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 Decem­ber 15th, there are 7 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, and Riemann Zeta Func­tion by Prop (RZ). The the­or­ies will be ab­bre­vi­ated as WSP, SL, EF, CSR2, FI, FP and RZ 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 $\tau$. WSP, SL, CSR2, FI and RZ 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 $\tau$ = \(\rho^{0.3}\).

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

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.

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 #

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

\(\gamma = \frac{\rho_3}{\sqrt[\rho_3]{\rho_3!}}\)

\(\dot{\rho_2} = a_1a_2a_3^{-ln{\rho_3}}\)

\(\dot{\rho_3} = b_1^{1.04}b_2^{1.04}\)

\(a_3 = 1.96\)

The first line is the main part of the equa­tion. We want to max­im­ize \(\dot{\rho_1}\) to in­crease $\tau$. 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:

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)

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, 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 #

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

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

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 $\tau$. 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 #

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

\(G(t) = g_r+g_i\)

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

\(\dot{q} = q_1q_2\)

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

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

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 #

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 #

\(\dot{\rho} = q_1^{1.15}q_2q\)

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

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

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 $\dot{\rho }$ 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 #

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:

g(x) Equa­tions #

$\lambda$ Equa­tions #

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 \(q_1 * q_2\). 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:

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 $\dot{\rho }$ 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 Ex­plain­a­tion #

In FI, you can un­lock mile­stones in 2 ways:

1. by gain­ing \(\rho\) like nor­mal, or
2. 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 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.

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:

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 $\rho$. 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 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=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, auto­buy­ing is also fine.

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 #

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:

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!

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 \(\zeta’\) 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 \(\zeta’\) is higher than all other local zeros. For ex­ample, all zer­oes from \(14.15t\) to \(25.025t\) either have less \(\zeta’\) or have a lower \(t\):\(\zeta’\) ra­tio. We want as much \(\zeta’\) 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 #