We consider the hyperboloidal initial value problem for the cubic focusing wave equation (-partial derivative(2)(t) + Delta(x))nu(t, x) + nu(t, x)(3) = 0, x is an element of R-3 Without symmetry assumptions, we prove the existence of a codimension-4 Lipschitz manifold of initial data that lead to global solutions in forward time which do not scatter to free waves. More precisely, for any delta is an element of (0, 1) we construct solutions with the asymptotic behavior vertical bar vertical bar nu - nu(0) vertical bar vertical bar L-4(t, 2t)L-4(B(1-delta)t) less than or similar to t(-1/2+) as t -> infinity, where nu(0)(t, x) = root 2/t and B(1-delta)t := {x is an element of R-3 : vertical bar x vertical bar < (1-delta)t}.