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 March 2025, there are 9 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, Fractal Pat­terns (FP) by XLII, Riemann Zeta Func­tion by Prop (RZ), Mag­netic Fields by Mathis (MF) and Basel Prob­lem by Py­thon’s Ko­ala (BaP). The the­or­ies will be ab­bre­vi­ated as WSP, SL, EF, CSR2, FI, FP, RZ, MF and BaP from now on. The choice for a three let­ter ab­bre­vi­ation for BaP was made to avoid con­fu­sion with a pre­vi­ous un­of­fi­cial cus­tom the­ory shar­ing the same ini­tials (Bin Pack­ing).

In or­der to bal­ance cus­tom the­or­ies with the main the­or­ies in the en­dgame, cus­tom the­or­ies have a low con­ver­sion rate (with two ex­cep­tions) from $\rho$ to $\tau$. WSP, SL, CSR2, FI, RZ and BaP have con­ver­sion rates of $\tau$ = ${\rho }^{0.4}$ while EF has a $\tau$ con­ver­sion rate of $\tau$ = ${\rho }^{1.6}$ and FP with a con­ver­sion rate of $\tau$ = ${\rho }^{0.3}$. Mean­while, MF is the only cus­tom the­ory to this day to have $\tau$ = $\rho$.

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, MF, 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, BaP, 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, FI and BaP 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.

Graph made by Hack­zzzzzz (Source)

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 #

$\dot{\rho }=q_1^{1.04}{q}_{2}q$

$\dot{q}=c_2{s}_n(\chi )/sin(\chi )$

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

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

The first line states that the rate of change in $\rho$ is $q_1^{1.04}{q}_{2}q$. Ini­tially it’s simply $q_1{q}_{2}q$ 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 $\dot{\rho }$ with all mile­stones are as fol­lows:

Class: vari­able_­de­scrip­tion; Cap­tion: Brief sum­mary of vari­able strengths of WSP; last_row: false;

WSP Strategy #

Early game the vari­able strengths are ordered as fol­lows:

$q_2\approx c_2>n>c_1>q_1$

Late game these be­come:

$n>q_2\approx c_2>q_1>>>c_1$

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.

Class: mile­stone­_rout­ing; last_row: false;

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 $\dot{\rho }$ 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 #

${\dot{\rho }}_1=\frac{\sqrt{{\rho }_2^{1.06}}}{e-\gamma }$

$\gamma =\frac{{\rho }_3}{\sqrt[{\rho }_3]{{\rho }_3!}}$

$\dot{{\rho }_2}=a_1{a}_2{a}_3^{-ln{\rho }_3}$

$\dot{{\rho }_3}=b_1^{1.04}{b}_2^{1.04}$

$a_3=1.96$

The first line is the main part of the equa­tion. We want to max­im­ize $\dot{{\rho }_1}$ to in­crease $\tau$. 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 $\dot{\rho }$ with all mile­stones are as fol­lows:

Class: vari­able_­de­scrip­tion; Cap­tion: Brief sum­mary of vari­able strengths of SL; last_row: false;

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”.

SLMS

4>3>1>2 (60s) → 1>2>4>3 (60s*) → re­peat

* between $e200-e300$, 1>2>4>3 should be 20s.

SLMS2

1>2>4>3* (30s) → 2>1>4>3* (60s) → 1>2>4>3** (30s) →
4>3>1>2** (60s) → re­peat

* $b_1$ & $b_2$ turned off.
** $a_1$ & $a_2$ turned off.

SLMS3

2>1>4>3 (20s) → 4>3>1>2 (60s) → re­peat

When to use strategies:

un­til e100: SLMS

e100 - e175: SLMS2

e175 - e200: SLMS3

e200 - e300: SLMS

For a more pre­cise de­scrip­tion of SLMS, check out the the­ory strategy sec­tion.

