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Scattering off the edge of a composite particle or finite-range interactioncan precede that off its center. An effective theory treatment with pointlikeparticles and contact interactions must find that the scattered experimentalwave is slightly advanced, in violation of causality (the fundamentalunderlying theory being causal). In practice, partial-wave or other projectionsof multivariate amplitudes exponentially grow with imaginary E, so that uppercomplex-plane analyticity is not sufficient to obtain a dispersion relation forthem, but only for a slightly modified function (the modified relationsadditionally connect different J). This limits the maximum precision of certaindispersive approaches to compositeness based on Cauchy's theorem leading topartial-wave dispersion relations. This may be of interest to some dispersivetests of the Standard Model with hadrons, and to unitarization methods used toextend electroweak effective theories. Interestingly, the Inverse AmplitudeMethod is safe (as the inverse amplitude has the opposite, convergent behaviorallowing contour closure). Generically, one-dimensional sum rules such as forthe photon vacuum polarization, one-dimensional form factors or the Adlerfunction, to name a few, are not affected by this uncertainty. Likewise,fixed-t dispersion relations were cleverly constructed to avoid it andconsequences therefrom are solid.

Smearing of causality by compositeness divides dispersive approaches into exact ones and precision-limited ones