The­or­ies 1-4

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

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 “coast­ing”.
Total $\tau$, found in the equa­tion or at the top of the screen, is a mul­ti­plic­at­ive
com­bin­a­tion of all $\tau$ from each the­ory.

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

Gradu­ation rout­ing #

Re­mem­ber to fol­low our rout­ing ad­vice from In­tro­duc­tion to Gradu­ation.

Class: gradu­ation_rout­ing;
last_row: false;

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, ${\rho }_n$, and a con­stant add-on to
ob­tain the value of the next tick, ${\rho }_{n+1}$. This gives us an equa­tion
equi­val­ent to $\rho =at+constant$, with a chan­ging value $a$ and a con­stant
that is the ini­tial value of 1. Later when we add the $c_3{\rho }_{n-1}^{0.2}$ term,
this is now say­ing that we are now adding each tick the value of $\rho$ 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 $c_4{\rho }_{n-2}^{0.3}$, with the value of $\rho$ two ticks
ago and a mul­ti­plier $c_4$ put to the power $0.3$. When we mul­tiply the
$c_1{c}_2$ term by the term $1+ln(\rho )/100$ chan­ging the con­stant ad­di­tion to
be­ing based on the value of $\rho$ from the pre­vi­ous tick with the value of
$1+ln(\rho )/100$. 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 $q_1\ast q_2$. 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 $\rho$
in­flu­ences the rate of change of $\rho$ 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 $10\rho$
and you have $11\rho$, buy­ing that level will re­duce ${\rho }_{n+1}$ to $1$. This re­duces your ${\rho }_{n+1}$ by roughly a factor of $10$.
There are $3$ terms that in­flu­ence the rate of change of $\rho$, and all are af­fected by the pre­vi­ous state of $\rho$. 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 $e150\tau$ 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 ($e25\tau$) and auto­buy rest un­til ee6k.

The video be­low is only good for early ${\tau }_1$ between $1e150$ and $1e250$.

T1 mile­stone route #

Class: mile­stone­_rout­ing;
last_row: false;

Class: mile­stone­_rout­ing;
last_row: false;

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 $term\text{*}dt$ to the next $x_{i+1}$ term, but with
con­tinu­ous ad­di­tion of $q_i\text{*}dt$ to the term above $q_{i-1}$.
These two val­ues of $r_1$ and $q_1$ are mul­ti­plied to pro­duce the de­riv­at­ive
of $\rho (t)$, shown by New­ton’s de­riv­at­ive nota­tion $\dot{\rho }$. This would give the
equa­tion of $\rho$ to be $\rho (t+dt)=\rho +\dot{\rho }\ast dt$. 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 the 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, yield­ing 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 #

Class: mile­stone­_rout­ing;
last_row: false;

Class: mile­stone­_rout­ing;
last_row: false;

The­ory 3 (30σ / 7k) #

T3 Over­view #

The basis of this the­ory and un­der­stand­ing how it works is based on
mat­rix mul­ti­plic­a­tion. The fol­low­ing color-cod­ing helps dis­plays
how mat­rix mul­ti­plic­a­tion works:

Ori­ginal Im­age (April 2021 - May 2025)

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 ${\rho }_1$ pro­duc­tion which is used for $\tau$. An ex­tra
di­men­sion roughly gives $50$ more $\tau$ pro­duc­tion as it adds an ex­tra term
to the ${\rho }_1$ 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 ${\rho }_2$ 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 ${\rho }_1$ 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 ${\rho }_1$. You will cruise un­til you get to a
3-4 mul­ti­plier. Pub­lish and turn back on ${\rho }_1$ cost­ing vari­ables and
re­peat.

Com­ment­ary

T3 mile­stone route #

Class: mile­stone­_rout­ing;
last_row: false;

Class: mile­stone­_rout­ing;
last_row: false;

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 #

$\dot{\rho }=c_1^{1.15}{c}_2+c_{3}q+c_4{q}^2+c_5{q}^3+c_6{q}^4$

$\dot{q}=8q_1{q}_2/(1+q)$

The first line states 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:

Class: break­down;

The­ory 4 strategy #

The strengths of each vari­able are as fol­lows:

Early game (be­fore 14k f(t)):

  $c_6$ > $c_5$ > $c_4$ > $q_2$ > $c_2$ > $q_1$ > $c_3$ > $c_1$

From 14k f(t) 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 $c_{3}q$ term is bad early on. Pub­lish at ~2.5-3 if pos­sible.

e25 to e175:

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

When $c_5$ and $c_6$ are un­locked, add $c_5$ and $c_6$ to the auto­buy vari­ables. DON’T auto­buy $c_3$, $c_2$, $c_1$! Pri­or­it­ize the $\dot{q}$ mile­stones over the $c_1$ ex­po­nents. Try to pub­lish between 12-20.

e175 to en­dgame:

Simply auto­buy $c_3$, $q_1$, $q_2$ ONLY. Buy 1 level of $c_1$ to start the the­ory. Pub­lish at ~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 $c_{3}q$ term is bad early on. Buy $c_1$ un­til its cost ex­ceeds 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 f(t):

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 op­tim­ize a bit more, you can buy $q_1$ un­til its cost ex­ceeds 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 ~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 that 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 e14k $f(t)$, we will un­lock cer­tain up­grades that make $c_1{c}_2$ bet­ter again.

e175 OR 14k f(t) 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 pri­or­it­ize 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 #

Class: mile­stone­_rout­ing;
last_row: false;

Class: mile­stone­_rout­ing;
last_row: false;

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.

Class: break­down;
last_row: false;