Guest Post: Vladimir Khachatryan, The Higgs Mass From The Four-Colour Theorem
    By Tommaso Dorigo | July 25th 2010 02:26 AM | 74 comments | Print | E-mail | Track Comments
    About Tommaso

    I am an experimental particle physicist working with the CMS experiment at CERN. In my spare time I play chess, abuse the piano, and aim my dobson...

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    Ashay Dharwadker
    is the founder and director of the Institute of Mathematics, Gurgaon, India.
    He is interested in fundamental research in mathematics, particularly in algebra, topology, graph theory and their applications to computer science and high energy physics. Based upon the new proof of the four color theorem, he has developed a grand unified theory for the Standard Model and gravitation. In particular, this leads to a mathematically precise prediction of the Higgs boson mass.
    Vladimir Khachatryan is a member of the Institute of Mathematics, Gurgaon, India and a PhD student at the Department of Physics and Astronomy, State University of New York, Stony Brook, USA. With an academic background in astrophysics, he currently works in the field of high energy nuclear theory, in an attempt to explain the experimental data revealed at RHIC and LHC. He is also interested in particle physics, especially in problems related to the Standard Model and beyond. In this guest post, Vladimir explains the reasoning behind the theory Ashay and him have developed. Beware: the mathematics is not for everybody; still, I believe this article was well worth the attention of the more knowledgeable among you here...

    We show that the mathematical proof of the famous four color theorem yields a perfect interpretation of the Standard Model of particle physics. One of the main applications of the proof is that we are able to calculate and predict the mass of the Standard Model Higgs Boson as described by Veltman and 't Hooft. The Higgs boson mass turns out to be MH0~ 126 GeV (A. Dharwadker and V. Khachatryan, Higgs Boson Mass predicted by the Four Color Theorem (2009), arXiv:0912.5189 [gen-ph] ). Let us describe the way in which we obtained this remarkable result, which is likely to be verified experimentally with the discovery the Higgs Boson at Tevatron and/or LHC in the near future, given the recent promising developments. The current experimental constraint on the existence of the Higgs boson is found to be in the energy realm from ~ 114 to 160 GeV. There is also a small possibility of its existence in the 170-185 GeV range, nevertheless, the lower energies are far more probable. In this brief note, we will not go into the complicated mathematical structure of our approach, instead we will attempt to show only the main steps which brought us to the conclusion that MH0 should be about 126 GeV.

    The four color theorem arises as a fundamental problem in topology when the surface of a sphere or plane is partitioned into finitely many contiguous regions called a map. Two regions in the map are considered adjacent if they share a whole segment of their boundaries in common. The theorem states that the regions of any such map can always be colored by using at most four different colors, so that no two adjacent regions have the same color. It can be stated formally as follows:

    For any subdivision of the plane or the surface of a sphere into finitely many non-overlapping regions, it is always possible to mark each of the regions with one of the colors 0, 1, 2, 3 in such a way that no two adjacent regions receive the same color.

    The theorem was first conjectured by Möbius in 1840, later by DeMorgan and the Guthrie brothers in 1852, and by Cayley again in 1878. It remained one of the most celebrated and long outstanding conjectures in mathematics after many mathematicians tried to prove it for over a century. The conjecture was finally verified, using an extensive computer search for potential
    counter-examples by Appel and Haken in the 1970's. However, this computer verification cannot be checked by humans, even in principle. In 2000, the theorem was proved mathematically by Dharwadker using algebraic and topological methods. It is a standard mathematical proof and can be verified by hand (A. Dharwadker, A New Proof of the Four Colour Theorem (2000), That mathematical proof of the four color theorem has a rich topological and algebraic structure and its most important and fundamental application is a physical interpretation of the proof which directly implies the existence of the Standard Model of elementary particles (A. Dharwadker, Grand Unification of the Standard Model with Quantum Gravity (2008), .

    The steps of the proof enable us to construct a specific type of Riemann surface (which is a mathematical construction based on the complex plane) and particle frame which forms the gauge. Then on this basis it is possible to specify well-defined rules in order to match the Standard Model in a one-to-one correspondence with the topological and algebraic structure of the particle frame. This correspondence is exact - it only allows the particles and force fields to have the observable properties of the Standard Model, giving us a Grand Unified Theory.

    In order to exhibit how we obtain the one-to-one correspondence of the particle frame with the Standard Model, we start from the fact that the quantum-mechanical behavior of a free particle is completely described by its wave function, which is a solution of the relativistic Schrödinger wave equation. We show initially that this wave equation is sufficient to describe all the particles of the Standard Model on the particle frame. At each point (x, y, z, t) of space-time, the value of the wave function corresponds to a point on the boundary of a disc D centered at the origin of the complex plane C. The disc D may be oriented in two ways, clockwise or counter-clockwise depending on whether we select the normal vector according to the right-hand or left-hand rule. We call D a Schrödinger disc. According to the proof of the four color theorem, we select a specific map m(4) inside the disc. The regions of the map are partitioned into four equivalence classes that form the cyclic group {0, 1, 2, 3} under addition modulo 4, according to the color 0 (blue), 1 (yellow), 2 (green) or 3 (red) each region receives. In figure 1 below we show the map m(4) inside the Schrödinger disc D.

