Hi Fredrick,
Brouwer's work is a good illustration of the three theory, as is your acknowledgement of 'nothing' in a trio which was originally perceived as a duo by Aristotle. The three theory would explain the two extremes as the two things that are opposites, (high-low), and the other, unseen force that regulates them (middle), also known as the golden mean in many disciplines.
The gender neutral bacteria you mentioned in the binary number system helps to add credence to the theory that there are three, but one may only be theoretical , even possible. In another post, I explained how when structures are broken down into their smallest components, they become an atom with a proton, neutron and an electron. This represents the + charge and the - charge as opposites, and with the neutron being the golden mean, in the smallest structure that everything is made from. The magnetism vs gravity part of my theory states that everything is magnetic, because the atom has electrical charge ,(+, neutral, -), and it is therefore is connected to electricity and magnetism as we know how well they are intertwined.
I had a thought about the binary system with the numbers 1 and 0, it could be with any number such as 4 and 7. Does this make the 1 and 0 less credible because it can be replaced with any other numbers or does it strengthen it, as it encompassed all the numbers and not just two? Someone said that they have a theory of 1/0 but you can't divide or divide by 0. Could it be that another number could divide by another number and the result shows every number from 1-9, like when you type all the numbers from 1-9 and add the same numbers together you get all the numbers back in the answer, but just jumbled up.
I also think that the the ultimate structure must contain duality, but should not be limited by it. This ties in with the theory because if you look at a tripod standing, it has more strength than something with two stands, it needs the other stand to support it, but it may not be obvious or seen.
I enjoyed reading 'In Search of A Cyclops'. I see that you also don't share the belief in singularities, I reason that if they did exist, then it would suck in everything in the universe into it, according to Einstein's theory of gravity. My theory states that singularities cannot exist, but it includes black holes and its counterpart, the white hole. It says that black and white holes are linked in space-time by heavy distortion, not by a total singularity. Even Professor Hawking has in recent years come to the conclusion that singularities don't exist. You have said that if a singularity cannot exist, then people would stop looking for a unified field. Why would they not try to find a unified field that doesn't include singularities, like mine? A unified field is something that unites everything, a theory of everything, that encompasses the whole universe and everything within it and evens its own structure.
In paragraph 8, you mention the prime numbers in table 1. Can you see that 1, 2 and 3 are the only trio of numbers next to one another for the whole numerical system? The three theory describes this as being that you can see the three items, but they are all seperate and with these 3 numbers you can work out the rest of the primes from them. If you can see one of the three, one extreme, then you must be able to see another one, as it is the other extreme. It follows that if you can see two, then you have to be able to see all three, because they are all linked in the ratio 2:1, as you rightly stated 'When two out of three parts of a sum are known, the remainder can be found also'.
You also mention that religion doesn't fit on the modern scientific framework, but they theory of everything must include religion as well, as it is apparent and acknowledged in the universe. The TOE should prove the existence of God, but what if, as religion claims, God made the universe and as the Bible says that he exists outside of time? It is discussed on this link-
http://www.beliefnet.com/story/148/story_14839_1.html.
Just as the reasoning that singularities cannot exist because they would break the rules of numerous theories, as the three theory states, 'extremely dense places do exist, but just not to the point where all the Laws of Nature collapse. This is because they are controlled by the Laws, not a controller of them'.
Your theory states that it is either 4 or 5 with a pyramid shape. What kind of things in nature and mathematics have structures like that? How does 4 or 5 explain phenomena in the universe and things like consciousness?
I have also found information that may allow double and even triple universes into a singular torus shape universe with the 'n-fold torus'.