Ashay Dharwadker - Proof of the Four Color Theorem and the Theory Of Everything

In 2000, Ashay Dharwadker discovered a purely topological and algebraic proof of the celebrated four color conjecture. To this day, it is the only proof of the four color theorem that has ever been published in a university level textbook. Dharwadker introduces the proof as follows.
From A New Proof of The Four Color Theorem by Ashay Dharwadker
"FOUR COLOR THEOREM.  For any subdivision of the plane into non overlapping regions, it is always possible to mark each of the regions with one of the numbers  0, 1, 2, 3,  in such a way that no two adjacent regions receive the same number.

STEPS OF THE PROOF:  We shall outline the strategy of the new proof given in this paper.  In section I on MAP COLORING, we define maps on the sphere and their proper coloring.  For purposes of proper coloring it is equivalent to consider maps on the plane and furthermore, only maps which have exactly three edges meeting at each vertex.  Lemma 1 proves the six color theorem using Euler’s formula, showing that any map on the plane may be properly colored by using at most six colors.  We may then make the following basic definitions. 

  • Define N to be the minimal number of colors required to properly color any map from the class of all maps on the plane.
  • Based on the definition of N, select a specific map m(N) on the plane which requires no fewer than N colors to be properly colored.
  • Based on the definition of the map m(N), select a proper coloring of the regions of the map m(N) using the N colors 0,1, ..., N-1.
The whole proof works with the fixed number N, the fixed map m(N) and the fixed proper coloring of the regions of the map m(N).  In section II we define STEINER SYSTEMS and prove Tits’ inequality and its consequence that if a Steiner system S(N+1,2N,6N) exists, then  N cannot exceed 4.  Now the goal is to demonstrate the existence of such  a  Steiner  system. In section III we define EILENBERG MODULES.  The regions of the map m(N) are partitioned into disjoint, non empty equivalence classes 0,1, ..., N-1 according to the color they receive.  This set is given the structure of the cyclic group ZN={0,1, ..., N-1} under addition modulo N.  We regard ZN as an Eilenberg module for the symmetric group S3 on three letters and consider the split extension ZN]S3 corresponding to the trivial representation of S3.  By section IV on HALL MATCHINGS we are able to choose a common system of coset representatives for the left and right cosets of S3 in the full symmetric group on |ZN]S3| letters.  For each such common representative and for each ordered pair of elements of S3, in section V on RIEMANN SURFACES we establish a certain action of the two-element cyclic group on twelve copies of the partitioned map m(N) by using the twenty-fourth root function of the sheets of the complex plane.  Using this action, section VI gives the details of the MAIN CONSTRUCTION.  The 6N elements of ZN]S3 are regarded as the set of points and lemma 23 builds the blocks of 2N points with every set of N+1 points contained in a unique block.  This constructs a Steiner system S(N+1,2N,6N) which implies by Tits’ inequality that N cannot exceed 4, completing the proof."

Then, in 2008, Ashay Dharwadker discovered the raison d'être for all the marvellous symmetry and structure in the proof. The proof described, in a very precise way, the complete Standard Model of particle physics!

From Grand Unification of the Standard Model with Quantum Gravity by Ashay Dharwadker
"We show that the mathematical proof of the four color theorem directly implies the existence of the standard model, together with quantum gravity, in its physical interpretation. Conversely, the experimentally observable standard model and quantum gravity show that nature applies the mathematical proof of the four color theorem, at the most fundamental level. We preserve all the established working theories of physics: Quantum Mechanics, Special and General Relativity, Quantum Electrodynamics (QED), the Electroweak model and Quantum Chromodynamics (QCD). We build upon these theories, unifying all of them with Einstein's law of gravity. Quantum gravity is a direct and unavoidable consequence of the theory. The main construction of the Steiner system in the proof of the four color theorem already defines the gravitational fields of all the particles of the standard model. Our first goal is to construct all the particles constituting the classic standard model, in exact agreement with 't Hooft's table . We are able to predict the exact mass of the Higgs particle and the CP violation and mixing angle of weak interactions. Our second goal is to construct the gauge groups and explicitly calculate the gauge coupling constants of the force fields. We show how the gauge groups are embedded in a sequence along the cosmological timeline in the grand unification. Finally, we calculate the mass ratios of the particles of the standard model. Thus, the mathematical proof of the four color theorem shows that the grand unification of the standard model with quantum gravity is complete, and rules out the possibility of finding any other kinds of particles."
References

University of Barcelona, Mathematical Physics Archives, 08-201  , 2008