Those diagrams are pretty simple, so the best way to illustrate them is to give an example:

Two cars start from separate locations, drive in separate directions.

Is it possible for the cars to collide?

Here we read the diagram as from bottom to the top, starting with "separate
locations",

then proceeding to "separate directions" (as two different functions of motion)

resulting in the same location.

Whether two cars can collide depends on their direction.

Two cars that have the same direction on a flat surface starting at different
locations,

will not collide. This can be written as

Another way to write the diagram above is [1,0,1].

This system of two moving cars, satisfy and
Havox symbols.

Often we just say "A" and "X".

Havox diagrams can be used to systematically describe all possibilities for any
correlation between systems.

For example, the scientific method relies on for a given output, a correct
model should give the same result as in experiments.

This is the of the Havox symbols.

But if one do one of these experiment and by coincidence a wrong model gives
the right prediction?

This is the of the Havox symbols.

When we can exclude out all other possibility, then we can be sure that our
model is correct.

When we say "differences" it can mean a lot of stuff.

Havox diagrams makes it possible to show exactly which meaning we use.

The diagrams are called "Havox" because of the 5 ways to tell situation A is
different from B.

Havox diagrams has two axis with discrete steps.

The center of it indicate that two values or states are equal.

The diagram can have multiple heights, the lowest is the input and the heighest
is output.

Here is a list of all possible values for a simple function that takes input
and produces an output:

Written as binary vectors:

"H" [1,1,1]

"A" [1,1,0]

"V" [0,1,1]

"O" [0,1,0]

"X" [1,0,1]

"Y^-1" [1,0,0]

"Y" [0,0,1]

"I" [0,0,0]

The first 5 possibilities are clear distinctions,

the following 2 are mathematical technical differences,

and in the last possibility, there are no differences.

Here are some examples:

f(x) = x^{2} ()

Since negative input produces the same output as positive input through the
same function, we draw two joining paths.

f(x) = ±√( x ) ()

Since the function returns two possible values using same input and with the
same function, we draw two splitting paths.

"The temperature on the sun affects the earth's gravitational field." ()

This is completly irrelevant, therefore we draw two paths that has nothing in
common.

It means, the input is different, the function is different and the result is
different.

You can stack multiple functions at top of each other:

f(g(x)) is a two-step recursive function.

First the input x is handled through g, then the output from g is handles
through f which produces the output.

Since the only positions in the diagram are either equal or non-equal, we don't
have to worry about values disappearing left or right.

If you combine any "Havox" differences (the first 5 that are clear distinct)
with each other, the number of possible ways to write the differences

increased with the Fibonnaci sequence.

If you have **n** functions, then you use following formula (φ is golden
ratio):

5^{-1/2} × (φ^{n+4} - (1-φ)^{n+4})

`x=1`

(phi()^(x()+4)-(1-phi())^(x()+4))/sqrt(5)