Distance Structure of Insertion Kernels and Local Port Response in Non-Reciprocal Su--Schrieffer--Heeger Chains

Abstract: Non-reciprocal lattices offer a flexible setting for directional wave transport, raising the question of how enhanced internal scattering translates into transmission between fixed external ports. We address this question for non-reciprocal Su--Schrieffer--Heeger (SSH) chains with staggered gain and loss by relating the reference-dependent insertion kernel $K_p=VG_pV$ to the coherent transmission intensity $T$. On a fixed finite graph, a strong-insertion expansion gives a contribution proportional to $γ<sup>{1-d}$, whose coefficient is the signed sum of shortest directed paths of length $d$. When this sum is nonzero, it sets the leading behavior independently of fixed diagonal losses; when it cancels, those losses can enter the first surviving higher-order term. For a fixed cross-sublattice endpoint channel, transmission has a strictly negative slope at every nonzero differentiable stationary point of the kernel magnitude. In the weak-insertion regime, the reference-system coefficient $χ=-\mathrm{Re}\,\mathrm{Tr}[(G_{0p}Z)<sup>2]$ predicts the initial transmission change for symmetric SSH ports, and $χ&gt;0$ together with a strictly stable baseline guarantees local stable enhancement. Finite-chain calculations and driven dynamics connect these results to observable response and distinguish asymptotic convergence from transient settling. A strictly dissipative short-chain example exhibits local enhancement without net power gain. Together, these results separate graph-path structure, coherent port response, and dynamical establishment, providing channel-specific criteria for interpreting internal scattering.