The­or­ies 5-8

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

This guide is cur­rently un­der­go­ing change. Keep in mind, strategies may change.

Gradu­ation rout­ing #

Re­mem­ber to fol­low our rout­ing ad­vice from the in­tro­duc­tion to gradu­ation.

Stu­dent rout­ing with R9 #

All rout­ing fol­lows the stu­dent cal­cu­lator (by Niedzielan, AfuroZamurai, and Milla) and star cal­cu­lator (by Eaux Ta­cous#1021). When you are not push­ing \(f(t)\) you should al­ways have the 9th re­search op­tion maxed (after The­ory 8). When push­ing \(f(t)\), you should be R9 seap­ing (be­low).

There is also the the­ory sim­u­lator by XLII, which works both be­low max mile­stones and after max mile­stones for all the­or­ies.

How to push F(t) with R9 swap­ping #

Mem­or­ize your stu­dent dis­tri­bu­tions with and without 10/​20/​30 R9 stu­dents. Use the stu­dent cal­cu­lator if needed. You will com­monly see people refer to this as R9 seap­ing as a long held name of the strategy.

1. Wait till \(f(t)\) stops grow­ing with stu­dents in R9 push­ing \(\tau\).
2. Start ac­cel (prefer­ably keep it between prestiges).
3. Po­ten­tially sit here to stack t for big­ger \(\varphi_2\) when you have stu­dents in \(\varphi_2\). Only do this when you are near a gradu­ation mark. This is not use­ful if you will not swap into \(\varphi_2\).
4. Re­spec all 10/​20/​30 stu­dents from R9.
5. Wait for the auto­prestige to prestige and swap back stu­dents to R9.
6. Re­peat.

This method al­lows you to push \(f(t)\) with al­most no loss of R9 up­time or push­ing power. This is harder with fewer levels of R9 but still helps if you get used to it.

R9 auto­prestige ex­pres­sion #

You can find the auto­prestige used for R9 Seap­ing here: Equa­tion. If you don’t have this ex­pres­sion, then you will have to manu­ally prestige each seap.

Ref­er­ence R9 Swap­ping Auto­prestige Ex­plan­a­tion

The­ory 1 #

You will not touch this the­ory un­til after ee14k. Once you be­gin push­ing T1 after ee14k, be­gin us­ing the The­ory Sim and the The­ory Sim Guide to give the best strategy and mul­ti­plier for the next pub­lic­a­tion.

The­ory 2 #

This the­ory will be used as overnight un­til 1e350 Tau where it will not be touched un­til after ee14k. See our earlier guide for an over­view for the­ory 2.

The­ory 3 #

See our earlier guide for an over­view for the­ory 3.

The­ory 4 #

See our earlier guide for an over­view for the­ory 4.

The­ory 5 (40σ / 9k) #

Vari­able over­view #

\(q_1\) & \(q_2\): Simple mul­ti­pli­ers that dir­ectly af­fect \(\rho\) pro­duc­tion. \(q_2\) is a doub­ling while \(q_1\) is not.

\(q\): The crux of T5 is to grow this value as fast as pos­sible, while in­creas­ing its max­imum value.

\(c_1\): In­creases the speed that \(q\) will ap­proach its limit. You need enough levels of \(c_1\) to al­low \(q\) to reach its limit, once \(q\) has reached its cap \(c_1\) has no ad­di­tional be­ne­fit un­til more \(c_2\) is pur­chased.

\(c_2\): Doubles the limit of \(q\) and halves the ef­fect of \(c_1\). Needed to bal­ance 2 parts of the equa­tion ap­pear­ing twice: \(c_1/​c_2\) and \(c_3^{1.1}-q/​c_2\). If you buy too much \(c_2\), it will make \(q\) growth ef­fect­ively noth­ing as \(c_1/​c_2\) ap­proaches \(0\). However, you still need to buy \(c_2\) when \(q\) ap­proaches \(c_2*c_3^{1.1}\) be­cause \(c_3^{1.1}-q/​c_2\) ap­proaches \(0\) mak­ing \(q=c_2*c_3^{1.1}\) the max­imum value of \(q\).

\(c_3\): In­creases the limit of \(q\) by \(2^{1+m/​20}\), where \(m\) is the num­ber of mile­stones, by in­creas­ing what \(q/​c_2\) frac­tion can reach. It does not have the prob­lems of \(c_2\) as lower­ing your \(\dot{q}\), mak­ing it an al­ways auto-bought vari­able.

