arXiv:2609.02363v2 Announce Type: replace-cross Abstract: This paper shows that the optimal gain of the risk-sensitive linear quadratic regulator (LQR) problem is invariant under input randomization when the cost penalizes the realized noisy input. This may appear counterintuitive at first glance because certainty equivalence does not hold for risk-sensitive LQR and input randomization inflates the effective process noise. Nonetheless, the gain is preserved because the input noise enters not only the system dynamics but also the cost functional, and its total effect eventually vanishes. Consequently, the optimal gain and the Riccati recursion need not be recomputed, and the increment in the optimal cost can be readily evaluated in closed form. This result facilitates the use of risk-sensitive LQR in applications that employ input randomization for privacy or exploration, such as dynamic watermarking for replay attack detection.