Geometric Feedback for Pollicott–Ruelle Resonances in Finite Feedforward Networks

We construct geometric feedback closures for finite directed acyclic multigraphs in a class of graph-directed open Axiom A suspensions. Vertices are realized by mixing basic sets and edges by heteroclinic return tubes. A marked closing tube converts the marked source-to-target paths into cycles; in the two-terminal spanning case its activation merges the vertex basic sets into a single mixing basic set. Such a merger cannot occur within a common locally maximal hyperbolic continuation, so the geometric family is one-sided at the closing parameter. Its normalized guarded return operator is nevertheless holomorphic in the closing weight. Suppose that the component resonances are simple and aligned, and let L be the maximum length of a marked source-to-target path. The marked entry of the feedforward resolvent has pole order at most L + 1; its coefficient at that order is the weighted sum of the longest paths. Pairing this coefficient with the closing channel gives a presentation-independent characteristic jet. If a longest path is Hamiltonian, the |V| colliding resonances split at scale z^{1/|V|}. The same coefficient occurs in the relative flat trace of primitive periodic orbits that traverse the closing tube once. The formulas persist under deformations that preserve the aligned simple component resonances and the closing channel.