  Bell's theorem is supposed to exclude all local hidden-variable models of
quantum correlations. However, an explicit counterexample shows that a new
class of local realistic models, based on generalized arithmetic and calculus,
can exactly reconstruct rotationally symmetric quantum probabilities typical of
two-electron singlet states. Observable probabilities are consistent with the
usual arithmetic employed by macroscopic observers, but counterfactual aspects
of Bell's theorem are sensitive to the choice of hidden-variable arithmetic and
calculus. The model is classical in the sense of Einstein, Podolsky, Rosen, and
Bell: elements of reality exist and probabilities are modeled by integrals of
hidden-variable probaility densities. Probability densities have a
Clauser-Horne product form typical of local realistic theories. However,
neither the product nor the integral nor the representation of rotations are
the usual ones. The integral has all the standard properties but only with
respect to the arithmetic that defines the product. Certain formal
transformations of integral expressions one finds in the usual proofs \`a la
Bell do not work, so standard Bell-type inequalities cannot be proved. The
system we consider is deterministic, local-realistic, rotationally invariant,
observers have free will, detectors are perfect, so is free of all the
canonical loopholes discussed in the literature.