T5 strategy #

The­ory 5 be­ne­fits the most from act­ive play and a lot of at­ten­tion mak­ing it the strongest the­ory un­til the very late game due to a very low mul­ti­plier de­cay rate. Here is what is known about op­timal mul­ti­plier: \(3\) un­til \(e25\); \(6\) to \(10\) dur­ing mid to late game. Pub­lish­ing at higher mul­tiples is not drastic­ally less ef­fi­cient and al­lows for slightly less act­ive play. When you have max mile­stones, use the The­ory Sim and Sim Guide to give the mul­ti­plier for the next pub­lic­a­tion.

Act­ive

Run­ning the act­ive strats, with some modi­fic­a­tions, will help you get this the­ory to \(e30\) eas­ily, but it will take some time. A step-by-step on how to pro­gress the the­ory is de­tailed be­low.

Be­fore e30, you should re­peat this after every pub­lic­a­tion:

1. Buy everything ex­cept \(c_2\).
2. Once \(q\) growth re­duces, \(c_2\) levels can then be pur­chased in­di­vidu­ally. Only buy when \(c_2\) is \(e1\) lower than your cur­rent \(\rho\).
3. When you are within \(e10\) of your last pub­lic­a­tion, auto­buy all but \(q_1\) & \(c_1\). You should then manu­ally buy \(q_1\) and \(c_1\) when it costs \(e1\) lower than the \(q_2\) doub­ling. Buy \(c_1\) only when \(q\) is not capped.
4. Re­peat un­til \(e25\). At \(e25\), push for \(e30\) with 0/​1/​0 mile­stone and start \(x6-10\) mul­ti­pli­ers.

After auto­buy at e30, you should re­peat this after every pub­lic­a­tion:

1. x1 (or x10 when above e200) buy \(c_2\) manu­ally and auto­buy the rest un­til within ~e10 of your pre­vi­ous pub­lic­a­tion. Your graph should re­semble a lin­ear func­tion on the graph.
2. Around your last pub mark within ~\(e10\), start auto­buy­ing \(c_2\) and stop auto­buy­ing \(c_1\) & \(q_1\). At this point:
   1. buy \(q_1\) up to \(15\%\) of the cost of the next doub­ling pur­chase (\(2^x\) pur­chase),
   2. and buy \(c_1\) after you pur­chase \(c_2\) up to \(e1\) lower than \(q\).
3. Once you reach the de­sired pub­lic­a­tion point, pub­lish.
4. Re­peat this for stonks.

Com­ment­ary

No com­ment­ary

T5 will al­ways give its best res­ults from act­ive play. However, after step 3, you can still get good res­ults while auto­buy­ing \(q_1\) and manu­ally pur­chas­ing \(c_1\) every 10-15min. This makes the the­ory slightly less act­ive and easier to deal with.

Warn­ing: Do not overnight this the­ory. It has ter­rible de­cay after passing a good pub­lic­a­tion mark and will not give good res­ults. T5i is only vi­able very late/​en­dgame.

T5 mile­stone route #

Ad­di­tional in­form­a­tion

Pur­chase \(c_2\) when \(1.5q > c_2*c_3^{m_3}\). \(m_3\) is the num­ber of mile­stone 3.

\(q\) be­gins to slow down when you reach \(2q > c_2*c_3^{m_3}\).

Strategy con­struc­ted by: Snaeky, Marks, Baldy, and Nerdy

The­ory 6 (45σ / 10k) #

T6 strategy #

This the­ory has the low­est de­cay of all the the­or­ies. It will be second place to T5 un­til about e750 and is the only the­ory that can get to \(>e1100τ\). You should overnight this after you get your T2 to \(e350+\). This is the best idle the­ory. Video of T6 at En­dgame

The op­timal pub­lic­a­tion mul­ti­plier is still un­known but em­pir­ic­ally seems to be about \(7\)-\(12\). Once all mile­stones, dis­able \(c_3\) \(c_4\) and auto­buy rest. For manual auto­buy \(q_2\), \(r_2\), \(c_2\), and \(c_5\) then manual buy rest with \(c_3\) and \(c_4\) still dis­abled. For idle/​auto, you are go­ing to just turn off \(c_3\) and \(c_4\).