    We now construct the Riemann surface which consists of 24 copies of oriented Schrödinger discs containing the map m(4) as shown below in figure 2. This Riemann surface is orientable since every orientation of a disc is carried over to the disc next to it.

    We define all the particles of the Standard Model by selecting certain regions (or their topological intersections) on this Riemann surface, in which context it is called a particle frame. Particle frames associated with space-time points constitute a vector bundle in mathematical terminology, and a section of the vector bundle, i.e. a particle frame at a space-time point, is called a gauge in the physics terminology. Thus, physical symmetries associated with sets of particles defined on a particle frame correspond to gauge transformations. The particle frame provides the general mathematical framework from which all the particles of the Standard Model will be defined, together with their basic physical properties: spin, charge and mass. Here we wish to describe our rules for matching the Standard Model in a one-to-one correspondence with the topological and algebraic structure of the particle frame.

    • The Fermion Selection Rule. Distinct particle frames with fermions defined on them cannot be superposed at a point in space-time because of the Pauli exclusion principle. A fermion-type particle will be selected from the particle frame as follows. First select a disc out of the 24 discs and then select a region of the map on that selected disc. In this way,
      there are two types of fermions and each type comes in three generations. Each generation consists of one lepton doublet and one quark doublet, as in the Standard Model.

    • The Boson Selection Rule. Many distinct particle frames with bosons defined on them can be superposed at a point in space-time, since the Pauli exclusion principle does not apply to the bosons. A boson-type particle will be selected from the particle frame as follows. First select a pair of fermion-type particles from the 24 discs (with selected regions of the
      same color respectively) such that the two discs have an intersecting boundary (a ray on the particle frame). Then select another pair of fermion-type particles with selected regions of the same color as before, however, in such a way that the corresponding ray on the particle frame is distinct. Thus, we may select the boson-type particle by choosing a pair of rays on the particle frame with a particular color. In particular, two pairs of fermion-type particles that define a boson are interpreted as creation and annihilation operators during interactions in which the boson is exchanged.

    • The Higgs Selection Rule. A Higgs-type particle is a scalar boson, i.e., it does not select a preferred direction in space like a vector boson. It is selected as the intersection of all 24 discs of the particle frame. This is the central point in our construction of the Riemann surface, and this selection of the Higgs-type particle is unique. The origins of the
      upper and lower sheets of the Riemann surface are interpreted as forming a Cooper pair and the Higgs particle undergoes Bose condensation, plunging into the lowest energy state possible.

    • The Spin Rule. The particle frame consists of four half-surfaces: the upper half of the upper sheet; the lower half of the upper sheet; the upper half of the lower sheet; the lower half of the lower sheet. Given a particle as a selection S of the intersection of a set of discs or as a pair of rays, count the number n of half-surfaces of the particle frame that intersect with a whole segment of S. Define s = n/2 to be the spin of a particle. We use this rule to explicitly calculate the spin of all the fermions and bosons on the particle frame.

    • The Electric Charge Rule. We first associate each color with a unique absolute value of the electric charge according to the scheme: 0 (blue)  →  0, 1 (yellow)  →  1/3, 2 (green)  → 2/3 and 3 (red)  →  1. The labeling scheme for the signs has the following representation: the upper half of the upper sheet has a + sign; the lower half of the upper sheet has a - sign; the upper half of the lower sheet has a - sign; the lower half of the lower sheet has a + sign. Given a particle as a selection S of the intersection of a set of discs or of a pair of rays, assign a signed electric charge to a particle according to this scheme. This is defined to be the electric charge of the particle. We use this rule to calculate the electric charge of all the fermions and bosons on the particle frame.

    The other rules may be briefly mentioned here: the weak isospin rule, the strong (color) charge rule, the mass rule, the equivalence rule, the antiparticle rule, the helicity rule and the CP transformation rule. We stress the importance of the mass rule by means of which we assign the
    rest mass of each particle of the Standard Model matched onto the particle frame. Finally, incorporating all the above rules we have:

    • The Standard Model Completion Rule. If all the particle frames corresponding to all particles in the universe were to be superimposed (hypothetically, of course) then the fermions and bosons should fit together perfectly according to the above rules, forming the complete Standard Model on the particle frame. The discs 1, ..., 24 of the particle frame represent the Schrödinger discs of the 24 distinct spin 1/2 fermions νe, νμ, ντ, t, c, u,
      e, μ, τ, b, s, dνeνμντtcueμτbsd in the Standard Model, respecting all the above rules as shown in figure 3(a) below. There cannot be any other fermions in our mathematical construction of the Standard Model. Also, each of the 24 pairs of rays of the particle frame represent four Schrödinger discs of a unique boson in the Standard Model: the photon γ (spin 1), the three vector bosons W+, W-, Z0 (spin 1), the gluon AS (spin 1) and the graviton g (spin 2), respecting all the above rules as shown in the figure 3(b)
      below. The branch point in the center of the particle frame represents the Higgs boson H0 (spin 0). There cannot be any other bosons in our mathematical construction of the Standard Model.

