Disproof of Bell’s Theorem

Why is this evidence so feared by the dark forces?

In 1964 John Stewart Bell proved a mathematical theorem and claimed that “no physical theory which is local and realistic (in the sense espoused by Einstein) can reproduce all of the statistical predictions of quantum mechanics.” However, in a series of papers written between 2007 and 2014 I constructed just such a local and realistic framework for physics. Consequently, today Bell’s theorem no longer has the fundamental significance for physics it was thought to have.

In a nutshell, my disproof of Bell’s theorem goes as follows: In 2007 I noticed that there is a serious error in the very first equation of Bell’s famous paper [Physics 1, 195 (1964)]. Bell simply made an elementary but fatal mistake, and this mistake lies in the formulation of the very first equation of his famous paper. The correct form of his proposed functions to reproduce the quantum correlations local-realistically must be

\displaystyle \pm\,1\,=\,{\cal A}({\bf a},\,\lambda):{\rm I\!R}^3\!\times\Lambda\longrightarrow S^3 \sim {\rm SU(2)} \ \ \ \ \ \ \ (1)

instead of

\displaystyle \pm\,1\,=\,{\cal A}({\bf a},\,\lambda):{\rm I\!R}^3\!\times\Lambda\longrightarrow S^0 \subset{\rm I\!R}, \ \ \ \ \ \ \ \ \ \ \ \ (2)

for them to provide a complete local description of the physical reality demanded by Einstein. The latter form, with {{S^0=\{-1,\,+1\}}} (or a subset of the real line) as the codomain of the functions {{{\,\cal A}({\bf a},\,\lambda)}}, is what Bell assumed in the first equation of his paper. But the complete description can be provided by such functions if and only if their codomain is an absolutely parallelized 3-sphere, {{S^3}}, which is a closed set of unit quaternions, homeomorphic to the fermionic rotation group SU(2):

\displaystyle S^3:=\left\{\,{\bf q}(\psi,\,{\bf a},\,\lambda):= \,\lambda\,\cos\frac{\psi}{2}\,+\,\lambda\,{\boldsymbol\beta}({\bf a})\,\sin\frac{\psi}{2}\; \Bigg|\;||\,{\bf q}(\psi,\,{\bf a},\,\lambda)\,||^2=1\right\}\!. \ \ \ \ \ \ \ (3)

Here the complete state \lambda of the spin system is represented by the initial orientation of the 3-sphere, and the results {\cal A}=-1 and +1 are limiting cases of {\bf q}(\psi,\,{\bf a},\,\lambda) for the rotation angles \psi=2\pi and 4\pi. Note also that a 3-sphere cannot be charted by a single coordinate system, but an anholonomic frame, made up of graded basis \{\,1\,,\;{\boldsymbol\beta}_1\,,\;{\boldsymbol\beta}_2\,,\;{\boldsymbol\beta}_3\,\}, can be defined on it, fixing each of its points uniquely. It is the impossibility of defining a global coordinate chart on {{S^3}} but possibility of defining an anholonomic frame on it that is ultimately responsible for the existence of EPR-Bohm correlation.


It is very important to note that the image points of both forms (1) and (2) of the above functions, namely the measurement results {{\pm\,1\,}} themselves, are the same:

\displaystyle {\cal A}({\bf a},\,\lambda)\,=\,\;either\;\;+\!1\;\;\;\;or\;\;\;-\!1\,,\ \ \ \ \ \ \ \ \



\displaystyle {\cal E}({\bf a})\,=\,\lim_{\,n\,\gg\,1} \left[\frac{1}{n}\sum_{k\,=\,1}^{n}\,{\cal A}({\bf a},\,{\lambda}^k)\right] \,=\,0\,. \ \ \ \ \ \ \ (4)

They are exactly what Bell assumed them to be in his first equation. Only the codomain of the two forms of the functions differ from each other. Failing to take the physically and mathematically correct codomain into account, Bell’s argument is thus indeed a non-starter. That is to say, his argument simply does not go through without the assumption of totally disconnected set \cup\,S^0 as the set of all possible measurement results (actual as well as counterfactual). What is more, once the correct form (1) of the measurement functions is recognized, the familiar singlet correlation predicted by quantum mechanics between the measurement results {\cal A}({\bf a},\,\lambda) =\pm\,1 and {\cal B}({\bf b},\,\lambda) =\pm\,1, namely

