On-Shell Perturbative Equivalence and Nonadiabatic Action Identifiability in a Pressureless Interacting Dark Sector

his paper studies which off-history interaction-action data can be identified from on-shell perturbations in the pressureless scalar-dark-matter class L = X - V(phi) - n h(Z,phi). On every regular nonzero-Z patch, the pressureless interaction action modulo a dynamically null shift is represented by the mass function M(Z,phi). Expanding background-equivalent completions transverse to the realized graph Z = Zbar(phi), the paper proves an on-shell filtration theorem: equality of the complete local perturbative flow map through order p is equivalent, under the stated regularity and well-posedness assumptions, to vanishing of the first p transverse coefficient functions. Each perturbative order therefore identifies one new transverse action jet. The work gives the transverse variable a nonadiabatic relative-clock interpretation, quantifies its superhorizon and squeezed suppression for adiabatic initial data, and derives a retarded Green representation in which the first quadratic unidentified profile enters as a linear functional. It also identifies the regularity boundary: smooth flat off-trajectory deformations can evade every finite perturbative order, while analytic and appropriate quasianalytic classes remove this beyond-all-orders residual.