c_6$ terms scale well with ‘q’. However we need enough rho to buy a lot of q. So in the be­gin­ning we buy only $c_1{c}_2$ as usual to ac­cu­mu­late enough rho so that we can buy $q_1{q}_2$ to stack q. Once we have enough q, the $c_4$, $c_5$, $c_6$ terms will outscale. Note that after ee14k ft, we will un­lock cer­tain up­grades that make $c_1{c}_2$ bet­ter again.

e175 OR 14k ft to ~e300 T4

We will do the ex­act same strategy as in the #start to e75 sec­tion above. This is be­cause $c_1{c}_2$ be­come really strong again and the $c_4{c}_5{c}_6$ terms take too long to outscale. Note that we still don’t buy $c_3$.

~e300 to en­dgame

At this point the $c_3$ term starts to be­come dom­in­ant. There­fore we will pri­or­it­ise buy­ing $c_3$, $q_1$, $q_2$. We will NOT buy any­thing else ex­cept 1 level of $c_1$ to start the the­ory. If you wish, you can buy $q_1$ at about 15% ra­tio to $q_2$ cost. It is also ok to auto­buy $q_1$. The $c_3$ term will re­main dom­in­ant un­til en­dgame.

T4 mile­stone route #

The­ory tier list (Pre-9k+) #

Be­fore you reach 9k, these are the re­com­men­ded val­ues for each the­ory. You may not hit the val­ues and have a dif­fer­ent dis­tri­bu­tion, but work on get­ting these the­or­ies up to these val­ues later. This list is in or­der of pri­or­ity.