    We briefly note that it is also possible to obtain the values of the Weinberg and Cabibbo angles on the particle frame. The Weinberg angle ΘW is a parameter that gives a relationship between the masses of the W+, W-, Z0 bosons as well as the ratio of the weak Z0 mediated interaction, called its mixing. In our model, the components of the weak Z0 field mix with the components of the weak W+, W- fields and the angle subtended by the mixing Schrödinger discs on the particle frame is exactly π/6 radians or 30 degrees, as shown in figure 4 below.

    Hence, ΘW = 30 degrees on the particle frame. This is in good agreement with the SLAC experiment, which estimates sin2ΘW = 0.2397, i.e. ΘW = 29.3137 degrees (this is a "running" value, depending on the momentum at which it is measured, with a significance of 6 standard deviations). The Weinberg angle is a measure of the strength of the weak force on the particle frame. For the Cabibbo angle we calculate the value θC = (4/9)ΘW ~ 13.33 degrees, which is proportional to the strength of the weak force on the particle frame.

    Finally, we show how the mass of the Standard Model Higgs Boson H0 can be calculated on the particle frame by using the above rules. By the Higgs Selection Rule, the Higgs particle is given as the intersection of all 24 discs of the particle frame. We may regard the Higgs particle as
    the intersection of the discs 1,...,12 of the upper sheet (the origin of the upper sheet), and the Higgs antiparticle as the intersection of the discs 13,...,24 of the lower sheet (the origin of the lower sheet). However, the Higgs particle and antiparticle are identified as the branch point
    of the Riemann surface. These 24 discs together represent the Schrödinger discs corresponding to the Higgs field and the blue branch point at the center represents the Higgs particle on the particle frame as shown below in figure 5.

    Thus, by the Higgs Selection Rule, the Higgs particle is a scalar boson. Furthermore, by the Spin Rule, the spin of the Higgs boson is 0; by the Electric Charge Rule, its electric charge is 0, by the Weak Isospin Rule, its weak isospin is 0; and by the Strong Charge Rule, its strong charge
    is neutral with Nc = 1.

    The Higgs boson has not been observed yet but it is an inevitable consequence of the Higgs-Kibble mechanism, whereby it attributes mass to all particles of the Standard Model, including itself. The Higgs Selection Rule and the Mass Rule give the following mechanism for obtaining the mass value of H0. Since the Higgs particle/antiparticle will be identified (as a Cooper pair), their combined mass would then be the sum of the masses of all the bosons defined on the particle frame. However, note that only the rest masses of the three gauge vector bosons W+, W- and Z0 contribute to the sum because all the other bosons are massless. Thus, we can have all types of bosons superposed on a single particle frame, and the single Cooper pair of the Higgs particle/antiparticle must be able to attribute energy/rest mass to all types of bosons on this particle frame, by the Higgs-Kibble mechanism.

    The particle frames of the bosons can be superposed at a point in space-time because they follow the Bose-Einstein statistics. Hence, this Cooper pair must have at least enough energy to attribute the sum of the rest masses of all types of bosons defined on the particle frame. On the other hand, the most important property of Bose condensation is that the Cooper pair of the Higgs particle/antiparticle must have minimum energy, so it can have at most the energy required to attribute the sum of the rest masses of all types of bosons defined on the particle frame. This must be the lowest energy state possible for the Higgs boson when it undergoes Bose condensation. Summarizing all these facts, we obtain the formula for the Higgs boson mass defined on the particle frame as follows:

    The Higgs boson mass value MH0 ~ 126 GeV results from the currently best known experimental mass values of the three gauge vector bosons from the Particle Data Group. In conclusion, we would like to emphasize that we have calculated the mass of the Higgs boson from the above formula by taking into account the whole topological and algebraic structure of the Riemann surface, the particle frame and the proof of the four color theorem.


    Any maths/physics student can immediately tell that this article is just a bullshit (although quite an elaborate one). So I must assume it was intented for a layman who should probably be shocked by it.

    Also Dharwadker's "proof" of the Four-color theorem is not peer-reviewed (kinda strange for such an outstanding and famous problem, isn't it?) and quite obviously fake as discussed many times on the internet. Although again, it is quite elaborate piece of bullshit using correct (but unreferenced) lemmas from group theory and topology, probably again to confuse anyone trying to understand the proof.

    The quality of this blog decays exponentially these days...

    Daniel de França MTd2
    Well, that theorem was proved a few times already:
    Whether theorem is valid doesn't tell anything about whether any particular proof is valid.

    Daniel de França MTd2
    You are complaining too much!
    This specific proof enables to construct the above model for the prediction of the Higgs Boson mass.

    Daniel de França MTd2
    Close to the value of Higgs mass range (around 123 GeV) that can be used to fix the muon anomaly without evoking new physics. It would imply a correction on hadronic cross sections, which is quite likely given the new measures of the proton radius:
    Entertaining crackpottery - Garrett Lisi on steroids. ;-)

    You forgot to say that the 125.992 GeV mass of the Higgs can be derived not only from the four-color theorem but also from the legendary Sabre Dance by one of the authors, Mr Khachaturian

    To me this looks like a completely arbitrary construction without any physical justification which was designed to capture some basic aspects of the SM. Why should anyone believe this has anything to do with reality?

