Cold Dark Matter Cannot Supply Local Dissipative Friction to Dark Energy: A Structural No-Go from the Influence Kernel

Preprint of a manuscript submitted to Physical Review D (June 2026). All numerical claims are reproduced by publicly archived code: DOI 10.5281/zenodo.20547191. Abstract: A substantial literature on interacting and dissipative dark energy postulates a local dissipative friction term in the quintessence equation of motion, with the friction coefficient tied to the dark-matter density - a late-time analogue of warm inflation. Separately, recent open effective field theories of dark energy parametrize dissipative dynamics sourced by an unspecified environment. We ask whether such friction can be derived, in the Caldeira-Leggett sense, by treating cold dark matter (CDM) as the environment, and we answer in the negative on structural grounds. Integrating out CDM density perturbations coupled to the field through gravity, with the canonical fluid normalization, yields an influence kernel obtained in closed form on a matter-dominated background. This kernel fails every standard requirement of a dissipative (Kubo/FDT) environment, for five independent reasons: (i) the bath has no real frequencies - pressureless density modes are Jeans-unstable at every wavenumber, so there are no oscillators to absorb energy; (ii) the effective spectral function has the monotonicity opposite to Ohmic, peaking at zero frequency and decreasing; (iii) the kernel is non-stationary, with an exact scaling law in place of time-translation invariance; (iv) there is no system/bath timescale separation, the bath memory spanning the age of the Universe; and (v) the kernel's overall magnitude is dominated by the smallest (nonlinear) scales and is therefore not calculable within linear theory. A small sound speed does not restore a local dissipative limit: an integration-by-parts bound shows that the oscillatory component of the aggregated kernel decays with conformal-time separation while the secular component grows, and numerically no restoration occurs across the tested range of sound speeds. What the coupling does produce is characterized by a moment expansion: a secular induced mass set by initial-time memory, alongside a genuine but never dominant friction-like component. Pressureless matter, having no restoring forces, cannot dissipate; it can only record.