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תקציר מקורי באנגליתarXiv:2610.02195v1 Announce Type: new Abstract: The generalized Schr\"odinger bridge on a graph moves mass between two distributions while charging a cost for the states visited. It has been approached by learning the rates of a controlled continuous-time Markov chain, with a temporal-difference penalty that restores the cost. A state cost folds into the reference process as a Feynman-Kac tilt. The cost-augmented bridge is then a plain bridge against the tilted reference, and the penalty is unnecessary. The bridge is computed exactly by alternating two endpoint rescalings, each one sparse matrix-exponential application; nothing is discretized in time or learned. The alternation converges at a rate set by the endpoint coupling alone. For a quadratic congestion cost on time-averaged occupanc