Neural correlates of the cognitive map are well documented, yet how a neural circuit turns a stored map into an obstacle-avoiding route remains unclear. Here we show that a hippocampal network with appropriate recurrent connections achieves optimal goal-directed navigation through its own relaxation dynamics. The recurrent weights represent the transition probabilities between the locations the neurons encode, walls are reflected by vanishing weights, and behavioral-timescale synaptic plasticity (BTSP) writes this connectivity during goal-free exploration. When a goal is recalled, a top-down current clamps the goal neuron and the population relaxes into a goal-locked activity field. We prove that this field is the desirability field of a linearly-solvable Markov decision process: the relaxed state is the exact optimum of a stated control problem, the KL-regularized objective of Todorov, and the optimal next-step distribution is carried by the branch synaptic currents. The identity extends to directed transitions, and a weak-cost limit returns the Green's function of the recurrent Laplacian. A place-tuned readout converts the field's log-gradient into a direction of travel, and the same synapses answer a predictive query when the source is placed at the current location. Simulations on mazes with complex obstacles show near-shortest routes, an identity that survives a thousand-fold range of recall amplitude, the loss of 80% of place cells, and tens of thousands of states, low-rank updates of every plan after a local change, and a transient that is anytime: the local action is ready long before the field converges. We hope this offers a general circuit principle for planning in abstract cognitive maps beyond spatial navigation.

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