Nonadiabatic Long-Mode Response and Transverse Action-Jet Tomography in a Pressureless Interacting Dark Sector

This paper studies how the hidden transverse action-jet hierarchy of a reconstructed pressureless interacting dark sector becomes accessible when the long perturbation is genuinely nonadiabatic. A prescreen of the calibrated family finds no useful critical-surface amplification on the relevant interval. Using the endpoint-regular transverse coordinate and two independent constraint-compatible homogeneous nonadiabatic seeds, the work shows that the long transverse transfer can remain finite as k_L approaches zero, while the comparable adiabatic long leg is gradient suppressed. For an order-r first-visible action-sensitive source with one quasi-static short leg, the resulting hierarchy scales schematically as epsilon^(r-1) kappa_S^(-2), rather than carrying a kappa^(-2r) suppression on every transverse leg. A fixed-output factorization lemma isolates the soft scaling, and the response-order analysis shows that, once lower jets are fixed, successive transverse derivatives first expose successive action-jet coefficients. The observable object is the chart-invariant contraction C_r (delta F)^r, not the coordinate coefficient C_r alone. The paper therefore provides a source-level tomography hierarchy and states explicitly the boundary to the subsequent primordial-population analysis.