    I'm not sure what Tommasso was thinking here =/

    Weak, Tomasso, weak.

    Yeah, this is a new low....

    This theory could be correct, but there is a little flaw that nobody, except me, has seen so far. You can’t use the four colour theorem. That’s a common mistake done by many of you. With the three plus one colour theorem is enough. Indeed, in QCD there is a problem dubbed confinement problem of quarks, and you should show that the sum of the three quark’s colours is white, then, you could show why all particles observed are white. To kill two bird with one stone, man. I’m sorry, but it’s virtually impossible that some of you had solved the Higgs boson problem, only the confinement problem, at most.
    BTW, Tommaso, my aunt has also developed a new Theory of Universe. The Best in the World. But it's necessary a little of propaganda, could help with this?

    Tommaso is advertising crackpots ? Hmm. Although I'm not a particle physicist to judge the theory of everything part, but the claimed "proof" of the 4 color theorem not published anywhere except a website is enough to suspect a crackpot.

    Why is this here ? =/

    If this were the first crackpot, advertised on the pages of this blog, one would have more reason for surprise.

    My own favorite among all crazy theories is the strand model from At least it reproduces the quark model and the three gauge groups. And it has no supersymmetry.

    an important paper about the 4-color theorem was recently published on the ViXra server.
    I think it could be very important for this research.

    Thank you !

    Don't be so hard on Tommaso. Isn't it cute how reasonable experimentalists can be totally lost when it comes to theory? In fact many unscrupulous theorists take advantage of them all the time. It's much worst when that affects experimental analyses rather than a simple blog.

    Those "unscrupulous" theorists asked Tommaso for his opinion about the possibility of Higgs Boson to be at mass region close to 126 GeV, and nothing else. But I am quite sure that experimentalists will not use any proof of the four color theorem or our constructed t-Riemann surface, in order to detect the Higgs Boson at Tevatron or LHC. Their experimental methods are much more powerful and the professionalism of experimentalists is very high, whereby any kind of prediction will not affect the final result of their experimental analyses.

    This is my opinion dear Anonymous friend !

    125,992 GeV?
    It is a boson. Has everybody forgotten that? A virtual particle. It's mass is fluctuating and depends more of the circumstances of the experiment, in fact it should be possible to push the mass-result a bit depending on the experimental energies. The environment is important for the result.

    A prediction is not working like this. 126 GeV? Where is the spread?

    Hey Ulla, just a note. Any particle has a width, inversely proportional to their lifetime; not just bosons; and the Higgs may be as real as any other particle -or as virtual as any other. I would not blame the above construction for having not an answer for what the Higgs lifetime is; besides, the Higgs lifetime is a known function of its mass in the standard model.

    As for the spread, it is not too hard to compute the propagation of uncertainties from the W and Z masses. If MH=MW+MZ/2, then sigma(MH)=sqrt(sigma(MW)^2+sigma(MZ)^2/4)= sigma(MW). So the uncertainty is of a few tens of MeV. This means that if we find the Higgs at 125 or 127 GeV this particular theory is off the table...

    People, get a life. I have explained it several times here, that I like the fact that conventional and "non-conventional" research may have a way to interact in forums and blogs, and as Jacques reminds us, I have hosted much more extreme ideas here in the past.

    The article in question is an arxiv preprint. If it passed Cornell's standards I do not understand why I should be stricter.

    'Cause you don't get as many eprints each day?

    Well said Tommaso !

    Yes, there is arXiv and then Tommaso and then the general physics moderators and then the community as a filter - any number of 'woo' articles get booted off of here for lacking a method to the ideas.    Some ideas are just interesting to talk and think about and the audience is smart (no idea what the mainstream media will do with anything, though).
    Maybe... But more troublesome to me is that I unfortunately get a lot of fools here, apparently.

    And let me add a more general comment here, since I am being pulled into it.

    There appears to exist a whole class of colleagues, amateurs, or simple readers who believe they should teach me how to run this blog, what to write in it, what to allow and what to discard, whether I should censor some comments, whether it is acceptable to discuss rumors or scientifically despicable, etcetera, etcetera.

    Maybe this is in part due to my openness in discussing my choices. Or maybe it is due to a misguided will to participate in the success of this site. Or maybe it is just driven by more personal feelings.

    This is what I summarized above by saying "get a life". Maybe now it is more clear what I mean.

    The "new proof" of the 4-color theorem is completely bogus as should be obvious even to non-mathematicians. It's simple: any real proof should somehow make non-trivial use of topology which is completely absent in this case. Even the fact that we consider planar graph (and, say, not the graphs on a Klein bottle) isn't being used materially. Worse, the "proof" is not wrong by mistake, it's an intentional deceit, a lot of irrelevant constructions thrown in to conceal the fact that there was never any beginning of an idea of a proof. All in all, quite disgusting.

