When is Noisy State Information at the Encoder as Useless as No Information or as Good as Noise-Free State?

For any binary-input channel with perfect state information at the decoder, if the mutual information between the noisy state observation at the encoder and the true channel state is below a positive threshold determined solely by the state distribution, then the capacity is the same as that with no encoder side information. A complementary phenomenon is revealed for a similarly defined quantity.