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The fabled violation of Bell's inequality by the probabilistic specificationsof quantum mechanics is shown to derive from a mathematical error. Theinequality, designed to assess consequences of Einstein's principle of localrealism, pertains to four polarization products on the same pair of photonsarising in a gedankenexperiment. The summands of the CHSH quantity $s(\lambda)$inhere four symmetric functional relations which have long been neglected inanalytic considerations. Its expectation $E[s(\lambda)]$ is not the sum of four``marginal'' expectations from a joint distribution, as quantum theoryexplicitly avoids such a specification. Rather, $E[s(\lambda)]$ has fourdistinct representations as the sum of three expectations of polarizationproducts plus the expectation of a fourth which is restricted to equal afunction value determined by the other three. Analysis using Bruno de Finetti'sfundamental theorem of prevision (FTP) yields only a bound for $E(s)$ within$(1.1213, 2]$, surely not $2\sqrt{2}$ at all. The 4-D polytope of coheringjoint $P_{++}$ probabilities at the four stipulated angle settings aredisplayed as passing through 3-D space. Aspect's ``estimation'' is based onpolarization products from different photon pairs that do not have embeddedwithin them the inhering functional relations. When you do actively embed therestrictions into Aspect's estimation procedure, it yields an estimate of1.7667, although this is not and cannot be definitive.

Quantum Violation of Bell's Inequality: a misunderstanding based on a mathematical error of neglect