Custom Theories

Guide written by Playspout and 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.

Custom Theory Basics #

Custom theories are theories made by players in the community. As of April 1st, 2024, there are 6 official custom theories that contribute up to e600 $\tau$ per theory; Weierstrass Sine Product made by Xelaroc (WSP), Sequential Limits by Ellipsis (SL), Euler’s Formula by Peanut, Snaeky, and XLII (EF), Convergents to Square Root 2 (CSR2/CS2) by Solarion, Fractional Integration (FI) by Gen and Snaeky, and Fractal Patterns (FP) by XLII. The theories will be abbreviated as WSP, SL, EF, CSR2, FI, and FP from now on.

In order to balance custom theories with the main theories in the endgame, custom theories have a low conversion rate (with one exception) from $\rho$ to $τ$ . WSP, SL, and CSR2 have conversion rates of $τ$ = $\rho^{0.4}$ while EF has a $\tau$ conversion rate of $τ$ = $\rho^{1.6}$ and FP with a conversion rate of $τ$ = $\rho^{0.3}$.

Which Custom Theories (CTs) should I do? #

In general, you want to be as efficient as possible since R9 does not affect custom theories. If you cannot be active, try not to do an active theory or do an active strategy. Some custom theories are more active than normal theories and it is highly suggested that if you are doing active strategy for a Custom theory (SL or FI before all milestones, CSR2, WSP, or early FP) that you do an idle main theory (such as t2, t4, or t6) so that you don’t miss out on $\tau/hour$.

If you have time for active strategies, try to do the CT with the highest active $\tau/hour$, or you can chase a spike in tau/hour, such as EF e50 $rho$ or FP e95 $rho$. You can check this with the sim.

For idle time, do the one with the highest idle $\tau/hour$, (or the longest publication time if you’re doing overnights), with preference toward EF, SL, FP past e1050, or FI when you only have 1 milestone to swap. For example, if SL has 2 $\tau/hour$ and CSR2 also has 2 $\tau/hour$, ideally we would pick SL. The reason we prefer SL, EF, FP and FI is because these theories contain multiple growing variables. This means the theories generally require less babysitting as the variables grow by themselves. The assumption of daytime idle is that we can check and publish a theory every 2 hours or so. If you can only check every 8 hours idle, please see the overnight strategy just above.

Weierstrass Sine Product (WSP) #

WSP Overview #

The very first official custom theory; WSP was developed by Xelaroc, who also came up with some of the strategies used in the theory. The idea behind the theory is to use the factorization of sine to increase $\rho$. There are multiple equations with this theory, and some may look daunting, so we’ll have a look at each one.

WSP Equation Description #

$\dot{\rho} = q_1^{1.04}q_2q$

$\dot{q} = c_2s_n({\chi}) / sin({\chi})$

$s_n({x}) := x\prod_{k=1}^{n}(1-\frac{x}{k\pi}^2)$

$\chi = \pi\frac{c_1n}{c_1+n/3^{3}}+1$

The first line states that the rate of change in rho is $q_1^{1.04}q_2q$. Initially it’s simply $q_1q_2q$ without any exponent. With milestones we add more exponents.

For the second line, the higher the $\chi$ (spelled ‘chi’, pronounced as ‘kai’), the higher the $s_n({\chi})$. We want to increase $\chi$ by increasing $n$ and $c_1$. The signs of $s_n({\chi})$ and $sin({\chi})$ will always match, so the fraction can’t be negative. Additionally, the $c_2$ variable is a milestone which is not initially available.

The third line is the most complicated. Generally we can factorize an equation when its graph touches the x-axis. For a sine curve, it touches the x-axis starting from x = 0, and repeats every x= $\pi$. These multiplied factors form the basis of the Weierstrass Sine Product. A simpler interpretation is that we can see ‘x’ appearing both outside and inside the products in the numerator. Since $\chi$ is ‘x’ here, the higher the $\chi$, the higher the $s_n({\chi})$ as stated earlier.

Finally, the actual $\chi$ equation: increasing $c_1$ and $n$ increases $\chi$. Note that from the fraction, we don’t want to increase only $c_1$ or only $n$. Rather we should increase both. Using standard strategies this should be no problem. The $n/3^{3}$ part in the denominator is a milestone term. This means that $n$ is better than $c_1$ as more $n/3$ milestones are accumulated.

WSP Variable Description #

Approximate variable strengths on $\dot{ρ}$ with all milestones are as follows:

Brief summary of variable strengths of WSP.
	Brief Description
q_₁	About 7% increase on ρ dot on average.
q_₂	Doubles ρ dot - instantaneous.
n	Initially about 50% increase similar to c_₁. Slowly ramps up to 4 times increase in ρ dot. At e400 ρ and higher, it is very close to a 4x increase.
c_₁	Initially about 50% increase. Tends to 0% increase as ρ increases. At e400 ρ the increase is not noticeable anymore. Early in WSP we still buy them throughout. Late in WSP we only buy for the first 20 seconds or so of each publication.
c_₂	Doubles ρ dot - over time

WSP strategy #

Early game the variable strengths are ordered as follows:

$q_2$ ≈ $c_2$ > $n$ > $c_1$ > $q_1$

Late game these become:

$n$ > $q_2$ ≈ $c_2$ > $q_1$ >>> $c_1$

Idle

Before you get e400 $\rho$ for idle, simply autobuy all.

