We demonstrate the existence of a certain genus four origami whose Kontsevich--Zorich monodromy is arithmetic in the sense of Sarnak. The surface is interesting because its Veech group is as large as possible and given by SL(2,Z). When compared to other surfaces with Veech group SL(2,Z) such as the Eierlegendre Wollmichsau and the Ornithorynque, an arithmetic Kontsevich--Zorich monodromy is surprising and indicates that there is little relationship between the Veech group and monodromy group of origamis. Additionally, we record the index and congruence level in the ambient symplectic group which gives data on what can appear in genus 4.

The authors would like to thank Carlos Matheus Silva Santos and Pascal Kattler for a careful reading of the manuscript. In particular, we thank Matheus for comments on the Lyapunov spectrum and the reference [7] and Pascal Kattler for pointing out a mistake in a previous computation of α(S). Lastly, we are immensely grateful to Pascal Kattler and Gabriela Weitze-Schmithusen for running the generators of Theorem 4.1 through their software [12]. A.S. was supported by the National Science Foundation Postdoctoral Fellowship under grant number DMS-2103136.