T6 mile­stone route #

The­ory 7 (50σ / 11k) #

T7 over­view #

T7 can be sum­mar­ized as a max­im­iz­a­tion prob­lem : given a sur­face in 3-di­men­sional space, you want to find its highest alti­tude by mov­ing along the sur­face, al­ways in the dir­ec­tion of steep­est as­cent (that’s ba­sic­ally a gradi­ent as­cent). The func­tion \(g(x,y)\) can be seen as a sur­face in \(\mathbb{R}^{3}\) (con­sid­er­ing the set of points \((x,y,g(x,y))\), see at­tached im­age). \((\rho_1,\rho_2,g(\rho_1,\rho_2))\) is a point on this sur­face. Our goal is to max­im­ize \(g(\rho_1,\rho_2)\), i.e. to find \((\rho_1,\rho_2)\) that max­im­ize \(g(\rho_1,\rho_2)\). No­tice that the func­tion \(g\) is un­boun­ded, i.e. you can’t find a proper max­imum (we say that the max­im­iz­a­tion prob­lem is ill-con­di­tioned); so one way to max­im­ize \(g(\rho_1,\rho_2)\) is to move \((\rho_1,\rho_2)\) to­wards the dir­ec­tion of steep­est as­cent. This is what is pre­cisely done by set­ting \(\dot{\mathbf{\rho}}\) (which is the dir­ec­tion the point \(\mathbf{\rho}=(\rho_1,\rho_2)\) will move to­ward) to \(\nabla g(\rho_1,\rho_2)\) (i.e. the gradi­ent of \(g\) eval­u­ated at \((\rho_1,\rho_2)\), which gives the dir­ec­tion of steep­est as­cent of \(g\) at the point \((\rho_1,\rho_2)\).

This is the graph of the func­tion \(g\), taken after the first four mile­stones have been un­locked (Note: here, coef­fi­cients like \(c_1,c_2\ldots\) are ig­nored. The ef­fect of those coef­fi­cients is simply mak­ing the graph steeper in \(x\) or \(y\) dir­ec­tion, de­pend­ing on the value of each coef).

T7 strategy #

The op­timal pub­lic­a­tion mul­ti­plier is \(4\)-\(6\). You will swap from 0/​1/​1 → 0/​0/​2 at near the 2/​3 mark of your next mile­stone. The strategy for manual buy be­fore 4 mile­stones is to only manual buy \(q_1\) and \(c_1\) cheap (e1 less \(\rho\)) and the rest full auto. After mile­stone 5, turn it on full auto­buy for idle. For act­ive, you will fol­low the strat de­scribed in the the­ory sim guide or watch the video be­low.

T7 mile­stone route #

The­ory 8 skip­ping #

T8 skip is sig­ni­fic­antly faster than buy­ing T8 right away (don’t worry, you will still buy it, just at 14k, not 12k). T8 is very slow un­til you get to about \(e60\) (it took the sim 16 hours to get that far into the the­ory without R9). We highly re­com­mend buy­ing t8 for the achieve­ment, then selling it right away and us­ing those stu­dents for \(\varphi\). You will need about e1350 \(\tau\) in or­der to get R9 (\(ee14k\)) without T8, which will help you get through T8 faster than be­fore. You will need to buy T8 again to get R9, and you should start R9 right away after that The cur­rent re­com­mend­a­tion for your \(\tau\) dis­tri­bu­tion for T8 skip is as fol­lows:

If your num­bers are dif­fer­ent, that is fine. You just need to reach 1e1350 \(\Pi\tau\) to get to R9 at 14k.

The­ory 8 (55σ / 12k) #

T8 strategy #

The op­timal pub­lic­a­tion mul­ti­plier is 2.5-5 de­pend­ing on how close you are to the next mile­stone. This the­ory is ex­tremely slow at the start which is why we skip un­til we ob­tain R9. It is also the only one with a \(1e20\) mile­stone step. It will speed up once you hit \(1e60\) and even faster at \(1e80\) and \(1e100\) etc. un­til ~\(e250\)-\(e300\). The worst part is the \(1e50\)-\(1e60\) grind. The grind to \(1e60\) will take a good bit of time but is faster with R9.

At the start, manual buy pri­or­it­izes \(c_2\) then \(c_1\) then rest. Once you get to 0/​0/​0/​2, pri­or­it­ize \(c_2\) and \(c_5\) then \(c_1\) then the rest. Once at 2/​0/​3/​0, you will pri­or­it­ize \(c_2\) and \(c_4\) then \(c_1\) then the rest after. This con­tin­ues to max at 2/​3/​3/​3.

T8 mile­stone route #

Note: the 2/​0/​0/​0 → 0/​0/​0/​2 swap hap­pens at e52 tau.