Once you have e400 $\rho$, $c_1$ starts to become extremely bad. Because of this, the new idle strategy would be to autobuy all for 20 seconds or so. Then turn $c_1$ OFF. Continue to autobuy the rest of the variables.

Active

For a simple active strategy before e400 $\rho$, simply autobuy $q_2$ and $c_2$ since they double the rates long term. $n$ and $c_1$ give approximately 60% boost (with $n$ becoming more powerful with milestones and vice versa for $c_1$). We will buy $n$ and $c_1$ when their costs are less than 50% of the minimum of $q_2$ and $c_2$.
For $q_1$, we will buy it when its cost is less than 10% of the minimum of $q_2$ and $c_2$. For example, if $q_1$ costs 1.2e100 and $q_2$ costs 1e101, we would not buy $q_1$ as it’s ‘too expensive’ compared to $q_2$.

For active strategy, $n$ starts to become more powerful than $q_2$. If their costs are similar, we will prioritize $n$ first. For example, if $n$ costs 1.4e101 and $q_2$ costs 1.2e101, we will buy $n$ first. Similarly to the idle strategy, we will buy $c_1$ only for the first 20 seconds or so. If you want more information on the different strategies pertaining to WSP, please see List of theory strategies

WSP milestone route #

All milestones into the 3rd/last milestone. Then into 2nd milestone, then into 1st milestone.
For milestone swapping, swap all milestones from 2nd and 3rd into 1st milestone. Usually you only do this when you’re about to publish.

0/0/1	→	0/0/2	→	0/0/3
0/1/3	→	1/1/3	→	2/1/3	→	3/1/3	→	4/1/3

Sequential Limits (SL) #

SL Overview #

SL, the second official custom theory, uses a variation of Stirling’s formula to approximate Euler’s number (e≈2.71828). As upgrades are bought, the approximation becomes more precise, increasing $\dot{ρ}$ and $\rho$ because $e-\gamma$ approaches 0. As with the first official custom theory (WSP), there are several equations in this theory. Let’s explore each one:

SL Equation Description #

$\dot{\rho}_1 = \frac{\sqrt{\rho_2^{1.06}}}{e - \gamma}$

$\gamma = \frac{\rho_3}{\sqrt[\rho_3]{\rho_3!}}$

$\dot{\rho_2} = a_1a_2a_3^{-ln{\rho_3}}$

$\dot{\rho_3} = b_1^{1.04}b_2^{1.04}$

$a_3 = 1.96$

The first line is the main part of the equation. We want to maximize $\dot{\rho_1}$ to increase $τ$ . The ‘1.06’ exponent is from milestones. The default is no exponent. From the equation, we can see that $\dot{\rho_1}$ is proportional to approximately $\sqrt{\rho_2}$. This means that if we quadruple $\rho_2$, we would approximately double $\rho_1$ long term. The denominator of the fraction has a gamma symbol ($\gamma$) which looks like the letter ‘y’. As our $\rho$ increases, our $\gamma$ becomes closer to ‘e’, so the denominator will decrease, which increases $\rho_1$. We will explore $\gamma$ in the next equation.

The second equation refers to Stirling’s approximation of Euler’s number ‘$e$’. As $\rho_3$ increases, $\gamma$ converges to Euler’s number. Long term we can approximate this convergence as linear. The implication is if we double $\rho_3$, $\gamma$ will be twice as close to Euler’s number, so $e-\gamma$ in the first equation will be halved.

The third equation relates $\rho_2$ with $\rho_3$ and some upgrades. The most interesting part is the exponent part containing $ln({\rho_3})$. The negative exponent actually implies that as $\rho_3$ increases, $\dot{\rho_2}$ DECREASES. If $\rho_3$ is high, $\rho_2$ doesn’t grow as fast (it still grows). This has implication on the first equation as well, since $\dot{\rho_1}$ depends on $\rho_2$, which depends on $\rho_3$.

The fourth equation relates $\dot{\rho_3}$ with some upgrades. This one is relatively simple; increase $b_1$ and $b_2$ to increase $\rho_3$. The ‘1.04’ exponents are from milestones.

The final equation simply states the value of $a_3$. The lower the better. Default without milestone is $a_3 = 2$.

SL Variable Description #

Approximate variable strengths on $\dot{ρ}$ with all milestones are as follows:

Brief summary of variable strengths of SL.
	Brief Description
a_₁	Value times 3.5 every 3 levels on average. This comes to about 52% increase in ρ_₂ dot per level. Since ρ_₁ is approximately square root of ρ_₂, overall this comes down to about 23% increase in ρ_₁ per level.
a_₂	Doubles in value every level. Doubles ρ_₂ long term. Increases ρ_₁ by 40% ish long term.
b_₁	Value times 6.5 every 4 levels on average. This comes down to about 60% increase in ρ_₃ dot. Toward the end of a publication, this translates to approximately 60% increase in ρ_₁.
b_₂	Doubles in value every level. Toward the end of a publication this doubles ρ_₁.

SL strategy #

All variables in SL are about the same in power, except for $a_1$ and $b_1$ (which are slightly worse than $a_2$ and $b_2$. Selectively buying variables at certain times (active) yields very little results. Therefore, we can get away with autobuy all for idle. Before autobuy, simply buy the cheapest variable. If you want more details on SL strategies, in particular the execution of various strategies, please see List of theory strategies.

