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:

\[q=a*t^n ↔ q’=n*a*t^{n-1}\]

In sim­pler terms, it works sim­ilar to how \(x_i\) up­grades work for \(f(t)\) equa­tion with con­tinu­ous ad­di­tion of the pre­vi­ous \(term*dt\) to the next \(x_{i+1}\) term, but with con­tinu­ous ad­di­tion of \(q_i*dt\) 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 \(qdot\) is pro­por­tional to the in­verse of \(q\) it­self! This means that the more \(q\) we have, the slower \(q\) grows, as \(qdot\) 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 c1, c2. DON’T buy c3, q1, q2! 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 c4, q1, q2 ONLY. Best pub­lic­a­tion mul­ti­plier is about 6-7.

When we un­lock c5 and c6, we can add these to the auto­buy vari­ables. DON’T auto­buy c3, c2, c1! Pri­or­it­ise the qdot mile­stones over the c1 ex­po­nents. Try to pub­lish between 12-20. See the idle sec­tion of mile­stone or­der be­low.

e175 to en­dgame

Simply auto­buy c3, q1, q2 ONLY. Buy 1 level of c1 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 c2. DON’T buy c3, q1, q2! The \(c_3q\) term is bad early on. Buy c1 un­til its cost ex­ceed about 15% of c2 cost. Pub­lish at about 2.5-3 if pos­sible. When we reach e25 rho, we get the c1 ex­po­nent mile­stone (note the dif­fer­ence between this strategy and the idle strategy). With the c1 ex­po­nent mile­stone, the c1c2 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 c1 and c2 (NO c3, c4, c5, c6, q1, q2!), 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 c1c2 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 c1 ex­po­nent mile­stones and un­lock all the terms (the first mile­stone). We should now have ac­cess to c6.
3. Auto­buy c4, c5, c6, q2.
4. If you want to op­tim­ise a bit more, you can buy q1 un­til its cost ex­ceed about 15% of q2. Oth­er­wise it’s ok to also auto­buy q1.
5. DO NOT auto­buy c1, c2, c3.
6. Pub­lish at about 10-20. Once pub­lished, re­mem­ber to take out the mile­stone point and put it back into the c1 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 c4, c5, c6 terms scale well with ‘q’. However we need enough rho to buy a lot of q. So in the be­gin­ning we buy only c1c2 as usual to ac­cu­mu­late enough rho so that we can buy q1q2 to stack q. Once we have enough q, the c4, c5, c6 terms will outscale. Note that after ee14k ft, we will un­lock cer­tain up­grades that make c1c2 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 c1c2 be­come really strong again and the c4c5c6 terms take too long to outscale. Note that we still don’t buy c3.

~e300 to en­dgame

At this point the c3 term starts to be­come dom­in­ant. There­fore we will pri­or­it­ise buy­ing c3, q1, q2. We will NOT buy any­thing else ex­cept 1 level of c1 to start the the­ory. If you wish, you can buy q1 at about 15% ra­tio to q2 cost. It is also ok to auto­buy q1. The c3 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.