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, 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 #

Class: mile­stone­_rout­ing; last_row: false;

Act­ive #

See SL Mile­stone Swap­ping Strategies

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

$e^{i\ast \theta }=cos\theta +isin\theta$, 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 $\tau$. 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 #

$\dot{\rho }=(a_1{a}_2{a}_3{)}^{1.5}\sqrt{t{q}^2+R^2+I^2}$

$G(t)=g_r+g_i$

$g_r=b_1{b}_{2}cos(t),g_i=i{c}_1{c}_{2}sin(t)$

$\dot{q}=q_1{q}_2$

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

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 $\dot{R}=b_1^2{b}_2^{2}co{s}^2(t)$ and $\dot{I}=c_1^2{c}_2^{2}si{n}^2(t)$

EF Vari­able De­scrip­tion #

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

Class: vari­able_­de­scrip­tion; Cap­tion: Brief sum­mary of vari­able strengths of EF; last_row: false;

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 $t{q}^2$ 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 #

Class: mile­stone­_rout­ing; last_row: false;

Class: mile­stone­_rout­ing; last_row: false;

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 $\dot{\rho }$ im­prove, in­creas­ing $\tau$. An ex­plan­a­tion of each sec­tion of the equa­tions is shown be­low:

CSR2 Equa­tion De­scrip­tion #

$\dot{\rho }=q_1^{1.15}{q}_{2}q$

$\dot{q}=c_1{c}_2^2|\sqrt{2}-\frac{N_m}{D_m}{|}^{-1}$

$N_m=2N_{m-1}+N_{m-2},N_0=1,N_1=3$

$D_m=2D_{m-1}+D_{m-2},D_0=1,D_1=2$

$m=n+lo{g}_2(c_2)$

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:

$N_2=2N_1+N_0->N_2=2x3+1=7$.

Then $N_3=2N_2+N_1->2x7+3=17$.

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 $\dot{\rho }$ with all mile­stones are as fol­lows:

Class: vari­able_­de­scrip­tion; Cap­tion: Brief sum­mary of vari­able strengths of CSR2; last_row: false;

CSR2 Strategy #

Idle #

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

Once all mile­stones are un­locked, auto­buy all!

Act­ive #

The act­ive strategies are sig­ni­fic­antly more in­volved. De­pend­ing on how act­ive you would 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 $\dot{\rho }$ straight away. We also have $c_2$ re­lated mile­stones, which in­creases the $q$ vari­able, which in­creases $\dot{\rho }$.

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 $\dot{\rho }$. 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 #

Class: mile­stone­_rout­ing; last_row: false;

Class: mile­stone­_rout­ing; last_row: false;

Frac­tional In­teg­ra­tion (FI) #

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 #

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

Class: strat; Cap­tion: $g(x)$ Equa­tions; last_row: false;

Class: strat; Cap­tion: $\lambda$ Equa­tions; last_row: false;

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 $\sqrt[\pi ]{}$ rad­ical is just $\dot{q}$/$\pi$ where $\dot{q}$ is just $q_1\ast q_2$. However, once $g(x)$ is ad­ded to the $\dot{\rho }$ equa­tion, the $\sqrt[\pi ]{}$ rad­ical be­comes ${\int }_0^{q/\pi }g(x)dx$ which can be es­tim­ated by rais­ing $q$ to the highest power of $g(x)$ by 1 and upon max­ing out the $g(x)$ mile­stone, it be­comes ${\int }_0^{q}g(x)dx$. The vari­ables $m$ and $n$ are simple mul­ti­pli­ers that do not change over time without pur­chas­ing them with $\rho$.

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 sub­tract­ing 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:

Class: vari­able_­de­scrip­tion; Cap­tion: Brief sum­mary of vari­able strengths of FI; last_row: false;

FI Strategy #

Idle #

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

Act­ive #

The act­ive strategies are a bit more in­volved. De­pend­ing on how act­ive you would 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 don’t have the be­ne­fits to $\dot{\rho }$ that $m$ and $n$ provide. If we only use $m$ and $n$ mile­stones, then we have low $q$, but have nor­mal $\dot{\rho }$. 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 Ex­plan­a­tion #

In FI, you can un­lock mile­stones in 2 ways:

1. by gain­ing $\rho$ like nor­mal, or
2. by 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 mile­stone that you pur­chased the up­grade 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 al­most al­ways the best thing you can do mid pub­lish, even if you need to sac­ri­fice a vari­able to do so, with one ex­cep­tion.

