No single correct decomposition
It would be reassuring if there were one true way to break a model's activations into features, but the paper's own results argue against that comfort. Change one hyperparameter and you get an equally defensible, differently-shaped dictionary.
These six connections chain that concession together: a smooth tradeoff curve with no distinguished bend, a size setting that decides which curve is even on offer while quietly creating features that never fire, and a downstream perplexity check that gives an abstract reconstruction number real stakes by showing what a modest-looking loss actually costs the model.
Six edges chain the paper's most honest concession. Sparsity loss trades off against reconstruction loss along one smooth curve with no knee, and because that curve has no distinguished bend, the sparsity-reconstruction tradeoff rules out a single canonical dictionary for sparse autoencoders: every hyperparameter setting is a defensible decomposition, not an approximation to one true dictionary. Expansion factor R sets the operating point on that same tradeoff curve, determining which curve is even on offer, while separately governing the emergence of dead features once dictionaries grow past roughly fourfold overcompleteness. Those dead features hollow out the count of dictionary feature: a row that occupies capacity and counts toward R but never fires, so nominal size overstates effective size. And because reconstruction loss alone is an abstract squared-error number with no inherent stakes, the reconstruction-perplexity metric independently corroborates it, showing a modest-looking loss still costs the model real predictive accuracy once substituted into an actual forward pass.