Milestone swapping - why it works #

For active, there is a milestone swapping strategy that is significantly faster than idling (approximately twice the speed). If we carefully examine the effects of each milestone, we can conclude the following:

1st milestone: Increases $\rho_2$ exponent and increases $\dot{\rho_1}$ straight away. The actual value of $\rho_2$ does not increase.
3rd/4th milestone: Increase $b_1$/$b_2$ exponents, and $\dot{\rho_3}$, and $\rho_3$. This also increases $\dot{\rho_1}$. However, the effect is long-term and not instantaneous unlike the effect of the 1st milestone.

We have different milestones which affect the same thing ($\dot{\rho_1}$), but one is instantaneous, while the other builds over time. This forms the basis of ‘milestone swapping’, swapping milestones at certain times to maximize $\rho_1$ per hour. If you’ve done T2 milestone swapping, this should be familiar.

We initially put our milestones in the 4th and 3rd milestones. Once our $\rho_3$ doesn’t increase quickly anymore, we switch milestones to the 1st one to gain a burst of $\dot{\rho_1}$. Once our $\rho_1$is not increasing quickly anymore, we switch back to the 4th and 3rd milestone!

Milestone Swapping Strategies #

(Courtesy of Gen).

x>x>x>x represent the max buy order of milestones not the amount allocated. For example, 4>3>1>2 means “Allocate everything into 4th milestone, then use leftovers into 3rd milestone, then into 1st milestone, then into 2nd milestone”.

From e75-e100 is 4>3>1>2 (60s) ↔ 1>2>4>3 (60s)

SLMS2 is 1>2>4>3 (30s) → 2>1>4>3 (60s) → 1>2>4>3 (30s) → 4>3>1>2 (60s), with $b_1$$b_2$ off during the first two, and $a_1$$a_2$ off during the last two

SLMS3 is 2>1>4>3 (20s) ↔ 4>3>1>2 (60s)

When to Use Strategies until e100: SLMS
e100 - e175: SLMS (100-175)
e175 - e200: SLMS3
e200 - e300: SLMS

(note that it depends also on the swapping durations, on the last range SLMS should be run with 60s [4/3/1/2] and 20s on [1/2/4/3] to be best). So from e200-e300, SLMS 4>3>1>2 (60s) ↔ 1>2>4>3 (20s)

Post e300+ $\rho$ #

At this point, the theory becomes very idle. We simply autobuy all variables. Publish at approximately 8-10 multiplier. If you wish to improve efficiency, you can disable $a_1$$a_2$ at about 4.5 publication multiplier and $b_1$$b_2$ at 6.0 multiplier until publish.

SL milestone route #

Idle

0/0/0/2

→

0/0/2/2

→

3/0/2/2

→

3/5/2/2

Active

Milestone Swapping (active)

How to read notation: 4/3/1/2 means put all points into 4th milestones, use leftovers into 3rd milestones, etc.

SLMS is 4/3/1/2 (60s) ↔ 1/2/4/3 (60s)

SLMS2 is 1/2/4/3 (30s) → 2/1/4/3 (60s) → 1/2/4/3 (30s) → 4/3/1/2 (60s), with $b_1$$b_2$ off during the first two, and $a_1$$a_2$ off during the last two

SLMS3 is 2/1/4/3 (20s) ↔ 4/3/1/2 (60s)

When to Use Strategies until e100: SLMS

e100 - e175: SLMS2

e175 - e200: SLMS3

e200 - e300: SLMS

Euler’s Formula (EF) #

EF Overview #

This custom theory, along with Convergents to Square Root 2, were released at the same time and is based on Euler’s Formula of

$e^{i*\theta} = cos{\theta} + isin{\theta}$, where ‘i’ is the complex number.

EF is unique, along with FP, in that all the milestone paths are locked, so there’s no choice in which milestones to take. This was deliberately done to prevent milestone swapping strategies and to balance the theory. Furthermore, the $\rho$ to $\tau$ conversion for this theory is uniquely at $\rho^{1.6}$ rather than the usual $\rho^{0.4}$ meaning that less $\rho$ is needed to get an equivalent amount of $τ$ . Due to the conversion rate, EF can feel extremely slow in comparison to other theories, but it is the fastest theory to e150 $\tau$ and has the largest instantaneous jump in $\tau$ out of all custom theories.

EF Equation Description #

$\dot{\rho} = (a_1a_2a_3)^{1.5}\sqrt{tq^2+R^2+I^2}$

$G(t) = g_r+g_i$

$g_r = b_1b_2cos{(t)}, g_i = ic_1c_2sin{(t)}$

$\dot{q} = q_1q_2$

$\dot{R} = (g_r)^2, \dot{I} = -(g_i)^2$

The first line is the main equation. We want to maximize $\dot{\rho}$. All the $a_n$ terms and their exponents are obtained from milestones. Parts of the square root term are also obtained from milestones. Note that the $R^2$ and the $I^2$ terms are effectively redundant at all stages of this theory; but due to them purchasing $a_2$ and $a_3$ respectively, they are very important.

The second line defines the graph shown. Since $G(t)$ is graphed on the complex over time, it is possible to have it show as a particle spiraling through space.

The third line describes $g_r$ and $g_i$, which are used to generate ‘$R$’ and ‘$I$’ currencies. This line by itself doesn’t do much.

The fourth line simply describes $\dot{q}$. This is used in the first equation directly.