It is im­port­ant to note, however, is that buy­ing or re­fund­ing $g(x)$ mile­stones will re­set your $q$, $q_2$ level and will change the $q_2$ cost func­tion. Sim­il­arly, buy­ing or re­fund­ing $\lambda$ mile­stones will re­set your $K$ and change the $K$ cost func­tion.

FI perma-up­grades are at 1e100, 1e450, and 1e1050 $\rho$ for the $g(x)$ mile­stone and 1e350 and 1e750 $\rho$ for the $\lambda$ mile­stone. Upon buy­ing these mile­stone, im­me­di­ately put a mile­stone from $q_1$ or $n$ into them de­pend­ing on how many mile­stone you have, ex­cept for the 3rd level of the $g(x)$ mile­stone.

The 3rd level of the $g(x)$ mile­stone is bad early on, and is only worth buy­ing at e1076ρ. Swap­ping to the 3rd level of the $g(x)$ mile­stone mid-pub is known as Per­maSwap, check the the­ory sim­u­lator to know if you should do this strategy.

FI Mile­stone Route #

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

Class: mile­stone­_rout­ing; last_row: false;

Class: mile­stone­_rout­ing; last_row: false;

Class: mile­stone­_rout­ing; last_row: false;

Class: strat; Cap­tion: Perma-Up­grade Costs; align: left;

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 $T_n$, Ulam-War­bur­ton cel­lu­lar auto­maton $U_n$, Si­er­piński tri­angle $S_n$. As each of the fractals grows, so does $\tau$. An ex­plan­a­tion of each sec­tion of the equa­tions is shown be­low:

FP Equa­tion De­scrip­tion #

Main Equa­tions #

The first equa­tion is for $\rho$, which is the product of $c_1{c}_{2}qr$ and the fractal term $T_n^7$, where $T_n$ is the nth term of the Tooth­pick Se­quence shown be­low. Its ex­po­nent starts at 7, but when you un­lock the $s$ mile­stone, it will change to $5+s$, where $s$ is an up­grade.

The $\dot{q}$ equa­tion is sim­ilar, but de­pends on Ulam-War­bur­ton Cel­lu­lar Auto­maton $U_n$ in­stead. Its ex­po­nent starts at 7, and changes to $7+s$ when you un­lock the $s$ mile­stone, mean­ing this mile­stone has no draw­back to $q$ un­like $\rho$.

$q$ growth also de­pends on the $A$ term, which it­self de­pends on $q_2$. For the ex­act for­mula, if $k$ is the level of $q_2$, then $A=(2^{2k+1}+1)/3$. This means that each level of $q_2$ tends to a x4 in­crease to $\dot{q}$.

The $r$ equa­tion de­pends on all fractals avail­able in FP.

Tooth­pick Se­quence #

This is the Tooth­pick Se­quence. We can’t really ex­plain it without get­ting tech­nical, but this se­quence grows as $n$ grows. It is im­port­ant to note that it grows faster right be­fore a new power of two, and slower right after a power of two. This trait is shared with the next fractal. These $n=2^k$ spikes have a lot of in­flu­ence on the the­ory speed, es­pe­cially on the second half of it.

If you want to learn more about the Tooth­pick Se­quence, you can search about it on the in­ter­net. You can find an an­im­a­tion of the fractal here.

Ulam-War­bur­ton Cel­lu­lar Auto­maton #

These equa­tions are used to de­scribe the Ulam-War­bur­ton Cel­lu­lar Auto­maton ($U_n$). This is the second main fractal used in FP. Like $T_n$, it grows faster right be­fore a new power of two, and slower right after a power of two.

The $w_n$ equa­tion can look in­tim­id­at­ing, but it is sim­pler to ex­plain than some of the other for­mu­las. $w_n$ is the Ham­ming weight of the bin­ary rep­res­ent­a­tion of $n$, which is the num­ber of 1s that ap­pear in its rep­res­ent­a­tion. Right be­fore a power of two, a num­ber has a lot of 1s on the left of its bin­ary rep­res­ent­a­tion, which means $w_n$ is higher, and as such $U_n$ grows faster with $n$. The op­pos­ite is true for right after a power of two.

You can find an an­im­a­tion of the fractal here after se­lect­ing it in “Main se­quence”.