\displaystyle {\cal E}({\bf a},\,{\bf b})\,=\,\lim_{\,n\,\gg\,1} \left[\frac{1}{n}\sum_{k\,=\,1}^{n}\,{\cal A}({\bf a},\,{\lambda}^k)\;{\cal B}({\bf b},\,{\lambda}^k)\right] \,=\,-\,{\bf a}\cdot{\bf b}\,, \ \ \ \ \ \ \ (5)

is relatively easy to reproduce in a strictly local and realistic manner. In fact, once the topology of the codomain of the measurement functions {{{\,\cal A}({\bf a},\,\lambda)}} is correctly taken into account in the above manner, the observed EPR-Bohm correlation predicted by quantum mechanics inevitably falls out as a classical, deterministic correlation among the points of an absolutely parallelized 3-sphere. This is a robust result. It is not difficult to prove that correlation between any two points {\bf q}({\bf a}) and {\bf q}({\bf b}) of S^3 is necessarily equal to \,-\,{\bf a}\cdot{\bf b}\,, with {\cal A}({\bf a},\,\lambda) =\pm\,1 and {\cal B}({\bf b},\,\lambda) =\pm\,1 being the limiting cases of {\bf q}(\psi,\,{\bf a},\,\lambda) and {\bf q}(\psi,\,{\bf a},\,\lambda), respectively. Thus all the perplexity generated by Bell’s impossibility claim is much ado about nothing. It stems from a topologically incorrect accounting of measurement results.


You can verify these facts in an explicit computer simulation of the model, as discussed in this paper. At least five explicit, event-by-event, numerical simulations of my local model for the EPR-Bohm correlation have been independently produced by five different authors, with codes written in Java, Python, Mathematica, Excel VB, and “R“. The above 2D-surface version of the simulation of my local model can be found here. I discuss two of these simulations in the appendix of this paper. A compact translation of one of these simulations (from Python to Mathematica) can be found here. Each simulation has given different statistical, geometrical, and topological insights into how my local-realistic framework works, and indeed how Nature herself works.

The most accurate simulation of my local model for the EPR-Bohm correlation can be found here.



From the above discussion Bell’s mistake is now very clear. Instead of investigating correlation among the points of a parallelized 3-sphere, Bell naïvely investigated correlation among the points of a real line to prove his celebrated theorem. But even an elementary topological consideration is sufficient to recognize the fact that correlation among the points of a real line cannot possibly be the same as those among the points of any topologically nontrivial space such as a parallelized 3-sphere.

But what about the famous Bell-CHSH inequality on which Bell’s supposedly iconic theorem is based? It turns out that it too can be violated local-realistically. In doing so, however, we gain a tremendous insight into the very origins of quantum correlations. This is discussed in greater detail on the next page, but first let us recall that for the set of four measurement directions, a, a', b', and b, the Bell-CHSH inequality, which claims an upper bound on local-realistic correlations, is given by

\displaystyle |\,{\cal E}({\bf a},\,{\bf b})\,+\,{\cal E}({\bf a},\,{\bf b'}) \,+\,{\cal E}({\bf a'},\,{\bf b})\,-\,{\cal E}({\bf a'},\,{\bf b'})\,|\, \leqslant 2\,, \ \ \ \ \ \ \ (6)

provided we assume the corresponding measurement functions to be of the incorrect form (2) assumed by Bell. Thus it is in this step—the crucial step in the derivation of Bell-CHSH inequality and the proof of Bell’s theorem—that the assumption of incorrect codomain gets smuggled-in. The four pairs of measurement results occurring in the above expression clearly cannot all occur at the same time. If, however, we conform to the usual assumption of counterfactual definiteness and pretend that they do all occur at the same time at least counterfactually, then we must specify the correct codomain for the functions {\cal A}({\bf a},\,\lambda) =\pm\,1 and {\cal B}({\bf b},\,\lambda) =\pm\,1 to represent them correctly. But for the measurement functions of the form (1) with the correct codomain S^3 the story turns out to be dramatically different from (6). Because then the measurement results \pm\,1 occur as points of an absolutely parallelized 3-sphere, which is the set S^3 of unit quaternions. As a result, we must specify the joint probability distribution for the occurrence of the four pairs of measurement results on such a 3-sphere, defined by the graded basis \{\,1\,,\;{\boldsymbol\beta}_1\,,\;{\boldsymbol\beta}_2\,,\;{\boldsymbol\beta}_3\,\}. Consequently, the upper bound on the correlations depends on the torsion {\cal T}_{\,{\bf a\,a'}} at each point of the manifold S^3 as follows:

\displaystyle |\,{\cal E}({\bf a},\,{\bf b})\,+\,{\cal E}({\bf a},\,{\bf b'}) \,+\,{\cal E}({\bf a'},\,{\bf b})\,-\,{\cal E}({\bf a'},\,{\bf b'})\,|\, \ \ \ \ \

\displaystyle \leqslant\sqrt{\,\lim_{\,n\,\gg\,1}\left[\,\frac{1}{n}\sum_{k\,=\,1}^{n}\, \big\{\,4\,+\,4\,{\cal T}_{\,{\bf a\,a'}}({\lambda}^k)\,{\cal T}_{\,{\bf b'\,b}}({\lambda}^k)\,\big\}\,\right]} \ \ \ \ \ \ \

\displaystyle \leqslant\,2\,\sqrt{\,1-({{\bf a}}\times{\bf a'})\cdot({\bf b'}\times{{\bf b}})\,}\, \ \ \ \ \ \ \

\displaystyle \leqslant\,2\,\sqrt{2}\,.\, \ \ \ \ \ \ \ \ \ (7)

This upper bound has been derived in detail in the last appendix of this paper. This derivation has also been numerically verified in a GAviewer by Fred Diether and myself. It is of course exactly what is predicted by quantum mechanics. Thus, with the topologically correct choice (1) for the measurement functions, not only the strong correlations (5) but also the upper bound (7) on the correlations turns out be exactly what is predicted by quantum mechanics.

More importantly, the above derivation reveals a crucial role played by the non-vanishing torsion in our physical space on the strength of the observed correlations. If the physical space in which the measurement events {\,{{\,\cal A}=\pm\,1\,}\,} and {\,{{\,\cal B}=\pm\,1\,}\,} are occurring has vanishing torsion, then the correlation between these events cannot be as strong as those predicted by quantum mechanics. But vanishing torsion is precisely what all derivations of Bell-CHSH inequality unwittingly assume (see, for example, the discussion in this paper). Note also that parallelizability of spaces by torsion is a topological concept. A toy example of how topology can strengthen the observed correlation can be found in the first appendix of this paper.

Needless to say, the above discussion is merely a sketch of my disproof. In particular, I have glossed over the important issue of how errors propagate within a parallelized 3-sphere (a detailed statistical analysis of which can be found in this paper). Moreover, in my book I have shown that, not only the simple EPR-Bohm-type correlations, but ALL quantum correlations can be reproduced purely local-realistically in an analogous manner. For more general cases, however, the codomain of the measurement functions such as {{{\cal A}({\bf a},\,\lambda)=\pm\,1}} must be extended to the absolutely parallelized 7-sphere. Thus quantum correlations in general turn out to be nothing but classical, local, and deterministic correlations among the points of an octonionic 7-sphere. On the next page I discuss how the parallelizing torsion {\cal T} within such a 7-sphere is responsible for ALL quantum correlations.

The argument I have presented above is experimentally testable. I have proposed a macroscopic experiment for this purpose. Further discussion about this experiment can be found on this page.


PS: A theoretical computer scientist, Paul Snively, has crystalized the essence of my argument in a logical sequence that I find quite interesting. According to Snively the logic behind my refutation of Bell’s theorem is:

algebra with operations lacking the closure property \rightarrow mathematical singularities \rightarrow partial functions \rightarrow logical inconsistency.

A brief discussion of what he means by this sequence can be found on this page of his blog.

He summarizes the main point thus: “Dr. Bell used scalar algebra. Scalar algebra isn’t closed over 3D rotation. Algebras that aren’t closed have singularities. Non-closed algebras having singularities are isomorphic to partial functions. Partial functions yield logical inconsistency via the Curry-Howard Isomorphism. So you cannot use a non-closed algebra in a proof, which Dr. Bell unfortunately did. … This is a sufficient disproof of Bell’s theorem.”

