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In this work, we formally derive the optimal guidance term for Stochastic Interpolants (SI) by analyzing the value function's time evolution via the Backward Kolmogorov Equation, establishing a modified drift that theoretically guarantees sampling from a target distribution. We apply this framework to real-time control through the Streaming Stochastic Interpolant Policy (SSIP), which generalizes the deterministic Streaming Flow Policy (SFP). Unifying this guidance law with the streaming architecture enables fast and reactive control. 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