Si­er­piński Tri­angle #

This is prob­ably the most fam­ous fractal used in FP. It can be ob­tained from an equi­lat­eral tri­angle, by re­curs­ively sub­divid­ing each tri­angle into 4 smal­ler identical tri­angles and re­mov­ing the middle one. Its for­mula is much sim­pler than the other two fractals.

FP Vari­able De­scrip­tion #

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

Class: vari­able_­de­scrip­tion; Cap­tion: Brief sum­mary of vari­able strengths of FP; last_row: false;

FP Strategy #

Check out the FP Quick Pur­chase Tester for vari­able checks mid-pub­lic­a­tion.

Idle #

For idle, we simply auto­buy all, however, it is very slow to start idle, and it is sug­ges­ted to be act­ive un­til e950 $\rho$. The idle strategy does­n’t change much. If you’d like to be more ef­fi­cient while still be­ing idle, you can stop buy­ing when $c_1\equiv 50\mathrm{mod}100$, or around when the last 2 di­gits in the level are 50 or more, then buy 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 are very dif­fi­cult to find and the only known buy­ing strategy is between c1 and c2. However, gen­er­ally you can fol­low this or­der of buy­ing $s>n=q_2>c_2\ge c_1>q_1>r_1$ 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 $q_1\equiv 0\mathrm{mod}10$ and $q_1$ may be stronger than $c_1$, which may be mid $mod100$ cycle. The vari­able re­la­tion­ships are as fol­lows:

$c_1$ and $c_1$ Buy­ing

Buy­ing $c_1$ ef­fe­ciently is the largest boost to rates you can do (out­side of MS).

The only known ra­tio cur­rently is $c_1$ to $c_2$ and, spe­cific­ally, it is $c_1$ price $<3/(\text{lvl}\mathrm{\%}100+2)\ast c_2$ price. But, for a more di­gest­ible strategy, you would want to:

When c1 mod 100 is $<92$, buy $c_1$ if $c_1$ is $(c_{1}mod100)$ times cheaper than $c_2$. When $c_1\mathrm{\%}100\ge 92$, wait un­til the sum to buy up to $c_1\equiv 1\mathrm{mod}100$ is cheaper than $c_2$. Buy $c_1$ up­grades as they be­come avail­able.

More hu­man way to do the second part is this: when $c_1\equiv 91\mathrm{mod}100$, switch to buy­ing x10, then see the cu­mu­lat­ive price to get $c_1\equiv 1\mathrm{mod}100$, and if that is be­low $c_2$ - it is time to buy $c_1$ up to $c_1\equiv 1\mathrm{mod}100$ us­ing auto­buy.

Note: the ac­tual ra­tio for part 1 is ac­tu­ally $(c1mod100)+0.67$, but that’s harder to play as a hu­man.

$q_1$ and $q_2$ Buy­ing

$q_1$ fol­lows a $mod10$ cycle, and adds ~100%, then ~50%, then ~33% and so on to $\dot{q}$. $q_2$ 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 $q_1\equiv 0\mathrm{mod}10$, you want to wait un­til $q_1$ up­grade price is twice as cheap as $q_2$, and so on.

Other vari­ables and what to do about them.

$s$ - al­ways buy on sight. $n$ - buy after $s$. $r_1$ - check how much per­cent­age in­crease it will give to $\dot{r}$, and then buy like nor­mal doub­ling chase, auto­buy­ing is also fine.

Over­all, We have $s$, $n$, $c_2$, and $q_2$, then $c_1$, $q_1$, and $r_1$. 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 $s>2$, 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 #

Class: mile­stone­_rout­ing; last_row: false;

Class: mile­stone­_rout­ing; last_row: false;

Class: mile­stone­_rout­ing; last_row: false;

FP Guide writ­ten by Snaeky, Hotab and Mathis S.

Riemann Zeta Func­tion (RZ) #

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 CT with a com­ple­tion time es­tim­ated be­low 70 days! The the­ory 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 $\rho$.

Its strategies range a lot in com­par­ison to other the­or­ies, however, RZ is not an idle the­ory at first and you must be act­ive be­fore about e700 $\rho$ due to its short pub­lic­a­tions. It also has a mile­stone swap­ping phase from e50 to e400 $\rho$. After e600, the en­tire dy­namic of the the­ory changes with the in­clu­sion of the black hole.

