The Fourth Continuous Bounded Cohomology of the Complex Symplectic Group

We prove that $H_{\mathrm{cb}}^4(\Sp(4,\CC);\RR)=0$. 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 $\ell^1$-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.