The fifth and final line use the results from the 3rd line, so effectively $\dot{R} = b_1^{2}b_2^{2}cos^2{(t)}$ and $\dot{I} = c_1^{2}c_2^{2}sin^2{(t)}$

EF Variable Description #

Approximate variable strengths on $\dot{ρ}$ with all milestones are as follows:

Brief summary of variable strengths of EF.
	Brief Description
$$\dot{ t }$$	Makes t increase faster. Since there are only 4 levels, after a certain point, this variable is effectively fixed.
q_₁	Standard variable. Doubles every 10 levels. Approximately 7% increase in ρ dot per level over time.
q_₂	Doubles in value every level. Also doubles ρ dot for each level bought, over time.
b_₁	Costs R to buy rather than ρ. Increases R by approximately 14% per level.
b_₂	Costs R to buy rather than ρ. Increases R by approximately 20% per level.
c_₁	Costs I to buy rather than ρ. Increases I by approximately 14% per level.
c_₂	Costs I to buy rather than ρ. Increases I by approximately 20% per level.
a_₁	Doubles approximately every 10 levels. Costs ρ to buy. With full milestones this variable increases ρ dot on average by about 11-12% for each level bought.
a_₂	Costs R to buy. Increases 40 folds for every 10 levels bought. However, note that some levels are much more impactful than others. Overall, this variable ranges from 10% to 700%+ effectiveness in ρ dot!
a_₃	Costs I to buy. With full milestones, this variable approximately triples ρ dot.

EF strategy #

Initially, you only have $\dot{t}$, $q_1$, and $q_2$ unlocked. Buy $q_1$ at about 1/8th cost of $q_2$, and buy $\dot{t}$ when it’s available. At e20 $\rho$ when autobuyers are unlocked, for idle, simply autobuy all. For active, continue to do what you were doing (buying $q_1$ at 1/8th cost of $q_2$). There are also more advanced strategies, in particular EFAI. For its description and execution, please see List of theory strategies.

The first 2 milestones are redundant by themselves. The $R^2$ term and the $I^2$ term are insignificant compared to the $tq^2$ term. Once you unlock the 3rd milestone ($a_1$ term) however, we can buy $a_1$ at 1/4th of $q_2$ cost.

EF milestone route #

2/0

→

2/3/0

→

2/3/5/0

→

2/3/5/2/0

→

2/3/5/2/2

Or
1 x2	→	2 x3	→	3 x5	→	4 x2	→	5 x2

Convergents to Square Root 2 (CSR2) #

CSR2 Overview #

This custom theory was released at the same time as Euler’s Formula. CSR2 is based on approximations of $\sqrt{2}$ using recurrent formulae. As the approximations improve, the $\dot{q}$ and $\dot{ρ}$ improve, increasing $τ$ . An explanation of each section of the equations is shown below:

CSR2 Equation Description #

$\dot{\rho} = q_1^{1.15}q_2q$

$\dot{q} = c_1c_2^2 |\sqrt{2} - \frac{N_m}{D_m}|^{-1}$, $N_m = 2N_{m-1} + N_{m-2}, N_0 = 1, N_1 = 3$ $D_m = 2D_{m-1} + D_{m-2}, D_0 = 1, D_1 = 2$ $m = n + log_2{(c_2)}$

The first line is self explanatory. The exponents on $q_1$ are from milestones. ‘$q$’ will increase during the publication.

For the second line, both the variable $c_2$ and its exponents are from milestones. The absolute value section on the right describes the approximation of $N_m$/ $D_m$ to $\sqrt{2}$. As $N_m$/ $D_m$ get closer to $\sqrt{2}$, the entire right section gets larger and larger (because of the -1 power).

The third and fourth lines are recurrence relations on $N_m$ and $D_m$. This means that the current value of $N_m$ and $D_m$ depend on their previous values. We start with $N_0$ = 1, $N_1$ = 3. The equation will then read as:

$N_2$ = 2$N_1$ + $N_0$ -> $N_2$ = 2 x 3 + 1 = 7. Then $N_3$ = 2$N_2$ + $N_1$ -> 2 x 7 + 3 = 17. Similar logic is applied to $D_m$ equations.

This occurs until we reach $N_m$ and $D_m$ reach whatever ‘m’ values we have. This is shown in the next equation:

The fourth equation relates ‘m’ as described above. We can see that as we buy $n$ and $c_2$, our $m$ will increase, so the 2 recurrence equations above will ‘repeat’ more often and $N_m$, $D_m$ will increase. From how $n$ and $c_2$ values are calculated, buying 1 level of $n$ or $c_2$ will increase $m$ by 1.

CSR2 Variable Description #

Approximate variable strengths on $\dot{ρ}$ with all milestones are as follows:

Brief summary of variable strengths of CSR2.
	Brief Description
q_₁	About 7% increase in ρ dot per level (instantaneous).
q_₂	Doubles ρ dot per level (instantaneous).
c_₁	About 7% increase in ρ dot per level; not instantaneous. This is the weakest variable.
n	Long term will multiply ρ dot by 6 times! However, it is not instantaneous.
c_₂	Approximately 22 times increase in ρ dot per level! Not instantaneous. This is the strongest variable by quite a lot.

CSR2 strategy #

Idle

For idle, we simply autobuy all. The idle strategy doesn’t change much. If you’d like to be more efficient while still being idle, you can remove milestones and stack them into the $q$ exponent milestones when you’re about to publish (from around e80 to e500). Don’t forget to change milestones back after publishing!

Once you have all milestones, autobuy all!

Active

The active strategies are significantly more involved. Depending on how active you’d like to be, there are several potential strategies. There’s the standard doubling chasing CSRd, which is just autobuy all except $c_1$ and $q_1$, where you buy them when they are less than 10% cost of minimum($c_2$, $q_2$, and $n$).