RZ Equa­tion De­scrip­tion #

These two equa­tions fol­low the ana­lytic con­tinu­ation of the Riemann Zeta func­tion along the crit­ical $1/2+it$ line, where all the “non-trivial” zeros of this func­tion should be loc­ated ac­cord­ing to the Riemann Hy­po­thesis.

The back­ground an­im­a­tion of the CT helps to un­der­stand the be­ha­vior of the $\zeta$ along the crit­ical line. You can see the back­ground as the com­plex plane, with the middle point be­ing zero, and the particle fol­low­ing the value of $\zeta$ at the given $t$. The fur­ther the particle is from the ori­gin, the higher $|\zeta (\frac{1}{2}+it)|$ is. The faster the particle travels, the higher $|{\zeta }^{\prime }(\frac{1}{2}+it)|$ is.

This particle de­scribes spir­als, and passes by the ori­gin at each of its turns.

We can see in the $\dot{\rho }$ equa­tion that $|\zeta (\frac{1}{2}+it)|/2^b+{10}^{-2}$ is on the de­nom­in­ator, which means $\rho$ grows faster when $\zeta (\frac{1}{2}+it)$ is close to zero. The ${10}^{-2}$ term pre­vents $\dot{\rho }$ from ex­plod­ing at each zero. The $2^b$ term helps the growth of $\rho$ when $\zeta (\frac{1}{2}+it)$ is away from zero.

$\delta$ grows faster as $|{\zeta }^{\prime }(\frac{1}{2}+it)|$ is higher.

RZ Vari­able De­scrip­tion #

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

Class: vari­able_­de­scrip­tion; Cap­tion: Brief sum­mary of vari­able strengths of RZ; last_row: false;

RZ Strategy #

Pre-e600 $\rho$ #

The op­timal pub­lic­a­tion mul­ti­plier is around 2-4 be­fore e50 $\rho$ and 4-8 after, but can vary if you are close to the next mile­stone. As al­ways, you can check with the sim.

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!

Also check out A Pos­sible Idle RZ The­ory By Time for more in-depth look at idle RZ strategies.

Act­ive #

For an act­ive buy­ing strategy, buy $c_1$ and $w_1$ and a 4x dif­fer­ence to $c_2$ and $w_2$ re­spect­ively. Read the next sec­tion for the mile­stone swap­ping strategies.

RZ Mile­stone Swap­ping Ex­plan­a­tion #

From e50 to e400 $\rho$, you will swap from 2>3>1 for re­cov­ery to 2>1>3 (ex­plan­a­tion for this nota­tion can be found here) for push­ing $\rho$ once you get e3 away from re­cov­ery. The sim can tell you when you should per­form this swap.

For a more act­ive re­cov­ery, you can swap from 2>3>1 to 2>1>3 when you are near or are at a 0. This strategy is known as Spir­alSwap. This is ex­tremely hard and may slow down pro­gress if you are not ac­cur­ate/​fast enough.

Post-e600 $\rho$ #

Black Hole (BH) is not a nor­mal mile­stone. Once you get BH, you will get 2 new but­tons ad­ded to your the­ory, one on the bot­tom right of your equa­tion screen that looks like a black hole; and one on the top right next to your pub­lish but­ton that looks like a back ar­row. The back ar­row but­ton will re­duce $t$ by 5 and will move $\zeta$ back to where it was at that $t$. The BH but­ton will bring up the BH menu. In the BH menu you can set a value where you want BH to ac­tiv­ate re­l­at­ive to $t$ and the game will auto­mat­ic­ally ac­tiv­ate BH, or you can ac­tiv­ate it manu­ally at any time by press­ing the “Un­leash a black hole” but­ton.

When BH is un­leashed, $t$ gets set back and frozen at the last 0 it en­countered. For ex­ample, when $\zeta$ crosses 0 at $14.15t$, that 0 is saved, if you Un­leash BH after $14.15t$ and be­fore the next 0 ($21.025t$), $t$ will be locked to $14.15$ and ${\zeta }^{\prime }$ will be locked at the value it was at at $14.15t$.

