The proof of the four colour theorem has an intricate mathematical construction that involves both topology and algebra, whose physical interpretation gives an exact one-to-one correspondence with all the properties of the standard model of particle physics together with gravitation and the standard model of cosmology, also describing the nature of dark matter and dark energy in the universe. So, we truly have a Theory of Everything for physics.
In 2008 we made two concrete predictions which were experimentally verified. The first was the prediction of the mass of the Higgs boson at 125.992 ~ 126 GeV. This was experimentally verified by the large hadron collider experiment in 2012. The second prediction was an explicit calculation of Einsteinís cosmological constant that gives the exact composition of the universe as consisting of 4.5% ordinary baryonic matter, 22.6% dark matter and the rest 72.9% dark energy. This experimental verification was done by the Wilkinson microwave anisotropy probe in 2010. So, together with these two predictions we have established our theory.
Let me give you a brief idea of how the physical interpretation of the four colour theorem works. Every particle in the universe is described by its Schrödinger wave equation whose solution lies on the boundary of a disk in the complex plane. We select a fixed four colour map inside that disk and we call such a disk a Schrödinger disk.
Now, we take 24 Schrödinger disks and form a 24 sheeted Riemann surface. This is the topology. The algebra comes in when the regions of the Schrödinger disks are labelled by the elements of a certain group algebra that occurs in the proof of the four colour theorem. For instance, the colours form the fundamental electric charges of all the particles and the other algebraic elements give the various gauge charges. Now, we open up this 24 sheeted Riemann surface to give a 2 sheeted Riemann surface with 12 Schrödinger disks on the upper sheet and 12 Schrödinger disks on the lower sheet. This 2 sheeted Riemann surface is called a particle frame.
With every particle in the universe we associate a particle frame. So, we have a vector bundle over space-time. This is what physicists call the gauge. All the different kinds of fermions are defined by selecting Schrödinger disks on the particle frame. The spin 1 bosons are defined as rays on the particle frame which are intersections of 4 Schrödinger disks. The upper sheet contains all the particles and the lower sheet contains their corresponding antiparticles.
The Higgs boson is the centerpiece of the mathematical construction. It is the branch point of the Riemann surface. The Higgs particle and antiparticle, which are essentially the same, spin 0 bosons, are identified at the center of the particle frame and the physical interpretation is that this forms what is called a Cooper pair of Higgs particle and antiparticle.
Let me give you an idea of how we obtain the formula for calculating the mass of the Higgs boson. The Higgs Cooper pair undergoes Bose-Einstein condensation and attains the minimum energy level possible. Then the mass of the Higgs boson is given simply as the sum of the masses of the spin 1 bosons divided by 2 to account for the Cooper pair formation.
Let me now give you an idea of how we calculate the cosmological constant. Letís ignore for a moment the contribution of mass of the massive bosons and consider only the fermions. For a particular fermion, there are 24 Schrödinger disks on the particle frame. Exactly one disk carries its electric charge. Because of the superposition, the other 23 Schrödinger disks of the particle frame carry a replicated gravitational charge. The 5 remaining disks of the half surface that contains the fermion constitute dark matter. So, this is not going to be visible but it will exert a gravitational force. The 18 remaining disks contribute dark energy corresponding to this fermion. Since this is true of every fermion in the universe at a particular time, this gives us the ratios for ordinary baryonic matter as 1/24, dark matter as 5/24 and dark energy as 18/24. That is basically the rough idea of how to calculate the cosmological constant. We need to make some corrections to account for the massive vector bosons and the massive Higgs particle.
Towards the end of his life Albert Einstein wrote a book called The Meaning of Relativity, in which he asked two questions which he could not answer during his lifetime. The first was to find a unified theory for the classic standard model and gravitation. The second was to explicitly calculate the cosmological constant. We have answered both the questions.