For the milestone swapping strategy, the general idea is to switch milestones from $c_2$ and its exponents, to $q_1$ exponent milestones whenever we are ‘close’ to a powerful upgrade. Please see the Theory Strategies section of the guide for how to perform milestone swapping.

CSR2 Milestone Swapping Explanation

This theory has a milestone swapping strategy before full milestones. We have $q_1$ exponent milestones, which increase $\dot{ρ}$ straight away. We also have $c_2$ related milestones, which increases the $q$ variable, which increases $\dot{ρ}$ .

The reason milestone swapping works is because the benefits of using $c_2$ related milestones (having high $q$) remain when you switch to $q_1$ exponent milestones. If we only use $q_1$ exponent, then we have really low $q$. If we only use $c_2$ related milestones, then we have high $q$, but low $\dot{ρ}$ . If we regularly swap them, we can increase $q$ through $c_2$ related milestones, then take advantage of the $q_1$ exponent milestones, while keeping the high value of $q$ we’ve accumulated earlier!

For a more detailed explanation on how to actually do the strategy, please see the Theory Strategies section of the guide.

CSR2 milestone route #

0/1/0

→

0/1/2

→

3/1/2

Or
2	→	3 x2	→	1 x3

Fractional Integration (FI) #

Overview #

This custom theory was released at the same time as Fractal Patterns. FI is based on Riemann–Liouville Integrals and allows you to approach the full integral as the fraction approaches 1. An explanation of each section of the equations is shown below:

FI Equation Description #

Base Equation

\dot{ρ} = t r \sqrt[π]{q / π}, \dot{q} = q_{1} q_{2}

\dot{r} = (\int_{0}^{π} g (x) d x -_{λ} \int_{0}^{π} g (x) d x^{λ})^{- 1}

λ \int_{0}^{π} g (x) d x^{λ} = \frac{1}{Γ (λ)} \int_{0}^{π} (π - x)^{λ - 1} g (x) d x

With $\dot{ρ}$ and $\dot{q}$ Equations Becoming:

\dot{ρ} = t r m n \sqrt[π]{\int_{0}^{q / π} g (x) d x}, \dot{ρ} = t r m n \sqrt[π]{\int_{0}^{q} g (x) d x}

\dot{q} = q_{1}^{1.03} q_{2}

g(x) Equations


	Equation
Milestone 0	$$1 - \frac { x^2 }{ 2! } + \frac { x^4 }{ 4! }$$
Milestone 1	$$x - \frac { x^3 }{ 3! } + \frac { x^5 }{ 5! }$$
Milestone 2	$$\frac{ x-\frac{ x^2 }{ 2 }+\frac{ x^3 }{ 3 }-\frac{ x^4 }{ 4 }+\frac{ x^5 }{ 5 }}{\ln( 10 )}$$
Milestone 3	$$1+x+\frac{ x^2 }{ 2! }+\frac{ x^3 }{ 3! }+\frac{ x^4 }{ 4! }+\frac{ x^5 }{ 5! }$$

$λ$ Equations


	Equation
Milestone 0	$$\lambda = \frac{ 1 }{ 2 }$$
Milestone 1	$$\sum_{ i=1 }^{ K }\frac{ 2 }{ 3^{ i } }$$
Milestone 2	$$\sum_{ i=1 }^{ K }\frac{ 3 }{ 4^{ i } }$$

The first equation is for $\rho$, which starts off simple, but gets more complicated as more milestones are reached and perma-upgrades are purchased. Initially, $\rho$ is fairly simple to calculate as $\dot{r}$ is just $1/2$, $\dot{t}$ is just the $t$ variable, and the $\sqrt[π]{}$ radical is just $\dot{q}$/ $π$ where $\dot{q}$ is just $q_1 * q_2$. However, once $g(x)$ is added to the $\dot{ρ}$ equation, the $\sqrt[π]{}$ radical becomes $\int_{0}^{q / π} g (x) d x$ which can be estimated by raising $q$ to the highest power of $g(x)$ by 1 and apon maxing out the $g(x)$ milestone, it becomes $\int_{0}^{q} g (x) d x$ . The variables $m$ and $n$ are simple multipliers that do not change over time without purchasing them with $ρ$ .

The second equation is for $\dot{r}$ , which seems simple at first, but gets more difficult to understand once we get to the fractional integral. The notation in game is rarely used, but it is used to save space. Tapping and holding the equation will give the full equation. When K increases, the fractional integral approaches 1, which makes the fractional integral get closer to, yet still smaller than, the full integral. By subracting the two, then dividing 1 by the difference, we get a very large number.

FI Variable Description #

Approximate variable strengths on their respective vardots with all milestones are as follows:

Brief summary of variable strengths of FI.
	Brief Description
q_₁	Grows by 50x every 23 levels. Mod23 levels are a 2.6x to $\dot{q}$
q_₂	Doubles $\dot{q}$ per level
K	Will double, triple, or quadrouple $\dot{r}$ depending on milestones
m	Will instantly increase $\dot{ρ}$ by 1.5x
n	Will instantly increase $\dot{ρ}$ by 3x every 11 levels

FI strategy #

Idle

For idle, we simply autobuy all. The idle strategy doesn’t change much other than we will not Milestone Swap. If you are able to check in every 30 minutes or so, you can manually buy $q_1$ and $n$. Just make sure that you autobuy $q_1$ when you are close to getting a mod23 boost.