Once you get Black Hole (BH), you will use it to push both $\rho$ to get to a good zero. Good zeros are zeros where ${\zeta }^{\prime }$ is higher than all other local zeros. For ex­ample, all zer­oes from $14.15t$ to $25.025t$ either have less ${\zeta }^{\prime }$ or have a lower $t$:${\zeta }^{\prime }$ ra­tio. We want as much ${\zeta }^{\prime }$ as pos­sible be­cause we can now per­man­ently max­im­ize the $\zeta$ func­tion for $\dot{\rho }$. We also want a good $t$ value for our pub­lic­a­tion.

To know which zero to use, please use the the sim. It will out­put the ex­act $t$ of the zero to use.

Al­ways set your BH ac­tiv­a­tion threshold to 0.01 above the value re­com­men­ded by the sim to en­sure that the Black Hole will cor­rectly lock to your zero. For ex­ample, if it re­com­mends t=3797.85, put your ac­tiv­a­tion threshold to 3797.86.

The op­timal pub­lic­a­tion mul­ti­plier is of­ten 5, but it is some­times higher de­pend­ing on the zero used or if you get a new $w_3$ dur­ing the pub­lic­a­tion. Check the sim to know the op­timal mul­ti­plier for your pub­lic­a­tion.

Vari­able buy­ing strategies stay the same as be­fore.

Don’t for­get to buy the $w_3$ per­man­ent up­grade after reach­ing e1000ρ! The first level of $w_3$ will not be avail­able right away, so you can buy the per­man­ent up­grade at the end of the pub.

RZ Mile­stone Route #

Class: mile­stone­_rout­ing; last_row: false;

Class: mile­stone­_rout­ing; last_row: false;

Mag­netic Fields (MF) #

MF Guide writ­ten by Mathis and Eyland­ing.

MF Over­view #

MF was re­leased on March 10th, 2025, along­side BaP. MF is the first phys­ics-in­spired of­fi­cial CT, spe­cific­ally Elec­tro­mag­net­ism.

MF has a unique mech­anic called “particle re­set”, a form of par­tial pub­lic­a­tion where you re­set $x$ to zero but in­crease $v_x$, $v_y$ and $v_z$ with the $v_i$ vari­ables you bought in-between. This mech­anic acts like a second prestige layer.

The ex­ist­ence of this mech­anic makes MF a very act­ive cus­tom the­ory at first, however it quickly slows down to longer pub­lic­a­tions where re­sets later in a pub­lic­a­tion take sev­eral hours to re­cover, of­fer­ing idle breaks.

While MF slows down quickly, reg­u­lar mile­stones sus­tain its rates, mak­ing it com­pletable in a bit over 6 months.

MF Equa­tion De­scrip­tion #

The MF equa­tions de­scribe the move­ment of a particle of con­stant mass $m$ and con­stant charge $q$ in­side a charged solen­oid of in­fin­ite length with a cur­rent $I$ and a dens­ity of turns $\delta$, cre­at­ing a mag­netic field $B$.

We con­sider a sim­u­la­tion where the particle starts at $x=0$ at $t_s=0$ with an ini­tial ve­lo­city given by the $v_i$ vari­ables. In these con­di­tions, the particle has a helix tra­ject­ory with a con­stant $x$ ve­lo­city, and an an­gu­lar ve­lo­city $\omega$. As you can see, the equa­tions for ve­lo­city in­clude $(t_s=0)$, which means here that the equa­tion only up­dates when $t_s=0$, that is when do­ing a “particle re­set”. As such, buy­ing $v_i$ vari­ables will have no ef­fect un­til you per­form a “particle re­set” where the sim­u­la­tion is re­set ($t_s$ and $x$ are set to 0), so that the ini­tial ve­lo­city can be ap­plied again.

The cur­rent is given by the last for­mula. The equa­tion is very sim­ilar to that of T5, but dif­fer­ent. Here, $I$ is capped at $a_2\times {10}^{-15}$, and $a_1$ only af­fects the growth speed of $I$.

Un­like in The­ory 5, buy­ing $a_2$ has no draw­back as it does not ap­pear in the de­nom­in­ator be­low $a_1$.

The cur­rent in­creases $B$ which it­self in­creases $\omega$.

