Cuil is a function that can be applied to a scenario.

0‽(x) = x = base scenario
1‽(x) = one level of abstractness above the base scenario
and so on in the form
N‽(x) = N levels of abstractness above the base scenario; where the limit of N is based upon N definitions given in the scenario.

We can define the Cuil level for the whole numbers in the following way: 0 - The event can possibly occur in the unchanged system (when the system is not specified, it is our world by default)
N - The event can possibly occur in the system following at least N abstract symbolic changes to the system.
So, the cuil value of an event is a lower bound for the amount of changes necessary for it to be possible.

The fractional part of a Cuil is the inverse of the highest probability that it will occur in the system after the number of changes specified in the number's integer part. If the event is almost certain to happen after N abstractions, the fractional part will be almost zero. If the event is possible, but very unlikely, then the fractional value will be nearly 1. As the value approaches a whole number, the possibility that the event will occur approaches zero. When it hits the number, it becomes impossible and another abstraction is needed.

One of the largest benefits of this over cuils as distance is that we now know that 4 cuils in a 3 cuil system will always be 7 cuils from our world.
Previously, this was not always true because one change could revert another. Now, since we know that all positive changes must make the system less consistent with our world, we no longer have that problem.

From SomewhatDisturbed @ reddit

Noting that the Cuil can be used to measure the levels of abstraction away from a situation, I would posit that they can also be used to measure the differences between any two chosen scenarios. For example, we could define the Cuil difference as the lowest value of X+Y, such that with scenarios A, B, we can find any such abstract scenario C such that X = Cuil(A,C) and Y = Cuil(B,C). More simply expressed, if we take any two scenarios, the Cuil distance is the smallest sum of any two values that lead to an identical abstract scenario from each of these two base scenarios.

For example, suppose the two original scenarios were as follows:

A: A duck crosses a road.

B: A car is driving down a road toward a chicken.

Suppose we examine this scenario:

C: A chicken imagines a car driving across a road.

Now A is approximately 2 Cuils from C (with the intervening step involving a car on the road, and possibly inclusion of the chicken), and B is approximately 1 Cuil from C (being a single abstract step from B). As such, the Cuil distance between scenarios A and B is 2+1 = 3. This result could be further refined by someone finding a suitable abstract scenario C that has the sum of Cuil(A,C) and Cuil(B,C) being less than three.

I submit this to reddyenumber4 for consideration in his Cuil theory.

We will now extend the above approach by giving the changes a direction - towards, or away from standard reality. We will define the + and the - abstractions.

A plus abstraction is an abstract symbolic change to the system that increases the entropy, chaos, and unpredictability in a system relative to system 0 (the real world by default).

A minus abstraction is an abstract symbolic change to the system that increases its predictability and consistency with the real world.

The fractional Cuil values are still defined by the probability of the event in the system, but they are obviously opposite in the negative cuils.

Let's say that we have an event that is 25% probable in our world, but it would be certain if our world was more self-consistent. Then, its cuil value is -0.75. Let's say that we have an event that is impossible in our world, but would be 47% probable after 2 optimal abstractions towards self consistency. Then, its Cuil value would be -2.53
As our world is often chaotic, there are a lot of events that have a negative Cuil value.

While not a firmly established branch of the theory, suggestions have been put forth that hyper- and para- reality can be measured in Cuils as well as the meta-reality of positive integer Cuils. Specifically, that negative Cuil denotes hyper-reality or „realer than real“ situations such as you might find in an Ice Cube song or a Brian DePalma movie, and imaginary values of Cuil serve to describe negating, distracting, or recursive aspects of the mesh in question. Some tentative examples are :

0‽ – You ask me for a hamburger, I give you a hamburger.
-1‽ – You ask me for a hamburger, I ask how you plan to pay for that hamburger.
-2‽ – You ask me for a hamburger, I give you the raw materials to cook and combine, indicating the kitchen.
-3‽ – You ask me for a hamburger, I hand you a saw and lead you to a steer . . .
. . . also:
i‽ – You ask yourself for a hamburger.
2i ‽ – You punch yourself before you can ask for a hamburger.
-4i‽ – You use a pocket knife to carve meat from your thigh, grind it and make a burger patty. You leave the patty out in the rain and walk away.

Under the above proposed framework imaginary Cuil provides a potential answer to the problem of researcher recursion – specifically that a theorist's consideration, without direct interaction, could increase the imaginary Cuil value of the subject without increasing the real abstraction of the subject. Critics have answered that this is a mathematical evasion of the recursion problem and consider it to invalidate the concept of imaginary Cuil.

This area of research is still in its infancy and no single system of analysis has yet been adopted by a majority of theorists.

