Recursive language-model loops often settle into recognizable attractor-like patterns. The practical question is how much injected text is needed to move a settled loop somewhere else, and whether that move lasts. We study this in 30-step recursive loops by separating the model from the context-update rule: append, replace, and dialog updates expose different histories to the same generator. The main result is that persistent redirection in append-mode recursive loops is memory-policy-conditioned. Under a 12,000-character tail clip, destination-coherent persistence plateaus near 16 percent and retained source-basin escape near 36 percent at dose 400; neither crosses 50 percent. Under a full-history protocol, retained source-basin escape crosses 50 percent near 400 tokens and saturates at 75-80 percent by 1,500 tokens; destination-coherent persistence first reaches 0.50 near 1,500 tokens (Wilson 95 percent CI [0.41, 0.61]). A four-step falsification battery (heterogeneity control, granularity sweep with hierarchical macro-merge, transition-entropy diagnostic, and long-horizon trajectory continuation) recasts the high-dose destination-coherent dip as a finite-horizon, endpoint-definition-sensitive feature rather than a stable structural asymmetry. Half the canonical magnitude is endpoint timing; the residual drops 73 percent from -0.143 at step 29 to -0.039 at step 79 under the frozen canonical cluster basis, bootstrap interval straddling zero. Replace-mode raw switching is near-saturated under the default protocol but largely reflects state-reset overwrite: insert-mode probes drop it to 12-32 percent. We report 37 experiments on gpt-4o-mini with within-vendor replication on gpt-4.1-nano. Recursive-loop evaluations should distinguish transient movement from durable escape, subtract stochastic floors, and treat context-update rules as safety-relevant design choices.