Perfect play in a finite nonlocal game requires infinite entanglement entropy

How much infinity can a finite nonlocal game require? We construct a finite two-player game with classical questions, classical answers and a Boolean winning rule. It has a perfect strategy on a tensor product of separable Hilbert spaces. Every perfect normal spatial strategy contains, under local isometries, the same pure entangled state with infinite entropy. For mixed strategies, both marginal entropies are infinite and the pure state remains a tensor factor. Thus infinite dimension can be strengthened to infinite entropy without losing spatial attainment. The proof constructs a rational Bell functional whose perfect strategies contain a state with Catalan Schmidt multiplicities, and compiles it into Boolean tests. For either construction, the minimum marginal entropy at loss ε is Θ(log(1/ε)). The lower bound holds for every normal spatial strategy, including mixed states and POVMs. It follows from polynomial decoding on the actual state and a finite-strip entropy estimate. Height and size cutoffs give the matching upper order. The sharp leading constant remains open.