DASH-LISTA: Disentangling History Feedback in Hybrid-ISTA Networks

History can improve an unfolded solver's learned function or become integral to a particular trained computation. We introduce DASH-LISTA to separate these roles through a state-preserving Hybrid-ISTA interface that controls correction information and injection location. Matched retraining, fixed-backbone replacement and equal-norm interventions measure architecture benefit, model dependence and directional effects. Under a shared feedback-gain cap, four-tap FIR improves on two-time and nonlinear current controls by $4.25\pm 0.66$ and $9.25\pm 0.90$ dB in three independently trained confirmation instances, each tested on 100,000 previously unused signals; all paired gains are positive. Original-gain effects remain tail-sensitive, with one short-history ordering reversal. Equal-norm reflection identifies useful older directions. In images, current replacement retains a dual-branch cost after 6,400 adaptation steps and relaxation to six independent bounded stage gains: about $0.026$ dB in each of two checkpoint cohorts. The same relaxation slightly improves refinement-only reconstruction. On test50, dual-branch reflection at stages four and five has a larger terminal effect through the direct path in all six checkpoints. Independent retraining yields nearly coincident image endpoint means. These results separate architecture benefit from checkpoint dependence and locate pathway-specific effects of older directions.