Active

The active strategies are a bit more involved. Depending on how active you’d like to be, there are several potential strategies. There’s the standard doubling chasing FId, which is just autobuy all except $q_1$ and $n$, where you buy them when they are less than 10% cost of minimum($q_2$, $K$, and $m$).

For the milestone swapping strategy, the general idea is to switch milestones from $q_1$, to $m$/$n$ milestones whenever we gain 3x to $q$ after purchasing $q_2$, or some gain adjusted for $\dot{q}$ from purchasing $q_1$. Please see the Theory Strategies section of the guide for how to perform milestone swapping.

FI Milestone Swapping Explanation

This theory has a milestone swapping strategy before full milestones. We have $q_1$ exponent milestones, which increases $\dot{q}$ .

The reason milestone swapping works is because the benefits of using $q_1$ related milestones (having high $q$) remain when you switch to $m$ and $n$ milestones. If we only use $q_1$ exponent, then we have really high $q$, however, we dont have the benefits to $\dot{ρ}$ that $m$ and $n$ provide. If we only use $m$ and $n$ milestones, then we have low $q$, but have normal $\dot{ρ}$ . If we regularly swap them, we can increase $q$ through the $q_1$ milestone, then take advantage of the $m$ and $n$ milestones to gain $\rho$, while keeping the high value of $q$ we’ve accumulated earlier!

For a more detailed explanation on how to actually do the strategy, please see the Theory Strategies section of the guide.

FI Milestone Routing Explaination #

In FI, you can unlock milestones in 2 ways:

by gaining $\rho$ like normal, or
buy purchasing the milestone upgrades for $\lambda$ and $g(x)$ in the permanent upgrades tab where you would normally buy publishing, buy all, and autobuy.

Buying the milestone upgrades will not give you a milestone, but will instead increase the max level of the miletone that you purchased the upgade for. For example, if you buy the $g(x)$ perma-upgrade for lvl 1, you will permanently unlock the first lvl of the $g(x)$ milestone. Moving milestones into these are always the best things you can do mid publish, even if you need to sacrifice a variable to do so.

FI perma-upgrades are at 1e100, 1e450, and 1e1050 $\rho$ for the $g(x)$ milestone and 1e350 and 1e750 $\rho$ for the $\lambda$ milestone. Apon buying these milestone, immediately put a milestone from $q_1$ or $n$ into them depending on how many milestone you have.

FI Milestone Route #

Colored milestones are perma-upgrade milestones that move into that upgrade.

→

1/1

→

1/1/0/1

→

1/1/0/2

1/1/0/1/1

→

1/1/0/2/1

→

1/1/1/2/1

1/1/0/2/1/1

→

1/1/1/2/1/1

→

1/1/0/2/2/1

1/1/1/2/2/1

→

1/1/2/2/2/1

→

1/1/1/2/2/2

1/1/2/2/2/2

→

1/1/3/2/2/2

→

1/1/2/2/3/2

1/1/3/2/3/2

Or
1	→	2	→	4x2

4-x1 5x1

→

3-x1 6x1

→

3-x1 5x1

3x2

→

3-x1 6x1

→

3x2

3-x1 5x1

→

Fractal Patterns (FP) #

FP Overview #

This custom theory was released at the same time as Fractional Integration. FP is A theory that takes advantage of the growth of the 3 fractal patterns: Toothpick Sequence (Tₙ), Ulam-Warburton cellular automaton (Uₙ), Sierpiński triangle (Sₙ). As each of the fractals grow, so does $τ$ . An explanation of each section of the equations is shown below:

FP Equation Description #

Main Equations

\begin{array}{r} \dot{ρ} = c_{1} c_{2} t T_{n}^{7} \to \dot{ρ} = c_{1} c_{2} q t T_{n}^{7} \\ \to \dot{ρ} = c_{1} c_{2} q r t T_{n}^{7} \to \dot{ρ} = c_{1} c_{2} q r t T_{n}^{5 + s} \end{array}

\dot{q} = q_{1} A U_{n}^{7} / 1000 \to \dot{q} = q_{1} A U_{n}^{7 + s} / 1000

\begin{array}{r} \dot{r} = r_{1} (T_{n} U_{n})^{\log (n)} S_{⌊ \sqrt{n} ⌋} \\ \to \dot{r} = r_{1} (T_{n} U_{n})^{\log (n)} S_{⌊ \sqrt{n} ⌋}^{2.8} \\ \to \dot{r} = r_{1} (T_{n} U_{n})^{\log (\sqrt{2 U_{n}})} S_{⌊ \sqrt{n} ⌋}^{2.8} \end{array}

A = (2 - U_{q_{2}} / T_{q_{2}})^{- 1}

Toothpick Sequence

T_{2^{k} + i} = \frac{2^{2 k + 1} + 1}{3}, if i = 0

T_{2^{k} + i} = T_{2^{k}} + 2 T_{i} + T_{i + 1} - 1, if 1 \leq i < 2^{k}

Ulam-Warburton Cellular Automaton

u_{0} = 0, u_{1} = 1, \dots, u_{n} = 4 (3^{w_{n - 1} - 1})

w_{n} = n - \sum_{k = 1}^{\infty} ⌊ \frac{n}{2^{k}} ⌋

U_{n} = \sum_{i = 0}^{n} u_{i}

Sierpiński Triangle

S_{n} = 3^{n - 1} \to S_{n} = 2 * 3^{n - 1} - 1

FP Variable Description #

Approximate variable strengths on $\dot{ρ}$ with all milestones are as follows:

Brief summary of variable strengths of FP.
	Brief Description
tdot	This is tdot
c_₁	c_1 is 150x over 100 levels for mod 100
c_₂	Simple Doubling
q_₁	Roughly is a 10x over 10 lvls mod10 for $\dot{q}$ change (don't ask)
q_₂	Quadruples $\dot{q}$ except for the first few levels
r_₁	is roughtly 10-20% every level to $\dot{r}$ . It is roughly a 2x over mod5 (don't ask again)
n	2^k=n is very nice, but n is very hard to describe
s	s is an additive to an exponent, whose value changes occationally.

