Is "Shannon-capacity of noisy computing" zero?

Towards understanding energy requirements for computation with noisy elements, we consider the computation of an arbitrary k-input, k-output binary invertible function. In our model, not all gates need to be noisy. However, the input nodes on the computation graph must be separated from the output nodes by a noisy cut. For this setup, we show that for the “information-friction” model proposed recently for energy consumed in circuits, and for binary-input AWGN noise in the computational nodes (with fixed communication schedule), the total (gate+ info-friction) energy consumption for reliable computation diverges to infinity as the target error probability is lowered to zero. Thus the capacity of noisy computing in this model is zero regardless of how “rate” of computation is defined: the required energy is unbounded for reliable computation regardless of how efficiently the available resources (e.g. gates, wires, etc.) are used.