    Since making my comment yesterday in response to Marcos (in a subsequent thread), I took another look at the four-color "proof", and I have to say that I agree with Yury's assessment. The manuscript does not contain even an attempt at a proof of the four-color theorem. It (apparently) does contain an algebraic construction of the Steiner system S(5,8,24), but no connection is made with planar graphs, much less with colorings. It is impossible to believe that the author is unaware of this. I hope his coauthor will take the time to explain to us what's going on here.

    Quoting from the post: "Thus, by the Higgs Selection Rule, the Higgs particle is a scalar boson. Furthermore, by the Spin Rule, the spin of the Higgs boson is 0; by the Electric Charge Rule, its electric charge is 0, by the Weak Isospin Rule, its weak isospin is 0; and by the Strong Charge Rule, its strong charge
    is neutral with Nc = 1."

    Unless the authors mean something different than the conventional meaning, this is not the SM Higgs boson. The Higgs as invoked in the standard model is in the doublet representation of SU(2), and thus has weak isospin = -1/2, not 0.

    Different than the conventional meaning of "weak isospin" I mean. Sorry :)

    Both are correct depending on how you interpret the Higgs particle. If the Higgs particle is interpreted as the branch point of the Riemann surface then it has weak isospin T=0 according to the weak isospin rule (see our paper). However, the Higgs mechanism for spotaneous symmetry breaking in the SM requires a isodoublet of Higgs fields Phi=(phi^(+),phi^(0)) consisting of a positively charged and a neutral spin 0 particle. The positively charged spin 0 particle must intersect one half-surface with a + sign (m_1=1) and the neutral spin 0 particle must intersect zero half-surfaces (m_2=0). Now the Weak Isospin rule tells us that T=(+m_1/2)+(-m_2/2)=(+1/2)+(-0/2)=1/2 (Reference: Gauge Theory of Weak Interactions By Walter Greiner, Berndt Müller).

    By the Weak Isospin Rule you can see how to define the weak isospins of fermions and bosons on the particle frame. You can see also by the Spin Rule how the spins are defined. On the particle frame select, for example, u quark, or t quark, or Z boson, and you will see that.

    What do you mean by "depending on how you interpret the Higgs particle." The term Higgs boson has a very specific meaning: calling it a Higgs boson means that it represents a remnant physical degree of freedom after a spontaneous breakdown of a gauge symmetry. That is the definition. If it is to be associated with the breakdown of electroweak symmetry, it had better actually interact under SU(2)_L. If you find T_3=0 then it is either a singlet and cannot break SU(2), or the original multiplet was in the adjoint, which would break SU(2)xU(1)->U(1)xU(1), which is NOT the standard model.

    This brings up another point in that between your paper and your reply you don't seem to be consistent in your definition of weak isospin as either referring to T or the T_3 eigenvalue. The W^-, for example, has T=1, T_3=-1, so you're apparently giving T_3 in your paper, whereas in your reply above you treat the Higgs doublet as a whole, which would give T, not T_3 (since the T_3 eigenvalues by definition are different for each member of the representation).

    Whatever scalar boson you have here with T_3 = 0, it is not the standard model Higgs, which has T=1/2, T_3=-1/2.

    Look at the Riemann surface which is the particle frame. On it we define the Higgs boson topologically as the branch point in the center, by the Higgs selection rule. The thing is that this branch point can be defined as the intersection of many different sets of Schrodinger discs (not necessarily all 24). So, to account for the different T_3 values we can simply select different sets of Schrodinger discs and take their intersection to obtain the same branch point but with different values of T_3. For the correct weak isospin T_3 value in the SM:

    1) Select a negatively charged spin 0 particle (e.g. the intersection of Schrodinger discs 7 and 9 with the red region selected).
    2) Select a neutral spin 0 particle (Goldstone boson) (e.g. the intersection of all remaining Schrodinger discs 1, 2, 3, 4, 5, 6, 8, 10, 11, 12, 13, 14, 15, 16 17, 18, 19, 20, 21, 22, 23, 24 with the blue region selected).

    Then (1) intersects one half-surface with a - sign (m_2=1) and (2) intersects zero half-surfaces with a + sign (m_1=0). The combined intersection of (1) and (2) is again the branch point of our particle frame which is the topological definition of the Higgs particle. Now the Weak Isospin rule tells us that T=(+m_1/2)+(-m_2/2)=(+0/2)+(-1/2)=-1/2 as in the SM.

    Of course, this complication is only in the case of the Higgs boson. For all other fermions and bosons the calculation of T_3 is straightforward and matches the SM.

    Thanks for the question Brian !

    Well, there is an error in the proof of the four colour theorem in the very first lemma (lemma 1). It is stated that 2E = 3V, E being the number of edges and V being the number of vertices in a 3-regular (flat) graph. This is true for the graph you get if you project a tetrahedron on the plane, It gives 4 vertices and 6 edges. However, any 3-regular extension with one extra vertex will give you 5 vertices and 9 edges.
    It is not the end of the proof because another proof of this lemma can certainly quite easily be given, but such a glaring error certainly doesn't give a lot of confidence in the rest.

    Well, the proof is garbage but that particular statement is correct - they only consider graphs with vertices of degree 3.

    OMG, they do indeed. I really thought they meant triangulated, and should have read better. It is of course not enough to consider graphs with vertices of degree 3. Could have stopped reading there.

