The Fourth Continuous Bounded Cohomology of the Complex Symplectic Group

We prove that $H_{\mathrm{c}\mathrm{b}}^4(\text{Sp}(4,\text{CC});\text{RR})=0$. Together with Blatz’s secondary stability and the rank-one theorem of Bucher–Savini, this gives degree-four vanishing for all complex symplectic and odd complex orthogonal groups. In normalized symplectic Gram coordinates, we establish a bounded-primitive estimate and compute the measurable cohomology of the projective action. An explicit rational cocycle has divergent periods on a family of finite orbit cycles of uniformly bounded ${\mathrm{\ell }}^1$-mass. A two-cone averaging construction extends the period estimate to measurable cochains and excludes bounded representatives of every nonzero degree-four measurable action class.