Optical Circuit Switches (OCS) make inter-pod wiring in AI clusters programmable, posing design questions on wiring choice, symmetry departure, and reconfigurations. We present a unified information-geometric framework to address these issues.A wiring is a point in a probability simplex over circuits within a degree polytope. Departure from a symmetric Fat-Tree prior
q \mathbf{q} q is measured by
\DKL ( ⋅ ∥ q ) \DKL(\cdot \,\|\, \mathbf{q}) \DKL ( ⋅ ∥ q ) . The cost
U U U is the smoothed DAG critical-path length, whose gradient equals path marginals of a Gibbs distribution via forward--backward recursion.For free energy
F β 1 = U + β 1 − 1 \DKL ( ⋅ ∥ q ) F_{β_1} = U + β_1^{-1} \DKL(\cdot \,\|\, \mathbf{q}) F β 1 = U + β 1 − 1 \DKL ( ⋅ ∥ q ) , we prove well-posedness, monotonic dependence on
β 1 β_1 β 1 , adaptation bounds via Jeffreys divergence, and a symmetry principle for static optimality. An entropic proximal-gradient method updates multiplicatively with a
∣ V ∣ |V| ∣ V ∣ -dim dual KL projection, achieving linear convergence.For time-varying traffic, a threshold policy yields regret
O ( Γ C V 1 T ) O(\sqrt{ΓC V_1 T}) O ( Γ C V 1 T ) against the switching-free optimum. Traffic changes are detected by a multinomial GLR--CUSUM detector with non-asymptotic false-alarm bounds. Multi-job generalized-mean fairness aligns with
α α α -fairness (
α ≥ 2 α\ge 2 α ≥ 2 ). Experiments on pipeline, data, and MoE workloads validate the framework against integer programming.