    You can always replace a vertex of degree n with a small n-gon. This replaces the vertex of degree n with n vertices of degree 3. Do this to all vertices of degree greater than 3. If the resulting map can be four-colored, then so can the original map (by simply shrinking the n-gons to points). So there is no problem with restricting to graphs whose vertices are all of degree 3. I should add that a couple years ago I attempted to read this paper and was not able to verify that it contained a proof.

    That statement is true for any 3-regular graph. Every vertex has degree three, therefore there are 3V neighbours. Obviously this counts every edge twice, so that indeed 3V = 2E. Also it shows that you can't get any 3-regular (or odd-regular, more generally) graph on odd number of vertices, so that your counter-example is bogus -- just try to draw it ;-)

    Of course, this isn't to say, that the "proof" of the Four-color theorem is correct. As I already said, it's quite elaborately built and there are many correct Lemmas with correct proofs, which are there just to confuse the reader. But as a whole it's just a bullshit.

    You're right. Wasted time enough I would say.

    Marco, I am sorry that you just wasted your time here.

    That is not what I meant Vladimir. In mathematics you always have the possibility to check the proof. I decided to give it a shot and form an opinion about it. If you do that you should be careful. As is evident from my wrong comments I wasn't. So yes, I think I wasted my time and that of others.

    Amateur Astronomer
    Proof of four color theorem is really not necessary to derive the prediction of Higgs mass in this article. Notice that the standard model was mapped into the Riemann surface with parameters enabled and constraints applied to limit the surface such that it expresses the complete standard model and absolutely nothing else. The four color theorem could be replaced by a different device such as a topographical construction with a conditional constraint on degrees of freedom, dimensionality, and number of intersections allowed at a single line. That is something more abstract than four color theorem. The use of four color theorem is an interesting attempt to attach a physical significance to the topology and eliminate a conditional logic sequence. It is the mapping of the standard model that regulates the topological construction, not the topology controlling the construction of the standard model. So the results can be obtained without the four color theorem, but with a conditional statement to qualify the Riemann construction. Maybe it’s just more fun with colors.
    No Jerry, it is not fun with colors. The steps of the proof of the four color theorem are necessary for the construction of that specific Riemann surface (called t-Riemann surface in our designations). For example, when we define the Mass Rule on the particle frame, we do that by using Lemma 15 and 17 of the proof (see our paper). Or when we define the Electric Charge Rule on the particle frame, we do that according to the labeling scheme of the t-Riemann surface (see the papaer again). What do I mean ? I mean that on the basis of this proof of the four color theorem it is possible to specify well-defined rules in order to match the Standard Model with the topological and algebraic structure of the particle frame.

    Well, if someone can do that by another way, I will only appreciate such an effort. It would be interesting. But we
    accomplished that by our way, namely using the steps of the proof of the four color theorem.

    Amateur Astronomer
    Thanks for the explanation Vladimir. I don’t object to using the 4 color theorem or to the way it was used. The prediction of the Higgs mass is more important to me, and that importance seems to be getting lost in the discussion. I intended to say that the prediction method looks more robust than is suggested in the comments about color theorem. Taking mathematics as a larger system containing physics as a subset, any of the physical models can be represented in a more generalized abstraction of topology by applying a conditional logic to the construction. Then Gödel's Incompleteness Theorem can’t be used as an argument against the conclusions. Some of the other comments are claiming incompleteness in one way or another. The physical interpretation is preferred when it is available. Most of the other comments are not accepting the physical explanations. Using some of the Lemmas for mapping is interesting but it would be more interesting if all of the Lemma were necessary and sufficient. I got the impression, maybe wrong, that some Lemmas were used and others were not. It looked more like a cut and paste operation instead of a generator algebra. You probably know that in mathematics a Lemma is a subsidiary proposition that is assumed to be true for the purpose of proving another proposition. Then your proof is based on assumptions unless the logic is completed far enough to verify the Lemmas. Maybe the discovery of Higgs at the predicted mass would provide the verification. Until then the conditional logic I suggested can replace the assumptions and remove most of the objections to your article. For the opponents I predict that the methods of approach are not likely to succeed, even if all of the arguments are found to be true, unless the Higgs is not found at the predicted mass energy.
    Jerry thanks for your answer as well !

    Ok ! I think you are correct, saying that "....unless the Higgs is not found at the predicted mass energy". But if it is not the case then it will not be the end of the world for us.

    Anyway, with you and with some people here it would be a good discussion, however, I am sorry I will have not time to do comments anymore. I am also busy as You all, dear friends.

    Good luck to You !

    Dear Friends, especially those of you who consider our work as nonsense. Please be aware that our prediction of
    the Higgs boson mass is not a prophecy of prophets. We haven't seen this mass value = 126 GeV in our dreams.
    People, who do research, usually at some stage of their work can come to some conclusions, after summarizing
    all their previously obtained results. Our case is one of these typical cases, nothing more. The conclusion about the Higgs Boson mass value can be correct or not which future experiments will show. But there is no need to blame Tommaso for the showing of the above note in this guest post.

    In this connection I wish to say the following. Please go ahead and develop your own approach, whereby you can do your calculations of the value of the Higgs boson mass. I would appreciate such your efforts much !

