The Bell and probability inequalities are not violated when noncommutation is applied according to quantum principles. (arXiv:1903.02116v2 [quant-ph] UPDATED)

The Bell inequalities in three and four correlations may be re-derived in a
general form showing that the corresponding number of data sets of +- 1's
identically satisfies them regardless of whether they are randomly or
deterministically generated. When the data sets become infinite in size in the
random case, and assuming convergence of the correlation estimates, the
inequalities become constraints on the correlation functions of the mutually
cross-correlated data sets. Replacing the correlations in the inequalities by
the corresponding physical model-based probabilities that produce them results
in inequalities in probabilities. Under the special assumption of a
wide-sense-stationary (WSS) random process, the Wigner inequality in
probabilities results in the three variables case. This inequality is violated
by probabilities that produce Bell states, since these states are inconsistent
with the assumption of a WSS process. They are also inconsistent with quantum
non-commutation as occurs in the case of more than one spin measurement on each
of two particles. When all the correlations or probabilities are computed
according to quantum principles, however, the corresponding version of the Bell
inequality is satisfied.

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