Let me add to Paul’s observation that, not only Bell’s original 1964 proof, but *ALL* of the proofs of *ALL* of the Bell-type theorems, including the proofs of their variants and generalizations such as Hardy’s theorem or GHZ theorem, use only scalar algebra. Therefore all such no-go “proofs” are logically inconsistent (i.e., they are nonsense).




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22 Responses to Disproof of Bell’s Theorem

  1. Fred Diether says:

    Hi Joy,

    I was just looking at this video on the Scientific American website about quantum entanglement presented on a blog by George Musser. I have to say that after understanding your model, I could immediately see where the argument for quantum entanglement breaks down in the video. I am wondering if anyone else reading this blog can point out those spots in the video?

    I remember on the FQXi blogs that George Musser noted to you that he would like to have a better discussion with you about your model. I think it is only fair that a counter opinion about quantum entanglement should be offered to SciAM readers and you should take George up on his offer.



  2. Hi Fred,

    I watched the video. I must admit I am deeply disappointed and disturbed by how misleadingly the idea of entanglement is presented in the video. With all due respect to George Musser (who I think is a talented young journalist), what he is doing is no different from “spreading the good word” of entanglement. The video is misleading from the start. The two men set out to witness entanglement firsthand in a physics lab. But you can no more witness entanglement of a photon than you can witness phlogiston in a combustible body. All one can ever see in a lab are correlations between the supposedly entangled photons. No one has ever seen, or could ever see, entanglement directly. If we could, then all controversies over the interpretation of quantum mechanics would end instantly.

    To be sure, we can prepare various states of the photons as shown in the video—say, a state in which the correlations exhibited by the photons is no better than that produced by coin tosses, or a state in which the correlations exhibited by the photons which can be explained by the concept of entanglement. But the latter is not the only explanation possible for the observed correlations, as we know from the results of this paper. I show in this paper how the observed correlations can be explained without the notion of phlogiston… I mean entanglement. They can be explained simply as classical, deterministic correlations among the points of a parallelized 3-sphere. So it is very disappointing indeed that Scientific American would continue to “spread the good word” of entanglement without even mentioning the much less mystical alternative. The day will come, however, when the idea of entanglement would be no more in vogue than the idea of phlogiston.



  3. Hi Joy,

    I appreciate your remarks and do hope to continue our discussions at some point. But I think you will admit that your views on entanglement are a minority opinion. Of course that doesn’t make them wrong! The history of science shows that minority opinions are often vindicated; all great truths are initially resisted. Nonetheless, Scientific American *has* to give higher weight to prevailing views — otherwise, the editors would be placing their own judgment above that of the majority of scientists, which is sometimes justified, but only in exceptional cases. To put it differently, once you convince your fellow physicists and philosophers of physics of your arguments, then you will have convinced us, too.


  4. Hi George,

    Thanks for dropping by and responding to my comments. I do agree that my views on entanglement are a minority opinion at the moment, and I do appreciate why Scientific American has to give higher weight to the prevailing views. But it seems to me that your video linked above gives a wrong impression of entanglement even from the prevailing point of view. It is understood by most experts (say, by Lucien Hardy, for example) that all we can see in the lab are correlations, not entanglement itself. The latter concept provides a good explanation for the observed correlations, but that is a different matter.

    But again, I appreciate that you have a short time at your disposal in which you have to explain the essence of the idea to lay viewers, without getting into technicalities.

    As for my own views on entanglement, I do hope that someday my arguments are better understood by my fellow physicists and philosophers of physics. In any event—leaving your journalist hat aside—please feel free to continue the discussion we started during the last FQXi conference.



  5. I hope our world lines will intersect soon, so that we can continue our chat. Please let me know if you have plans to visit NYC.


  6. T H Ray says:

    Hi Joy,

    I’d like to expand on the phlogiston example you used. Shorly before Lavoisier showed that combustion is caused by rapid oxidation, there were theoretically two kinds of phlogiston, deduced from the evidence of combustion — “positive” phlogiston in the form of rust, which apparently added matter to a substance such as iron, and “negative” phlogiston which apparently subtracted matter from a substance such as wood, which was reduced to ashes.

