Goals as Reverse‑Time Active‑Inference Agents - A Schrödinger‑Bridge Formulation for Bidirectional Control

Active‑Inference and the continuous‑time Schrödinger bridge solve the same optimization: steer a noisy system from an initial probability cloud to a target one while straying as little as possible from its natural dynamics. The bridge is governed by a single scalar potential whose gradient splits into two complementary drifts—one pushes the present toward the goal, the other lets the goal propagate backward through time. Because both drifts minimize the same free‑energy functional, the target distribution itself behaves as a reverse‑time Active‑Inference agent. If the potential factorizes into an internal–active part and a sensory–external part, the familiar Markov‑blanket partition remains intact in both temporal directions. Under mild smoothness and constant‑diffusion assumptions, any Schrödinger‑bridge algorithm therefore doubles as a turnkey Active‑Inference controller and places teleological talk on a rigorous, time‑symmetric footing.