    Sincerely Yours

    P.S. If You read the book "G. ’t Hooft, In search of the ultimate building blocks, Cambridge University Press (1997)", then many things in our work will be understandable for You.

    Vladimir, can you please comment on how you (or your collaborator) managed to prove the 4-color theorem without anything resembling investigation of properties of planar graphs?

    Yuri, the four color theorem in our paper is stated in terms of maps on the plane and not in terms of planar graphs.

    Yes, I noticed that. Doesn't change the fact that no property of planar graphs (or of maps on the plane) is being used in the proof.

    I'm actually extremely curious - do you really fail to realize that this proof of the 4-color theorem is bullshit? And if you don't what are your reasons for publishing that kind of stuff? I'd imagine continuing with any conventional academic career with this on you resume would be next to impossible. So what's the plan here?

    The proof is based on the specific planar map m(N) whose definition is quite sophisticated. Both the universal and existential quantifiers are used to select the map and its coloring from all possible planar maps and all their possible colorings. So, obviously all the required properties of planar graphs in the context of the theorem are being used in this definition itself. Get a life, Yury, (and don't give up your day job!).

    Touched by your worrying about my well-being, but I'm doing pretty good, thank you.

    As for the rest of your comment, I can only assume that you are kidding. "Existential quantifiers" is a nice touch.
    At any rate, I don't think that sorry piece of writing deserves any further discussion. I addressed my question to Vladimir, as I'd like to know his reasons. I'm not interested in discussing the matter with anyone else.

    I have a specific mathematical objection to the four-color proof in Section 2.1 of the manuscript. On page 8, the authors define 0, 1, ..., N-1 to be the subsets of the set of regions that are colored with a particular one of the colors, 0, 1, ..., N-1. (In contrast to the authors, I use italics rather than underlining to represent subsets - I couldn't get underlining to work.) They then make the following statement:

    Identify the set {0, 1, ...,N−1} with the underlying set of the N-element cyclic group ZN under addition modulo N.

    This is the point where the proof goes really wrong. By making this identification, the authors throw away all information about the structure of the map and its coloring. The fact that the color labels can be cyclically permuted says nothing about whether the coloring was valid, whether it used the minimal number of colors, or even whether there was a map in the first place. Perhaps the authors don't really mean what they seem to be saying? Unfortunately, they do. I have read through the rest of the proof carefully, and nothing about the map other than this cyclic group is used in the remainder of the argument.

    This is obviously a deadly flaw. The proof is meant to be proof-by-contradiction: If a map requiring five colors existed, then an impossible Steiner system could be constructed. Therefore there can be no such map. But all the authors really assume in their construction is the existence of the cyclic group of order 5. Since there is no problem with that assumption, there must, logically speaking, be at least one additional mistake somewhere in the construction. (Otherwise a false conclusion could not have been reached.) I have not had the patience to discover which step in the algebra doesn't work for N>4, but of course there must be such a step.

    I do not think there's a step there that does work in any non-trivial sense. It's not as much wrong as simply incoherent.

    As a proof of the four-color theorem, I agree that it's incoherent. Nevertheless, the proofs of the lemmas that I have checked so far are valid, leading me to wonder whether the construction of the Steiner system for the case N=4 does in fact work. If it does work, is it equivalent to any of the known constructions of Steiner (5,8,24), or is it new? If it works and is new, then that's especially disturbing to me. An author sophisticated enough to produce a genuinely new construction has to be well aware that the main claim of the paper is false. On the other hand, that the same author also claims to have produced a grand unified theory and to have solved graph isomorphism is more suggestive of the self-delusion one often encounters in people who make grandiose claims.

    You have misunderstood the definition of the map m(N). It is not just any map!

    I believe I have understood the definition of m(N). To restate my point:

    Logically speaking, this proof attempts to derive a contradiction from the existence of a map requiring more than four colors. That's a perfectly valid structure for a proof to have; the Appel-Haken proof can be thought of that way, although it proceeds very differently. For the strategy to work, however, the map m(5) (or m(6), but six colors were ruled out when the five color theorem was proved more than 100 years ago) must be used in obtaining the contradiction. You can't get a contradiction just from the number 5, or from the cyclic group Z5 alone. After all, the existence of those objects is not in question. It is only the existence of m(5) that is in doubt. As I said above, the construction in this proof uses only Z5; the map m(5) plays no role.

    I gave this manuscript more attention than I normally would. Recently I've been working in combinatorial design theory. Steiner (5,8,24) is one of the most beautiful objects in mathematics, and it would have been a major triumph for our field if had been used to prove such an important result in graph theory. Alas, it was not to be.

    No, you don't understand the basic definition if you write the symbol "m(5)" and try to give a "meta-mathematical" argument based on such an object. How do you propose to show that "m(5)" it is well-defined? If you want to base your argument on this object, shouldn’t you first show that this object is well-defined? I only wanted to make it clear to the readers that your prejudiced comments and biased conclusions are unjustified and not based on any well-defined mathematical argument. Yes, the construction of the Steiner system is quite wonderful. You would appreciate it more if you followed the definitions and lemmas of the proof.

