The forward and reverse Kullback-Leibler (KL) divergences arise as limiting objectives in learning and inference, yet they induce markedly different inductive biases that cannot be explained by expectation terms alone. We introduce the Surprisal-Rényi Free Energy (SRFE), a normalized Rényi-Chernoff log-moment functional of the likelihood ratio. Although SRFE is algebraically related to Rényi divergence for fixed values of the order parameter, its endpoint-symmetric normalization yields a distinct free-energy viewpoint for analyzing how forward- and reverse KL behavior emerges near singular limits. We show that SRFE recovers the forward and reverse KL divergences as endpoint limits and derive local expansions around both regimes in which the variance of the log-likelihood ratio appears as a first-order correction away from the KL objective. These expansions reveal an explicit mean-variance tradeoff governing departures from KL-dominated behavior. We further establish a Gibbs-type variational characterization in terms of weighted KL projections and prove that SRFE controls large deviations of excess code length through Chernoff-type bounds, giving a Minimum Description Length interpretation of the functional. We evaluate SRFE on two synthetic settings (two-dimensional Gaussian-mixture variational inference and multi-class classification) and the long-tailed CIFAR-10 and CIFAR-100 (CIFAR-LT) datasets. On CIFAR-LT, retraining the best configuration of each loss over five seeds, SRFE matches cross-entropy accuracy on CIFAR-10-LT and gives mixed results on CIFAR-100-LT, improving few-shot accuracy at one of three imbalance factors. Together, these results position SRFE as a variance- and tail-sensitive, normalized Rényi-Chernoff free energy that clarifies the geometric, statistical, and large-deviation structure underlying the transition between forward- and reverse-KL limits.
On a chain of $N$ states, any local learning rule with bounded communication range needs $Θ(N)$ rounds before the loss can influence the earliest parameters, and this latency is usually attributed to depth. We show that it is a property of the communication geometry rather than of depth. For $N=m^d$ states placed at constant density in $d$ dimensions, a light-cone argument gives an $Ω(N^{1/d})$ lower bound on the number of bounded-range rounds that any local algorithm needs before its gradient estimates can be sign-correct at every parameter. We construct a unit-range, degree-$2d$ directed acyclic graph that contains a Hamiltonian computation of length $N-1$ and in which every state lies within directed distance $d(m-1)=Θ(N^{1/d})$ of the output, matching the lower bound. In the realizable-circuit model, depth and credit depth are therefore independent parameters: represented depth $Θ(N)$ and causal credit latency $Θ(N^{1/d})$ coexist, with a separation of order $N^{1-1/d}$. SpatialPC realizes this separation within predictive coding. Credit is propagated by adjoint prediction dynamics (APD), a local update whose $K$-round state on a settled acyclic graph is the truncated reverse-path sum $a^K=\sum_{k<K}(J^\top)^k b$, with error exactly $(J^\top)^K a^\star$. Graph geometry fixes when credit can first arrive at a state; the Jacobian-weighted path tail fixes how many further rounds accurate credit requires. A sparse two-dimensional Moore compiler retains the $N-1$ backbone with $\tfrac32N-2$ unit-range edges and has measured worst-node reachability $\sqrt N+2$ up to $N=65{,}536$, where a 65,535-edge backbone is reached in 258 local hops. With $T=2\sqrt N$ rounds, SpatialPC keeps finite-round task gradients aligned with settled gradients (cosine $0.93$ on MNIST and $0.97$ on Fashion-MNIST at $N=256$) and reaches $97.6\%$ and $88.1\%$ test accuracy after 20 epochs.