Variable-Length Markov Chains on Finite Quivers: Boundary-Window Identifiability, Exact Depth, and Local Rank Comparison

Variable-length Markov chains on finite quivers provide a natural framework for context-dependent stochastic growth under incidence constraints. I study quiver-valued variable-length Markov chains observed through finite boundary windows and develop a first-order theory of visible-depth identifiability via stationary visible one-step transition laws and their restricted differentials on prescribed tangent blocks. For visible depth $m$, the main object is the stationary one-step informative map $q_{\mathcal{Q}}^{(m)}$. In the edge-homogeneous regime, once the local visible support is fixed and the representation hypothesis holds, all admissible visible depths encode the same edge-level extension law and hence have the same first-order rank. In the exact-depth regime of context length $r$, the depth-$r$ boundary process is the canonical finite-state Markov chain, smaller visible windows are deterministic truncations, and every coarser informative map factors $C^1$-smoothly through the depth-$r$ informative map on the relevant affine transition-array neighborhood. Hence rank cannot increase beyond depth $r$. After quotienting a tangent block by directions already invisible at depth $r$, I characterize strict coarse-depth loss exactly by coarse rank deficiency, equivalently by strict rank drop from depth $r$ to depth $m$ on the original block. I also give subspace-based and global selected-coordinate criteria, a global one-coordinate branching criterion, and an explicit depth-two example. Under full fine-depth rank and strict coordinate-rank loss at every smaller depth, a global coordinate-rank theorem yields $m_*(T,\theta_0)=r$. Reduced local coordinates remove stochastic redundancies, first-order criteria are invariant under $C^1$ reparameterization, and the statistical and LAN consequences remain conditional on additional estimation and likelihood-level hypotheses.