The Dynamical Persistence of Galactic Spirals: A Response Matrix Framework for Live Halo Wakes

The longevity of grand-design spiral structure in embedded baryonic disks remains a fundamental tension in galactic dynamics. While classical frameworks, such as density-wave and swing-amplification models, successfully describe recurrent transient instabilities, they often rely on the simplifying assumption of a static, rigid dark matter halo. This work presents a formalized analytical framework that treats the galaxy as a fully coupled, two-component system using a modified linear response matrix formulation. Because the collisionless dark matter halo relaxes via phase-mixing on timescales distinct from the dissipative baryonic disk, it retains a trailing gravitational asymmetry known as a "halo wake". We demonstrate that while individual halo wakes decay via phase-mixing over intermediate timescales (~100 Myr), they function as a dynamical memory buffer that continuously modulates the disk's swing amplification threshold. This coupled, regenerative feedback cycle biases pattern-speed evolution and extends the global coherence lifetime of grand-design spirals over secular timescales.