    These conclusions are of course contradictory. The contradiction is of the same form [as certain mistaken claims about] your framework. Just as “phlogiston theorists” were mistaken about the true nature of combustion, so are these [claims] about the constructive nature of a continuous measurement process that obviates the identification of discrete outcomes with physically real, nonlocal results.


  7. Hi Tom,

    Thank you for your comments. I did not know that there were two kinds of phlogiston.

    I am terribly sorry, however, because I had to edit out parts of your message. Names of certain individuals whom I cannot respect are not allowed on this blog. I hope you will understand. I have tried to preserve the integrity of your message as much as possible.

    As for the mistaken claims of those individuals you mention, it is important to note that equations (1.22) to (1.26) on page 10 of my book, as well as similar set of equations in this paper, have been explicitly verified (in great detail) by Lucien Hardy, Manfried Faber, and several other high profile and exceptionally competent physicists and mathematicians around the world. In fact, any competent reader with only basic skills in mathematics should be able to reproduce equations (1.22) to (1.26) of my book rather effortlessly.

    It is also worth noting that all of the so-called arguments against my disproof to date are based on an elementary logical fallacy—the Straw-man Fallacy. What they do is replace my argument (or model), say X, with a grossly distorted misrepresentation of it, say Y, and then pretend—by refuting their own distortion Y (by resorting to deliberate dishonesty or out of sheer incompetence)—that they have undermined my actual argument (or model) X. Such a dishonest strategy defies reason at its very core (for more details, see this paper).

    Unlike Bell himself, some of the followers of Bell are naïve, uninformed, and dishonest.



  8. T H Ray says:

    Hi Joy,

    The presence of positive and negative phlogiston is implied in the link you gave. The author just failed to point out the contradiction between two *physical* results from a theory that can accommodate only a singular physical outcome. If phlogiston is assumed to be physically real, it must be causal, and if causal, attributable to only one result. It doesn’t do to say that in the one case phlogiston causes rust and in the other case, causes ashes. If rust and ashes have a common cause, it can’t be phlogiston.

    By the same token, non-vanishing topological torsion analogous to the rapid oxidation that causes both rust and ashes, informs us that continuous functions imply classical orientation entanglement — (and not the entanglement of real physical quantities whether we call the phenomena phlogiston or quantum entanglement) — that guarantees a unitary cause.

    In other words, if one assumes phlogiston, an experimental outcome of + 1 (rust) or – 1 (ashes) explains nothing. Both (contradictory) outcomes can’t result from the same physical cause. Your topological model, OTOH, has the same attributes as chemical oxidation in principle: a continuous process drives the measurement result from a deeper underlying physical cause of nondegenerate torsion.

    Einstein faced the same problem in explaining how spacetime is physically real (” … independent in its properties, having a physical effect but not influenced by physical conditions…”) because one does not think of continuous phenomena as physically real. One intuitively wants to ascribe “reality” to the measurement apparatus rather than the measurement process.

    Your critics are not subtle thinkers. They allow that 2 and 3 dimensional simulations are adequate for n-dimension phenomena. Thus, they reject *any* topological model out of hand; global results, as topology implies, have to be nonlocal by definition. I think that until one can grasp the analytical nature of your framework, it’s an uphill battle. Einstein had the advantage of large scale validation for continuous spacetime. Small scale classical experiments are tougher, less forgiving. Just as one has to accept a 4-dimension physical spacetime continuum for general relativity to make sense, one has to accept a parallelized physical 7-sphere for your framework to make sense. And just as Minkowski space-time requires a great deal of mathematical background to grasp, your framework requires that much more.



  9. Hi Tom,

    You make some very good points. Sadly, as you put it, my “critics are not subtle thinkers”, to say the least. Moreover, dogmas, even within science, are not easy to overcome, as we have witnessed over the past few years. But thank you for your thoughtful remarks, especially concerning the phlogiston analogy.



  10. igael says:


    I have on my website 2 formulae that work with good performance at the degree resolution level. There is an online and checkable javascript implementation for the ones that haven’t a tool to emulate probability functions and to compute correlations.

    any advice is welcome :)

  11. Hello Igael,

    What you have is a nice demonstration of well known facts. It is well known (since 1970’s) that by reducing the efficiency of detection, to, say, 86%, one can reproduce the cosine correlation. Such a demonstration does not contradict Bell’s argument, and it does not have fundamental significance for physics.



