One-shot zero-error communication with normal entanglement is Σ⁰₂-complete

Can a finite classical channel require an infinite entangled resource to send a fixed number of messages without error? We classify the one-use feasibility problem for rational channels. With an arbitrary normal spatial resource it is Σ⁰₂-complete. The same classification holds when the resource must have finite marginal entropy. A separate channel admits a normal zero-error code, while every normal code achieving its specified message count has infinite marginal entropy on both sides. The proof extends Mančinska's finite-dimensional game-to-channel correspondence to trace-class operators. For the game's channel, a normal code attaining the clique bound (one message per local question) corresponds to perfect play with summable Schmidt amplitudes. Four successive constraints, each doubling a ratio of Schmidt weights, are critical and exclude such a code; five suffice for both reductions. As a corollary of Mančinska's finite-dimensional correspondence and MIP*=RE, we obtain Σ⁰₁-completeness for finite-dimensional resources. We also prove Π⁰₁-completeness for commuting operators. Our capacity statements are upper computability bounds; the hardness results concern one use.