FP strategy #

Idle

For idle, we simply autobuy all, however, it is very slow to start idle, and it is suggested to be active until e950 $ρ$ . The idle strategy doesn’t change much. If you’d like to be more efficient while still being idle, you can stop buying $c_{1}$ around mod%100 50 lvls, or around when the last 2 didgets in the level are 50 or more, then but them in chunks of no more than 13. When you reach e700, you will need to milestone swap to be able to get any good progress, however, you only need to swap every 20-30 minutes to get some good results.

Once you have all milestones, autobuy all!

Active

The active strategies change constantly depending on your milestones and there is no definitive active strategy like most other actives that we know of currently due to the complexity of the theory. For example, exact ratios of when to buy variables is very difficult to find and the only known buying straegy is between c1 and c2. However, generally you can follow this order of buying s>n=q2>c2>=c1>q1>r1 but the longer your publish goes, the weaker q2 gets overall and will eventually become less valuable than c2. There are also edge cases where q1 is mod%10=0 and may be stronger than c1, which may be mid mod%100 cycle. The variable relationships are as follows:

C1 and C2 Buying

BUYING c1 EFFICIENTLY IS THE LARGEST BOOST TO RATES YOU CAN DO (outside of MS).

The only known ratio currently is c1 to c2 and, specifically, it is c1 price < 3/(lvl%100 + 2) * c2 price. But, for a more digestible strategy, you would want to: When c1 mod 100 is < 92, buy c1 if c1 is (c1 mod 100) times cheaper than c2. When c1 mod 100 is >= 92, wait until the sum to buy up to c1 mod 100 = 1 is cheaper than c2. Buy c1 upgrades as they become available.

More human way to do the second part is this: when c1 mod 100 == 91, switch to buying x10, see the cumulative price to get c1 mod 100 = 1, and if that is below c2 - it is time to buy c1 up to mod 100 = 1 using autobuy.

Note: the actual ratio for part 1 is actually (c1 mod 100) + 0.67, but that’s harder to play as a human

q1 and q2 Buying

q1 follows a mod 10 cycle, and adds ~100%, then ~50%, then ~33% and so on to $\dot{q}$ . q2 always quadruples the $\dot{q}$ (except the first few purchases)

This plays roughly like doubling chase, but in this case you have to adjust ratios slightly - for example, if q1 mod 10 is 0, you want to wait until q1 upgrade price is twice as cheap as q2, and so on.

Other variables and what to do about them.

s - always buy on sight. n - buy after s. r1 - check how much percentage increase it will give to $\dot{r}$ , and then buy like normal doubling chase

Overall, We have s, n, c2 and q2, and we have c1, q1, and r1. The latter work roughly like doubling chase to the former most of the time, with additions of what was said about them beforehand.

FP Milestone Swapping Explanation

FP has a milestone swap that involves 1 milestone. This is the milestone that adds s as an exponent (e700 rho). The swap arises from the idea that initially, Tn power drops from 7 to 5 + s in the rho equation, and s is less than 2. Because of this, it makes sense to swap this milestone in for q growth, and swap it out for rho growth.

The swap is really hard to describe in terms of how long to keep it in and out but what can be said qualitatively:

At first, you follow very fast swaps to recover rho, and swaps gradually become slower and slower.
As s grows, it makes sense to keep the milestone swapped in longer.

Milestone swap ends when s becomes > 2, and dies out when you can recover to that point very fast. Past ~e950 rho, recovery takes ~1-3 minutes of idle time.

Milestone swap saves a LOT of time.

FP milestone route #

→

2/2

→

2/2/3

→

2/2/3/1

→

2/2/3/1/1

→

2/2/3/1/1/1

Or
1 x2	→	2 x2	→	3 x3	→	4	→	5	→	6

FP Guide written by Snaeky and Hotab

Riemann Zeta Function (RZ) #

Guide Writing is in progress. Not everything here is accurate, or from RZ at the moment.

RZ Overview #

This Custom Theory was the first solo launch CT since SL (has it really been over 2 years!). RZ is a very fast, very active CT with a completion time estimated at 100 days. The function follows the Zeta function over the critical line. Rumors say that reaching 1e1500 will be a proof of the Riemann Hypothesis, or if you prove it yourself, we will just give you the $\rho$.

RZ Equation Description #

\dot{ρ} = \frac{t c_{1}^{1.25} c_{2} w_{1}}{| ζ (\frac{1}{2} + i t) | / 2^{b} + 10^{- 2}}

\dot{δ} = w_{1} w_{2} w_{3} \times | ζ^{'} (\frac{1}{2} + i t) |^{b}

Function Description Under Construction please be patient.

