[Submitted on 26 Oct 2026]
Abstract:On a chain of $N$ states, any local learning rule with bounded communication range needs $Theta(N)$ 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 $Omega(N^{1/d})$ 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 $d(m-1)=Theta(N^{1/d})$ of the output, matching the lower bound. In the realizable-circuit model, depth and credit depth are therefore independent parameters: represented depth $Theta(N)$ and causal credit latency $Theta(N^{1/d})$ 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
Submission history
From: Benjamin Prada [view email]
[v1] Mon, 26 Oct 2026 18:47:06 UTC (378 KB)
