Theories

**Precursory note:** This is sloppy, incomplete, and much in need of formatting. What I have here & will eventually add here are two very different things. To anyone who edits this, my goal is to have all the "p" formulas expressed as simple algebraic functions so that people who don't understand calculus (or other advanced math concepts) can still understand what's happening in each equation, as well as just providing other details that aren't provided elsewhere. Feel free to add to incomplete sections, as I don't have enough time to finish this in one sitting

Theories are a sort of mini-equation unlocked by graduating students. They are representative of the "tau" variable in the main equation

Each theory #n has an assigned variable "taun", which is representative of the maximum value of the theory's formula. The formula for tau is obtained by adding 1 to each "taun", and then multiplying all 9 numbers together

## Theory #1 - Recurrence Relations

**Cost:** 20 sigma

**Minimum f(t) to unlock:** ee5000

The Recurrence Relations theory is based around a fairly simple recurrence relation, where each term can be expressed in terms of previous terms. There are two main variables in this theory - *p* & tickspeed

The tickspeed represents how much the n in *pn* increases each second, & it is always given by the formula *q1q2*

The *p* variable is the equivalent of f(t) in this theory, and it is equal to the displayed formula minus the total amount spent on upgrades

**Base formula:** pn+1 = pn + *c1c2*

* In other words, *p1p2* is added to your *p* value every tick
* If *t* is your total tick count, *pt* = *c1c2t*

**Milestone Additions:**

* Increase *c1* exponent by 0.05 (up to 0.15)
* Multiply *c1c2* by 1 + ln(pn)/100
* Add the term

c
3

p

n
−
1

0.2

{\displaystyle c\_3p\_{n-1}^{0.2}}
* Add the term

c
4

p

n
−
2

0.3

{\displaystyle c\_4p\_{n-2}^{0.3}}

If you have all these and are buying all upgrades, the contribution of each term will be around 10% for c1c2, 30% for c3 and 60% for c4.

## Theory #2 - Differential Calculus

**Cost:** 5 sigma

**Minimum f(t) to unlock:** ee6000

The Differential Calculus theory adds the notation

o
˙

{\displaystyle \dot{o}}

, which is to be interpreted as the rate of change of *o* over one second.

**Base formula:**

p
˙

{\displaystyle \dot{p}}

 = *q1* + *r1*

where

q
1

˙

=
a

q
2

{\displaystyle \dot{q\_1} = a q\_2}

and

r
1

˙

=
b

r
2

{\displaystyle \dot{r\_1} = b r\_2}

, *a* and *b* depending on upgrade level

* *p
˙

/

10

{\displaystyle \dot{p}/10}* is added to your *p* value every tick
* *qi* and *ri* work in a similar way to *x1-x8* in the main screen

**Milestone Additions:**

* Increase *q1* exponent by 0.05 (up to 0.15)
* Increase *r1* exponent by 0.05 (up to 0.15)
* Unlock

q
3

{\displaystyle q\_3}

, then

q
4

{\displaystyle q\_4}
* Unlock

r
3

{\displaystyle r\_3}

, then

r
4

{\displaystyle r\_4}

## Theory #3 - Linear Algebra

**Cost:** 5 sigma

**Minimum f(t) to unlock:** ee7000

Here we have a simple matrix multiplication. However, we now have two currencies: *p1* and *p2*. Tau depends solely on *p1*

**Base formulas (written explicitly):**

p
˙

1

=

c

11

b
1

c

12

b
2

{\displaystyle \dot{p}\_1 = c\_{11}b\_1 + c\_{12}b\_2}

p
˙

2

=

c

21

b
1

c

22

b
2

{\displaystyle \dot{p}\_2 = c\_{21}b\_1 + c\_{22}b\_2}

* b
1

{\displaystyle b\_1}

,

c

11

{\displaystyle c\_{11} }

 and

c

21

{\displaystyle c\_{21} }

 are bought with

p
1

{\displaystyle p\_1}

.
* b
2

{\displaystyle b\_2}

,

c

12

{\displaystyle c\_{12} }

 and

c

22

{\displaystyle c\_{22} }

 are bought with

p
2

{\displaystyle p\_2}

.

**Milestone Additions:**

* Add a new dimension
* Increase *b1* exponent by 0.05 (up to 0.1)
* Increase *b2* exponent by 0.05 (up to 0.1)

## Theory #4 - Polynomials

**Cost:** 5 sigma

**Minimum f(t) to unlock:** ee8000

## Theory #9 - Convergence Testing

This theory is unique in that it needs 40 students and doesn't provide any tangible benefits in the form of tau (only story progression). Therefore, it should be refunded after completion.

It consists of 7 stages called "lemmas". They are like smaller theories, but with noticeably odd formulas and no publications. Since rho growth isn't guaranteed and you might get stuck, a reset button is available for each lemma.

Instead of a full walkthrough (boring!) , here are some interesting insights:

### Lemma 1:

The sin(q)+½ term is only there to introduce waviness and averages to ½ over time, so c₂ and c₃ are equally strong. With good timing, you can get negative rho for a moment.

#### Lemma 2:

q is a parasite that grows over time and with each currently active upgrade. Active play, be quick and efficient.

#### Lemma 3:

Odd values of c₁ make the first term negative.

#### Lemma 4:

q is a parasite again, c₃ needs to keep up.

#### Lemma 6:

c₃-c₄ can get arbitrarily close to 0, but never exactly.