Reconfigurable intelligent surfaces (RIS) can be reciprocal, with a symmetric unitary scattering matrix, or non-reciprocal, with a general unitary one. The question is when a reciprocal RIS achieves the same optimal performance as a non-reciprocal RIS under general performance metrics.

Performance metrics are modeled as functions of the product of the RIS scattering matrix and the transmitter-RIS channel. Lemmas characterize which matrix products unitary and symmetric-unitary matrices can realize. These lead to necessary and sufficient conditions, refined to account for transmit precoding. All statements were formally verified in Lean 4 with assistance from Claude.

A reciprocal RIS performs as a non-reciprocal RIS if and only if some optimal non-reciprocal RIS makes the transmitter-RIS-transmitter round-trip channel symmetric. With precoding, this condition only needs to hold for the round-trip channel seen by the transmitted streams. It therefore holds when the transmitter-RIS channel has rank one or when single-stream transmission is optimal.