    From the proof:

    * Define N to be the minimal number of colours required to properly colour any map from the class of all maps on the sphere. That is, given any map on the sphere, no more than N colours are required to properly colour it and there exists a map on the sphere which requires no fewer than N colours to be properly coloured.
    * Based on the definition of N, select a specific map m(N) on the sphere which requires no fewer than N colours to be properly coloured.
    * Based on the definition of the map m(N), select a proper colouring of the regions of the map m(N) using the N colours 0, 1, ..., N-1.

    The proof purports to use this m(N) to obtain a Steiner system S(N+1,2N,6N). This is impossible if N>4. The author then concludes that N=4.

    My way of stating the argument is a wholly innocent paraphrasing: We know that either N=4 or N=5 from the five color theorem. Suppose N=5. Then, assuming the author's argument, S(6,10,30) exists. Contradiction. In this context it is completely appropriate to talk about m(5). There are no issues of well-definition here: Given the author's definition of m(N) and assuming, in order to derive a contradiction, that N=5, the map m(5) is defined.

    But if you are uncomfortable with my rephrasing, I'm happy to restate my previous post in terms of m(N) rather than m(5). The point still stands: The definition of m(N) is not used anywhere in the proof. Only the cyclic group ZN is used in the construction.

    No, you don't understand the definition at all! A well-defined "m(5)" would imply a false statement, which is not allowed in mathematics. Therefore, your "paraphrasing" is wrong since the construction invokes the map m(N). On the other hand, your last meta-mathematical statement is clearly absurd because the definition of the map m(N) is used everywhere in the proof.

    Ok, Anon - one simple question: where exactly in the proof is the fact that m(N) can not be colored using N-1 colors used? And do try to stay away from meta-mathematical statements - just point me to the relevant part of the proof.

    Yury is better at sticking to the point than I am. His question is the one I really want to know the answer to. The debate about the validity of proof by contradiction is trivial and irrelevant.

    Are most of the commentators of this article clearly so arrogant to presume they can dismiss this (or any) mathematical article because their gut tells them that it is not correct. A claim like this has a clear experimental means of being refuted if incorrect, so wait until it is confirmed or not before criticizing the work done. Otherwise, show that something is wrong with the mathematics involved (mathematically and not by shrugging it off as impossible).
    The mass of the charm and anti-charm quark in Table 2 says 1337 (leet speak for "elite") instead of ~1270 MeV. Obvious joke.


    The only references to works by the authors are to private websites

    A search for the second author turns up nothing but an article about this very 'theory', by the authors of this paper

    The same article says that Dharwadker is the founder and director of this Institute of Mathematics. An institute that Google maps cannot find using the address given for it. Slightly suspicious - after all, anyone can declare themselves an institute!

    The fact that they declare exactly thirty degrees to be 'in good agreement' to an experimental 29.3137 (significance of 6 standard deviations) seems odd to me. Maybe it is acceptable, but it looks like a pretty big discrepancy to me (This is in section 4.4)

    I glanced through the paper quickly and equation (4.8) caught my eye as being pretty hokey. It's doesn't come out of their model and is largely numerology. Also their introduction to the Cabibbo angle at the beginning of Section 4.5 is lifted from wikipedia.

    The fact that there is an opaque explanation that yields equation (5.7) rather than a mathematical derivation makes me think it's crankery. They also talk a lot about Higgs Cooper pairs, but a Cooper pair is formed between two fermions. The Higgs boson is a boson.

    Try saying the second author's last name out loud...another clue.

    > Try saying the second author's last name out loud...another clue.

    I'm not sure I get it... Catch-a-train? (I'm not a native speaker, please forgive me if it's obvious.)

    > The only references to works by the authors are to private websites

    Well, there are also these two:
    one of which is in collaboration with other 3 people, who have in turn other publications:

    Anyway, I agree that the website smells funny. (*)
    The Bogdanov Brothers created a net of websites of fake "institutes of physics" which cross-referenced, here at least there is only one...

    (*) of course this is not a proof of incorrectness of the work advertised there. I admit I didn't read the post until the end, as I'm not proficient enough in mathematics to discern the correct proofs from carefully crafted wrong proofs, hence I will not comment on the scientific content here.


    As valid as all of the other objections may be, perhaps Anonymous should wait before insulting someone's surname and heritage. I've seen no evidence that the authors are not who they claim to be.

    Khachatryan or Khachatrian (Armenian: Խաչատրյան) is an Armenian surname.

    Higgs Boson Mass correctly predicted in 2009 using the Four Color Theorem. WOW!!!!

    Cogratulation, Higgs Boson Mass predicted very perfect. How about neutrino mass in this model?

    Here's to all of the idiot skeptics who posted on this page prior to July 4, 2012, who therefore have their lazy and uninformed opinions recorded in history. I only wish the majority of you were brave enough to post using your real identities, as the authors of the theory did. Instead, the skeptics coward away anonymously after hurling their venom. But you know who you are, so that's all that potentially matters.

    It's irony that you post your criticism of anonymous commenters anonymously, right?
    You don't like that, do ya Hank? Touche. Go fly a kite.