Simple refutation of Joy Christian’s simple refutation of Bell’s simple theorem

I point out a simple algebraic error in Joy Christian’s refutation of Bell’s theorem. In substituting the result of multiplying some derived bivectors with one another by consultation of their multiplication table, he confuses the generic vectors which he used to define the table, with other specific vectors having a special role in the paper, which had been introduced earlier. The result should be expressed in terms of the derived bivectors which indeed do follow this multiplication table. When correcting this calculation, the result is not the singlet correlation any more. Moreover, curiously, his normalized correlations are independent of the number of measurements and certainly do not require letting n converge to infinity. At the same time, his unnormalized or raw correlations are identically equal to −1, independently of the number of measurements! Correctly computed, his standardized correlations are the bivectors −a · b − a ∧ b, and they find their origin entirely in his normalization or standardization factors. I conclude that his research program has been set up around an elaborately hidden but trivial mistake. In at least 11 papers on quant-ph author Joy Christian proposes a local hidden variables model for quantum correlations which disproves Bell’s theorem “by counterexample” in a number of different settings, including the famous CHSH and GHZ versions. Fortunately one of these papers is just one page long and concentrates on the mathematical heart of his work. Unfortunately for his grand project, this version enables us to clearly see a rather shorter derivation of the desired correlations, which exposes an error in his own derivation. The error is connected with an unfortunate notational ambiguity at the very start of the paper. In a nutshell: the same symbols are used to denote both a certain fixed basis (βj, j = 1, 2, 3) in terms of which two other bases are defined, (βj(+1), j = 1, 2, 3) and (βj(−1), j = 1, 2, 3), as well as to express the generic algebraic multiplication rules which these latter two bases satisfy. This tiny ambiguity, though harmless locally, is probably the reason why later on, when apparently using the multiplication tables for the new bases, he silently shifts from the derived bases to the original special basis.