Resonances extracted in truncated partial-wave analysis are effective mixtures of angular momenta

In truncated partial-wave analysis, one fits observables that are bilinear in the amplitudes rather than the amplitudes themselves. Truncation is therefore not merely a restriction of the amplitude basis, but of the bilinear interference terms admitted in the fit. As a result, the coefficients extracted in a truncated analysis are generally not projections of the coefficients of the full amplitude. Instead, they are determined by a coupled nonlinear fit in bilinear space and depend on combinations of the full coefficient set that contribute to the retained moments. We demonstrate this in a minimal scalar toy model, where a Hermitian bilinear generated by a Legendre expansion truncated at order 2 is approximated by one truncated at order 1. Even in this simplest case, the fitted low-order coefficients depend on bilinear combinations involving higher-order contributions of the original amplitude, providing a concrete mechanism for truncation-induced angular-momentum mixing. The implication is that quantities extracted in truncated partial-wave analyses that carry resonance information should not, in general, be identified with direct projections of the corresponding quantities of the exact infinite problem. They are truncation-dependent effective mixtures generated by the restricted bilinear fit. Although shown here for scalar scattering, the same algebraic mechanism applies to Legendre-moment analyses and to photoproduction observables. The statement is established at the level of fitted bilinear quantities and reconstructed truncated coefficients; the corresponding interpretation in terms of analytically continued resonance poles is not developed here.