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The algebraic derivation of the numerical limits of Bell inequalities ineither three or four random variables is independent of the assumption ofrandomness.The limits of the inequalities follow as mathematical consequencesof their created algebraic structures independently of application to random ordeterministic variables.The inequalities should be called identityinequalities.A final correlation reuses data from the previous correlations andthus leads to the inequality limits.It generally has a different functionalform from the previous correlations, whether derived as a counterfactualmathematical result, or in a way enabling comparison with experiment.Thesealgebraic facts and their consequences are central to understanding theinequalities use, but have not been widely recognized.Logically consistentapplication of the inequalities to Bell experiments is challenging, given thatthe number of mathematically assumed random variables is greater than thenumber of physical variables produced per experimental realization.Given Bellsrejection of the use of sequential, alternative paths, three experimental runsare here considered to enable acquisition of data to be rearranged forcomputation of statistical crosscorrelations.Predicted quantum mechanicalcorrelations then satisfy the inequality.Since mathematically inconsistent useis sufficient to cause inequality violation, the conclusion that violationimplies the nonexistence of underlying variables in the entanglement processdoes not follow.

An alternative view: satisfaction of the four variable Belli inequality using quantum correlations