Dark-Sector Memory as a Collision Model: Friction Requires Springs, Memory Requires Re-Encounter

Integrating out cold dark matter (CDM) density perturbations gravitationally coupled to a quintessence field yields an influence kernel that supports no local dissipative friction but carries a secular induced mass [companion papers]. In this paper we show that this kernel has, identically, the structure of a repeated-interaction (collision) model, K(t,t') = c rho_c(t) G_delta(t,t') rho_c(t'), i.e. a write weight, a record propagator, and a read weight. Three results follow. First (structural), friction requires springs: a nonzero Kubo friction coefficient at frequency Omega>0 requires ancilla spectral weight at real Omega; free-falling and unstable (Jeans-type) records do not dissipate and instead generate secular induced mass. Second (the main result), memory requires re-encounter: ballistic records saturate and produce an erasable memory, M0 ~ ln t / t2/3 -> 0, whereas the Dyson resummation of gravitational re-encounters reconstructs exactly the growing mode and the scaling law of the companion no-go paper; the unerasable memory of the dark sector is the geometric series of record re-collisions. Third (exact in the linear-Gaussian sector), the unconditioned mean obeys the memory equation of motion and is blind to record readout, whereas conditioning produces zero-mean white innovations and monotonically decreasing conditional variance; because readout is quantum-nondemolition with respect to the system, it induces no net average change in system energy. Mean-level observables see only the kernel, and we close this direction within linear theory: the joint statistics of system and records are Gaussian and fixed by the kernel, the collision weights, and the readout noise, so the conditioned structure inherits the gravitational screening of the kernel. Structure beyond the kernel becomes observable only through nonlinear record dynamics or non-gravitational couplings.