Perfect bidirectional-cbit transmission criterion

For a two-qubit unitary $U$, let $\beta$ be an actual interaction-content witness: $U$ is locally equivalent, up to global phase, to $U_\beta=\exp[-i\sum_{k=1}^3\beta_k\sigma_k\otimes\sigma_k]$, where $\sigma_1,\sigma_2,\sigma_3$ are the Pauli matrices $X,Y,Z$ and $\pi/4\ge\beta_1\ge\beta_2\ge|\beta_3|$. Transmission means one use of the gate with normalized pure product inputs, without ancillas or prior shared entanglement. Encoding and decoding axes are free local choices fixed before the unknown input. Classical messages are encoded in orthogonal states and decoded with certainty; qubit transmission preserves every superposition with the same encoding and recovery. In simultaneous tasks, a receiver may choose its local decoder according to its own sent classical bit. Then

The third coordinate is unrestricted within the canonical chamber, $|\beta_3|\le\pi/4$. The two input encodings are chosen before either bit; each receiver may use the sent bit to select a decoding observable.