Fi­nally, $\rho$ growth is af­fected by vari­ables $c_1$ and $c_2$, the po­s­i­tion $x$ of the particle, its an­gu­lar ve­lo­city $\omega$ and its total ve­lo­city $v$, cal­cu­lated as $\sqrt{v_x^2+v_y^2+v_z^2}$. Be­cause ${\sin }^2(\theta )+{\cos }^2(\theta )$ is al­ways 1, $v$ is in­de­pend­ent of time. $C$ is an ad­just­ment con­stant that com­pensates the para­met­ers be­ing less than one, it only changes with mile­stones by an amount in­dic­ated in-game.

MF Vari­able De­scrip­tion #

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

Class: vari­able_­de­scrip­tion; Cap­tion: Brief sum­mary of vari­able strengths of RZ; last_row: false;

MF Strategy #

Keep in mind that strategies are still un­der de­vel­op­ment and could change in the fu­ture.

When to pub­lish #

The op­timal pub­lic­a­tion mul­ti­plier slowly in­creases the later you are in the the­ory, and also de­pends on your last re­set.

It ranges from 10-50 early to 100-500 at the end of the the­ory.

Check the sim for more ac­cur­ate res­ults.

When to re­set the particle #

There is­n’t an ex­act rule yet on how of­ten you must per­form a particle re­set. A good baseline is to re­set every 1e9 $\rho$, which is every two $v_2$ levels, but it var­ies slightly from that. For ex­ample, early in the CT you want to re­set a bit more of­ten.

It is also im­port­ant to stop re­set­ting at an ap­pro­pri­ate point, you want to only re­set once after re­cov­er­ing to your pre­vi­ous pub­lic­a­tion mark.

We re­com­mend us­ing the sim to check the $v_i$ levels bought with each re­set to give you a clearer idea.

Vari­able buy­ing strats #

For vari­able buy strats, you can save a bit of time with act­ive $c_1$ buy­ing.

You can also save time by not buy­ing $a_1$ when $I$ is very close to its cap ($a_2\times {10}^{-15}$) and not buy­ing $a_2$ when $I$ is far away from its cap (which typ­ic­ally hap­pens near the end of the CT).

For more de­tails, check out the the­ory strategy sec­tion.

MF Mile­stone Route #

MF has a locked mile­stone path, like EF and FP.

Class: mile­stone­_rout­ing; last_row: false;

Class: mile­stone­_rout­ing; last_row: false;

Class: mile­stone­_rout­ing; last_row: false;

Basel Prob­lem (BaP) #

BaP Over­view #

BaP was re­leased on March 10th, 2025, along­side MF. It is based on the Basel Prob­lem, a fam­ous math­em­at­ical prob­lem solved by Euler about the con­ver­gence of the series $\sum _{n=1}^{\infty }\frac{1}{n^2}$, which con­verges to $\frac{{\pi }^2}{6}$.

BaP is an idle-friendly cus­tom the­ory (ex­cept for a bit of mile­stone swap­ping), and has sev­eral sim­il­ar­it­ies with T2.

BaP is much slower than the other CTs early, so it is bet­ter to not push it un­til your other CTs are slow enough. However, BaP holds a secret, a mile­stone un­locked at e1000$\rho$ that al­lows to com­plete the re­main­ing e200$\tau$ in un­der a week! for a total com­ple­tion time of about 5 months.

BaP Equa­tion De­scrip­tion #

The $\dot{\rho }$ equa­tion fea­tures 3 terms: $t$, $q_1$ and $r$.

$t$ is a vari­able with con­stant growth once all $\dot{t}$ vari­ables are bought.

The $q_i$ vari­ables work the same way as with T2, the bot­tom layer has a con­stant growth, then the growth of each other layer is af­fected by the value of the layer be­low, with factors be­ing the $c_i$ vari­ables (ex­cept $c_1$).

Fi­nally, we have the $r$ equa­tion. At the start of the the­ory, it is the par­tial sum of the in­verse of the squares, which con­verges. As such, there is no point to buy $c_1$ past a cer­tain point. After get­ting the first mile­stone, the $r$ equa­tion changes to be the in­verse of the re­mainder of the sum. As we omit more and more of the first terms of the sum, the re­mainder con­verges to zero, mak­ing $c_1$ use­ful again. Past early­game, we ap­prox­im­ate $\dot{r}=c_1$.

