The long-standing debate over whether the complete set of observables in pseudo-scalar meson photoproduction consists of eight or merely four elements continues to persist. From the perspective of amplitude analysis, it is argued that all eight observables are necessary to completely determine the others. On the other hand, proponents of partial-wave analysis, working with theoretically precise data of infinite accuracy, claim that only four observables are needed. However, this claim is not acceptable from an experimental viewpoint, as all data in the real world contain some uncertainty. This paper illustrates that the controversy is artificial and is due to additional mathematical assumptions used in partial-wave analysis. Our research advances this discussion by moving from exact synthetic numerical data to also synthetic, but more realistic data in partial-wave analysis and shows that the claimed reduction in observables is unjustified. Consequently, the final conclusion is that the complete set of observables in pseudo-scalar meson photoproduction, whether using amplitude analysis or partial-wave analysis with practical data, must consist of eight observables.

The new single-channel, single-energy partial wave analysis method based on a simultaneous use of amplitude and partial wave analysis called AA/PWA, developed and tested on $\eta$ photoproduction in ref. Svarc et al, PRC 102, 064609 (2020), is applied to the $K^{+} \Lambda$ photoproduction for the center-of-mass energy range of 1625 MeV < W < 2296 MeV. A complete set of multipoles has been created. The advantages of the method have been confirmed, and a comparison with the only existing single-energy partial wave analysis of $K^{+} \Lambda$ photoproduction given in refs. Anisovich et al PRL 119 062004 (2017) and Anisovich et al Eur. Phys. J. A 53: 242 (2017) is presented. We confirm the size and shape of Bonn-Gatchina multipoles, but we do not confirm the unambiguous interpretation of the structure in the $M_{1-}$ multipole as a $N(1880) \frac{1}{2}^{+}$ resonance. The decisive role of the self-consistency of the world database is emphasized.