A degree-three counterexample for the normalized upper Laplacian

Does a low codimension-one degree force zero into the spectrum of the normalized upper Laplacian of an infinite simplicial complex? We construct a countable pure three-dimensional complex with maximum triangle degree three whose upper Laplacian has spectrum bounded away from zero on the full degree-weighted triangle cochain space. This disproves both assertions in AIM Problem 643, including the stronger claim that degree at most the dimension forces a nonzero square-summable kernel. Every edge of the example has infinite triangle incidence. The proof uses disjoint square-summable fillings to control selected triangle values, then the gradient estimate on a binary tree to control all remaining coordinates.