Underlying algebraic organization of the scaling sequence

Determine whether the algebraic sequence R^a → R^{a^3} → R^{a^6} → R^{a^{12}}, with a = (z_m/R)^{1/6}, reflects a deeper organizing principle underlying the multiple self-similar scalings observed during post-breakup recovery of the quasi-two-dimensional Hele-Shaw fluid interface.

Background

The study identifies four successive self-similar regimes in the recovery of a fluid interface after breakup, characterized by the R_{1/6}-, R_{1/2}-, R_1-, and R_2-scalings. Their associated horizontal scales can be rewritten algebraically as R^a, R^{a^3}, R^{a^6}, and R^{a^{12}}, where a = (z_m/R)^{1/6}. The authors explicitly leave unresolved whether this striking algebraic pattern has any deeper dynamical or theoretical significance.