  12. Igael says:


    Please note that I’m working on new variants with 89% pairs ( then 94% of arm efficiency ) and probably more. I use 80 servers , raising to ‘powers’ above without Markov takes a lot of resources

    But, you are true , even if I haden’t yet seen theses formulae on sites or books.
    It is not so clear when you read QM curses or books on EPR to understand that the debate is not closed. How to start developping applications in such conditions ?

    Please, let me know where I could find the formulae known since 70 ? I found only the Maudlin scheme B ( it seems to my algo 9 without Markov and has a limit around 66.6 % pairs and not 63.6 as I read in some articles ) ?

    This is an amazing topic !
    Thank you.

  13. Igael,

    I had this paper in mind: Hidden-Variable Example Based upon Data Rejection, by P. Pearle, Physical Review D 2, 1418–25 (1970).



  14. Igael says:

    Hi Joy

    Many thanks.

  15. Olav Helgeland (emeritus professor Materials Science) says:

    Hi Joy.

    BLOG: (revise or cut as you wish)

    I found your decorative blog.

    Checking ‘Internet’ it appears that you have been through some nasty ‘entanglements’ and have passed it in ‘local reality’, free and well, – with honour.
    Good luck for your further work!

    engaged Emeritus.no


    If you check your e-mail box for 03.03.2014, Re: Bell’s theorem, with two enclosed items, you will find the background for my contact.
    I would appreciate v.m. a reply and a comment to my paper. ( I can repeat the e-mail if you have lost it).

    w.f.g. Olav

  16. KL Rajpal says:

    Linear Polarization http://vixra.org/pdf/1303.0174v5.pdf
    Einstein was right when he did not agree with the EPR experiment conclusions and had said, “spooky action at a distance” cannot occur and that, “God does not play dice”.

    Electron Spin

  17. John R. Cox says:

    Dr. Christian,
    I would imagine Tom Ray might keep in touch with you, so you might know that the current FQXi essay contest has an entry by Edwin Eugene Klingman in which he challenges constraints inherent in Bell’s formulation, which produce the same distribution curve against the Bell straight-line eigenvector that your topological challenge results in. Not that I blame either you or Tom for not continuing to subject yourselves to the abuse of the blogosphere, I think it wise. I recall warning R. Gill that his posturing in Vienna made him a perfect fall guy for professional spooks. I had wondered if he ever made it back from his ‘sabatical’ to the Baltic, hope you actually collected the 10K Euros. Kudos, jrc

  18. Brian Rosenblum says:

    Dr Christian:

    You are a diseased little turd swirling around the toilet drain en route to an obscurity even deeper than the one currently defining you.


  19. Thank you, Mr. Rosenblum.

    If your prophesy is right, then there is no reason for you to be concerned. :-)

  20. Martin says:

    Dr. Christian,

    As I understand (interpret) your theoretic conjectures, the problem arises from the interpretation of what a particle represents: QT versus your description – point vs 3D object (sphere), wave of probability with instantaneous communication between points in ST versus local ST with light speed limitation. I have always wondered how points in QT (Copenhagen Interpretation) can possess complex properties and give rise to complex interactions? QT begs for existence of hidden variables.
    Do waves of probability also govern virtual particles? And how could we unite the Copenhagen interpretation with your view?
    The answer could very well lay in an existence of a real De Broglie wave made of two sets of virtual photons (intersecting at opposite directions and at the speed of light) which express their circular polarization pattern only at a “point” at which the particle (the Higgs expression?) is located. Those photons would cancel to randomness in all other positions along their path of propagation, and shifting their frequency would lead to particle speed change. In this view matter would be a stable interference pattern. Particles would form as rain drops form in clouds and would exist intermittently pulsating to the beat frequency of the waves that build them.
    So, does this expression fit yours?
    Also you could have a look at the work of Dr. Yves Couder: https://www.youtube.com/watch?v=W9yWv5dqSKk


  21. We have finally launched the long-awaited website for the Einstein Centre for Local-Realistic Physics: http://einstein-physics.org/.

  22. I have posted a simplified version of my proposed macroscopic experiment to test Bell’s theorem here: http://libertesphilosophica.info/blog/wp-content/uploads/2016/05/PropExp1.pdf.

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