According to Article 5 complex numbers should be in the form of Re^(i*theta)*PHI where PHI is a function of ep and represents infinitesimally small fluctuations. I was trying to include PHI in article 5 but PHI seems to be based on time. According to FS time is located at inf of space (the x-axis). Then the complex number Re^(i*theta) turns into a wave Re^(i*(k.r-wt)), when time (which is defined at inf and so is mathematically expressed by ep objects) is added.
This also says something about the discreteness of time as seen on the x-axis.
UPDATE 1:
The leverage mechanism in R is a process that gradually removes inf from the set of R+{inf}. According to Article 3, There should be a mechanism that pushes finite to the inf zone. The leverage mechanism indicates that removing inf from the set is also a gradual process. In other words, to remove inf one can’t simply remove an element from the set. Instead, a process is required so that inf shrinks more and more and eventually inf is completely removed at the limit point of the sequence.
Things are a bit different for complex numbers in the form of Re^(i*theta)*PHI. Since PHI represents time, it cannot be entirely removed from the x-axis, therefore the leverage mechanism in Riemann is not a gradual process to remove inf from the set. this will be explained in Article 5.
According to Article 1, x and ct in Mink are centred at different points. PHI, therefore, should be centred at inf, and not zero on the projective plane.
I think I can now start the new version of articles 4 and 5 using the OOP structure.
UPDATE 2:
Each mathematical theorem consists of a set of inputs and an output (the result which is expressed by the theorem). For instance, if conditions A, B, C, D, etc hold then Z is true. So A, B, C, D, etc are inputs and Z is the output. As a simple example, according to the Pythagorean theorem in a right triangle (condition A), hypotenuse equals s_(a^2+b^2). This theorem has one input and one output. Therefore the structure of theorems is very similar to functions in programming. Each function (theorem) may call many other functions to produce the output. The aim of the article I’m now writing is to take another step and construct math structures based on OOP which incorporates objects (which in turn includes functions as well).
UPDATE 3:
As a result, math entities such as i^2=-1 will turn into objects which possess properties. It’s no longer sufficient to say i is an entity such that i^2=-1. Instead, i should be constructed (for instance in article 4, i is a rotation in the zero zone). As discussed before i can be constructed in different ways. Such constructions cannot be achieved by functional math structures.