Not defined yet.
Keep in mind that:
negative*negative=positive
positive*negative=negative
positive*positive=positive
a*b=b*a
a*b≠0 if a and b are nonzero.
a*b=0 if a or b are zero.
1 is the identity.

An interesting and supportive point to this mathematic is that multiple hyper-real events in combination produce an irreal (positive-Cuil) event, for example: a thug collecting on your parents' mob debt (-2‽) on the same day you're served a divorce notice (-1‽) produces an vaguely manic euphoria (2‽), wherein you begin to develop an epic musical based on your problems. The converse is the controversial but hypothetically testable situation of irreality coinciding with hyper-reality: on the same day you find out your liver is becoming sentient (3‽) you are also told your entire face is cancerous (-2‽), producing a deep and unbearable lucidity (-6‽). Some researchers have contested the „default hyper-reality“ conclusions of this algebra.

An interesting possibility has been raised by separating and separately multiplying the fractional Cuil in an interaction. Let's say the situation is an intense and unencouraged cramp in your arch muscles (-1.85‽) while a family of ferrets assaults your pantry (3.15‽). (-1 * 3 * (0.85 * 0.15)) This produces an intense realization you should be more active, say by reworking you back door (-2.87‽). These fractional multiplications curb the exponential growth of some Cuil calculations and recognize the fact that as things get weirder to us they also become more intensely real. Simultaneously the fractional mathematics will not take us to a 0‽ point without an unadulterated 0‽ event, merely make the experience more intense and reinforce the baseline reality even if ony by the example of the exceptions.

Trigonometric identities and Cuils mix in a variety of ways. The complexities go up the more advanced mathematics you apply to it. For example, what is illustrated below are two types of trigonometric identities pertaining to Cuils.

1. Basic sin²+cos²
sin²(θ)+cos²(θ) = 1

1. Basic sin²+cos² in Cuils
sin²(‽)+cos²(‽) = 1.618

2. Law of Cosines
a² = b² + c² - 2bc cos(α)
b² = a² + c² - 2ac cos(β)
c² = a² + b² - 2ab cos(γ)

2. Law of Cosines in Cuils
a² = b² + c² - 2bc cos(‽)
b² = a² + c² - 2ac cos(‽)
c² = a² + b² - 2ab cos(‽)

Notice how these translate into formulas and direct numbers. The reason for this is unknown. If this could be calculated into philosophical terms, perhaps the formulas would not be situations. They could be translated into simple words which progressively change across the dictionary sporadically. It is unknown what the Cuils rationalize out to be. Further research is necessary.

Suppose you seek to find the derivative of a Cuil. dy/dx(x‽), where x is any number from -∞ to +∞, is illustrated by the following boxes of rules.

dy/dx(1‽) = 1‽^-1 = 0.1‽ = You smell a hamburger, and you think about buying one.
dy/dx(2‽) = 2‽^-2 = 0.25‽ = You smell a hamburger, and you suddenly think about a raccoon.

Therefore, as the Cuil increases, the exponent increases downwards. This is only true when 1 ≤ ‽ ≤ 9, as illustrated below.

dy/dx(10‽) = 10‽^1/10 = 1.2589‽ = You ask for a hamburger, but I am actually Portuguese, and do not understand you.

As Cuil reaches 10 and continues to increase, the exponent becomes 1/x where x = the number of Cuils mentioned in the problem. Finding derivatives of negative Cuils, at least on the theoretical level, is shown in the box below.

-1‽ = You ask me for a hamburger, I ask how you plan to pay for that hamburger.
dy/dx(-1‽) = -1‽¹⁰ = -1‽ = You ask me for a hamburger, I ask how you plan to pay for that hamburger.
dy/dx(-2‽) = -2‽²⁰ = -1,048,576‽ = You ask me for a hamburger. I teach you the history of farming from 20,000 BC to the present day and demand that you profess a region of study. You become a Peruvian farmer and live in the Andes for the rest of your life. At the age of 76, you open a hamburger stand in the small village of Nuevo Chichen Itza.

As Cuil goes to the negatives, the exponent is the Cuil multiplied by -10. This allows for „realer than real“ reality shifts, as mentioned in the section concerning negative Cuils. It is as of yet unknown what happens if a Cuil coefficient exceeds 1.618×10⁷¹, or what situation it will yield.

Further details about the derivatives, intergals, and anti-derivatives of Cuils have yet to be defined by science.

• Define operations on Cuils.
• Complete definition of the derivatives of Cuils.
• Try to define the changes to keep them from being arbitrarily complex.

One proposed theory for measurement of Cuil used 3D Spherical Polar graphing techniques from Calculus 2 with a little bit of axis redefinition to combine current understanding of Cuil with the Grand Mesh Theory of the philosophy page.