RZ Variable Description #

Approximate variable strengths on $\dot{ρ}$ with all milestones are as follows:

Brief summary of variable strengths of RZ.
	Brief Description
c_₁	Doubles every 8 lvls
c_₂	Doubles ρ dot per level (instantaneous).
w_₁	Doubles every 8 lvls
w_₂	Doubling
w_₃	Doubling
b	Is capped at 6 lvls maxing out at 3 (+0.5/lvl)

RZ strategy #

Idle

For idle, we simply autobuy all. The idle strategy doesn’t change much. If you’d like to be more efficient while still being idle, you can remove milestones and stack them into the $c_1$ exponent milestones when you’re about to publish (from e50 to e400). Don’t forget to change milestones back after publishing!

Once you have all milestones, autobuy all!

Active

Active stragies are still being developed. Right now, Buy $c_1$ and $w_1$ and a 4x difference to $c_2$ and $w_2$ respectively. With a milestone swap from from e50 to e400 e3 $rho$ from recovery and publish between 7 and 10 multi.

RZ Milestone Swapping Explanation

Strategy in development, please be patient.

RZ milestone route #

0/1/0

→

0/1/1

→

3/1/1

→

3/1/1/1

Or
2	→	3	→	1 x3	→	4

Table of Con­tents

Table of Con­tents

Guides

Guides

Re­sources

Re­sources

Guide Ex­ten­sions

Guide Ex­ten­sions

More Guides ⓧ

Table of Con­tents ⓧ

Other Re­sources ⓧ

Guide Ex­ten­sions ⓧ

Cus­tom The­or­ies

Cus­tom The­ory Ba­sics #

Which Cus­tom The­or­ies (CTs) should I do? #

Wei­er­strass Sine Product (WSP) #

WSP Over­view #

WSP Equa­tion De­scrip­tion #

WSP Vari­able De­scrip­tion #

WSP strategy #

Idle

Act­ive

WSP mile­stone route #

Se­quen­tial Lim­its (SL) #

SL Over­view #

SL Equa­tion De­scrip­tion #

SL Vari­able De­scrip­tion #

SL strategy #

Mile­stone swap­ping - why it works #

Mile­stone Swap­ping Strategies #

Post e300+ \(\rho\) #

SL mile­stone route #

Idle

Act­ive

Euler’s For­mula (EF) #

EF Over­view #

EF Equa­tion De­scrip­tion #

EF Vari­able De­scrip­tion #

EF strategy #

EF mile­stone route #

Con­ver­gents to Square Root 2 (CSR2) #

CSR2 Over­view #

CSR2 Equa­tion De­scrip­tion #

CSR2 Vari­able De­scrip­tion #

CSR2 strategy #

Idle

Act­ive

CSR2 Mile­stone Swap­ping Ex­plan­a­tion

CSR2 mile­stone route #

Frac­tional In­teg­ra­tion (FI) #

Over­view #

FI Equa­tion De­scrip­tion #

Base Equa­tion

g(x) Equa­tions

λ Equa­tions

FI Vari­able De­scrip­tion #

FI strategy #

Idle

Act­ive

FI Mile­stone Swap­ping Ex­plan­a­tion

FI Mile­stone Rout­ing Ex­plain­a­tion #

FI Mile­stone Route #

Colored mile­stones are perma-up­grade mile­stones that move into that up­grade.

Fractal Pat­terns (FP) #

FP Over­view #

FP Equa­tion De­scrip­tion #

Main Equa­tions

Tooth­pick Se­quence

Ulam-War­bur­ton Cel­lu­lar Auto­maton

Si­er­piński Tri­angle

FP Vari­able De­scrip­tion #

FP strategy #

Idle

Act­ive

C1 and C2 Buy­ing

q1 and q2 Buy­ing

Other vari­ables and what to do about them.

FP Mile­stone Swap­ping Ex­plan­a­tion

FP mile­stone route #

Riemann Zeta Func­tion (RZ) #

Table of Contents

Table of Contents

Resources

Resources

Guide Extensions

Guide Extensions

Table of Contents ⓧ

Other Resources ⓧ

Guide Extensions ⓧ

Custom Theories

Custom Theory Basics #

Which Custom Theories (CTs) should I do? #

Weierstrass Sine Product (WSP) #

WSP Overview #

WSP Equation Description #

WSP Variable Description #

Active

WSP milestone route #

Sequential Limits (SL) #

SL Overview #

SL Equation Description #

SL Variable Description #

Milestone swapping - why it works #

Milestone Swapping Strategies #

SL milestone route #

Active

Euler’s Formula (EF) #

EF Overview #

EF Equation Description #

EF Variable Description #

EF milestone route #

Convergents to Square Root 2 (CSR2) #

CSR2 Overview #

CSR2 Equation Description #

CSR2 Variable Description #

Active

CSR2 Milestone Swapping Explanation

CSR2 milestone route #

Fractional Integration (FI) #

Overview #

FI Equation Description #

Base Equation

g(x) Equations

$λ$ Equations

FI Variable Description #

Active

FI Milestone Swapping Explanation

FI Milestone Routing Explaination #

FI Milestone Route #

Colored milestones are perma-upgrade milestones that move into that upgrade.

Fractal Patterns (FP) #

FP Overview #

FP Equation Description #

Main Equations

Toothpick Sequence

Ulam-Warburton Cellular Automaton

Sierpiński Triangle

FP Variable Description #

Active

C1 and C2 Buying

q1 and q2 Buying

Other variables and what to do about them.

FP Milestone Swapping Explanation

FP milestone route #

Riemann Zeta Function (RZ) #

RZ Overview #

RZ Equation Description #

RZ Variable Description #

Active

RZ Milestone Swapping Explanation

RZ milestone route #