$\dot{\rho }$ is also mon­itored by the $a$ ex­po­nent, which will al­ways be less than 1, but you will be able to in­crease it with mile­stones, and, later, with a vari­able called $n$.

BaP Vari­able De­scrip­tion #

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

Class: vari­able_­de­scrip­tion; Cap­tion: Brief sum­mary of vari­able strengths of BaP; last_row: false;

BaP Strategy #

Keep in mind that strategies are still un­der de­vel­op­ment and could change in the fu­ture.

Basel Prob­lem is an idle-friendly CT ex­cept dur­ing its MS phases.

When to pub­lish #

BaP pro­gress is hard car­ried by its mile­stones, which means the best time to pub­lish can vary a lot. For $a$ mile­stones, it is bet­ter to push past them to col­lect the massive boost it gives with all the $q_1$ and $r$ you stacked wait­ing for the mile­stone. On the con­trary, for $q$ mile­stones, it is bet­ter to push for them, buy the match­ing per­man­ent up­grade and pub­lish right away, as you can en­joy the boost right away without wait­ing for the boost to climb all the way to $q_1$.

To know when to pub­lish, please check the sim. Check out the BaP Quick Pur­chase Tester for vari­able checks mid-pub­lic­a­tion.

Idle #

For idle, you auto­buy all. For more ef­fi­ciency, turn off auto­buy when your $\rho$ is around x25 away from your pub­lic­a­tion mark or the next mile­stone.

Act­ive #

BaP act­ive strategies take ad­vant­age of act­ive $c_1$ buy­ing. $c_1$ is a unique vari­able with a low cost scal­ing and that gains a massive x1024 boost every 64 levels, when ($c_1$ level) % 64 = 1.

For the strategy, you want to chase those boosts and auto­buy $c_1$ when you are close to the next boost (when the cu­mu­lat­ive cost of $c_1$ pur­chases un­til the boost is be­low x2 of other vari­ables). When you are not chas­ing a boost, you can buy $c_1$ at ($c_1$ level % 64)/​2 ra­tio to other vari­ables.

Mile­stone Swap­ping Strategy #

Mile­stone Swap­ping is pos­sible when you don’t have enough mile­stone points to buy all the mile­stones. In that case, you can swap between $a$ and $q$ mile­stones.

MS is only rel­ev­ant when you need to stack $q_i$ lay­ers, which typ­ic­ally hap­pens when you un­lock a new $q_i$ layer. The cycle goes:

Put the mile­stone point in the $q_i$ layer $\to$ wait for $q_i$ and $q_{i-1}$ to grow $\to$ put the mile­stone point back in the $a$ mile­stone.

You gen­er­ally want to start a cycle once you buy a new $c_i$ and $c_{i+1}$ which boost $q_{i-1}$ and $q_i$ re­spect­ively.

BaP Mile­stone Rout­ing Ex­plan­a­tion #

Like FI, in BaP, you can un­lock mile­stones in 2 ways:

1. by gain­ing $\rho$ like nor­mal, or
2. by pur­chas­ing the mile­stone up­grades for $a$ and $q$ in the per­man­ent up­grades tab

Buy­ing the mile­stone up­grades will not give you a mile­stone, but will in­stead in­crease the max level of the mile­stone that you pur­chased the up­grade for. For ex­ample, if you buy the $a$ perma-up­grade for lvl 1, you will per­man­ently un­lock the first lvl of the $a$ mile­stone.

While, for most mile­stones, you un­lock the per­man­ent up­grade at the same time you get the mile­stone point for it, there are 6 ex­cep­tions: $q$ mile­stone levels 3,4,5 and $a$ mile­stone levels 4,5,6 in which you un­lock the mile­stone level be­fore you un­lock the mile­stone point, mean­ing you have a va­cant mile­stone space. This cre­ates an op­por­tun­ity for mile­stone swap­ping between the $a$ and $q$ mile­stone, however, in real­ity, MS is only ap­plic­able where you un­lock a new $q$ mile­stone level, as, when you un­lock a $a$ mile­stone level, it’s gen­er­ally best to put your mile­stones into it since you have already built enough $q_i$, and these MS phases are short any­ways.

BaP Mile­stone Route #

BaP has 20 mile­stones, the most out of any of­fi­cial the­ory to this day.