Dictionary expansion factor (R) — sets the operating point on → Sparsity-reconstruction tradeoff (no single correct decomposition)

The tradeoff curve traced out by sweeping alpha (Appendix B, Figures 6-7) is measured at a fixed dictionary size; R determines which curve you are on in the first place. Appendix C.1 (Figure 8) shows that shrinking R below 1 -- an undercomplete dictionary -- still yields interpretable features, but each such dictionary explains less of the model's overall activation variance than a larger one, because a smaller R simply gives the tradeoff less capacity to work with. Conversely, larger dictionaries push the whole curve toward explaining more variance at a given sparsity level, without any single feature becoming less interpretable on average. So where alpha moves you along a fixed tradeoff curve, R redraws the curve itself -- the two hyperparameters are not interchangeable knobs on the same axis.