Quintessence with Memory: Perturbative Non-Degeneracy and the Unerasable Imprint of Early Tracking

Integrating out cold dark matter (CDM) density perturbations gravitationally coupled to a quintessence field leaves the field with a memory: an influence kernel, obtained in closed form on a matter-dominated background from canonically normalized fluid perturbations and the retarded growth Green's function alone. The kernel supports no local dissipative friction (pressureless matter has no restoring forces) but it carries a secular, memory-dominated induced mass. We ask whether this memory is observable, and we answer twice in the affirmative. First, a structural result: while at the background level the memory term is exactly absorbable into a redefinition of the quintessence potential (the familiar dark degeneracy), at the perturbative level it is not. The memory operator is diagonal on power-law modes with closed-form eigenvalue, and its logarithmic mode dispersion D(t) = -ln(t/t*) is nonzero, grows with cosmic age, and is exactly independent of the kernel's overall normalization and of any time-dependent prefactor; it therefore ports unambiguously to a LCDM background, where the archive freezes while D keeps growing. Any local potential has D=0: the dispersion is a theorem-grade discriminant for the memory class. A moment-expansion shadow yields a kinematic mass-to-friction ratio -1/(Ht), equal to -3/2 in the matter era for any potential class. Second, a phenomenological result: because the memory acts mode by mode, with coupling proportional to 1/k^2 and no mode sum, its magnitude is fully calculable, free of the ultraviolet sensitivity that afflicts the homogeneous drag. For thawing quintessence the induced mass today is small, |M0|/H0^2 ~ 4e-2 at horizon scales. For a field that tracked during the early matter era, the operator's strength reaches 10-100 H0^2 at large scales, essentially independently of when the field left the attractor. We verify the consequence in the exact coupled linear system (synchronous gauge, validated against the textbook LCDM growth suppression and tolerance-stable): two fields with the same potential and the same present-day background, one that tracked and exited, one that thawed in place, differ today in their gauge-invariant dark-energy density contrast by a factor 5-9 at k/(a0 H0)