We measure how much one extra recurrence is worth to a looped (depth-recurrent) language model, in equivalent unique parameters. From an iso-depth sweep of 116 pretraining runs across recurrence counts r ∈{1, 2, 4, 8} spanning ∼50× in training compute, we fit a joint scaling law L = E + A (N_once + r^φ N_rec)^-α + B D^-β and recover a new recurrence-equivalence exponent φ= 0.46 at R^2 = 0.997. Intuitively, φ tells us whether looping a block r times is equivalent in validation loss to r unique blocks of a non-looped model (full equivalence, φ=1) or to a single block run repeatedly with no capacity gain (φ=0). Our φ= 0.46 sits in between, so each additional recurrence predictably increases validation loss at matched training compute. For example, at r=4 a 410M looped model performs on par with a 580M non-looped model, but pays the training cost of a 1B non-looped one. On a five-axis downstream evaluation, the gap persists on parametric-knowledge tasks and closes on simple open-book tasks, while reasoning tasks are not resolvable at our compute budgets. For any looped LM, our φ converts the design choice of r into a predictable validation-loss cost, and future training recipes and architectures can be compared by how much they raise φ above 0.46.
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