From Recursive Depth to Dynamical Systems: A Formalization of Eglash's Recursive-Depth Measure of Generativity

We give a rigorous formal-language and dynamical-systems foundation for Eglash's notion of recursive depth, a measure of generativity from the theory of generative justice (Eglash, 2025). First, we show that the recursive-depth equation is not merely analogous to, but is, a parsing score: it equals the root inside score of a weighted (probabilistic) context-sensitive grammar, with each of its symbols (d, r, P, E, W, M) receiving a coherent grammatical interpretation, and it admits a closed form as a sum over root-to-leaf descent paths. The environmental-response factor E forces the weighting to depend on derivation context, placing the parse at the (weighting-sense) context-sensitive level of the Chomsky hierarchy. Second, we give a mechanical, structure-preserving conversion of any such grammar into a multitype branching process, under which derivations and population evolution are equal in distribution; the spectral radius $\lambda$ of the resulting fertility operator gives a dynamical criterion separating regeneration ($\lambda > 1$) from extraction ($\lambda < 1$). A consequence invisible to the static score is that recursive depth is not regeneration: an acyclic nesting has $\lambda = 0$, and genuine self-maintenance requires the loop network to close into a cycle of net gain exceeding one. The bridge places the analytic methods of both generative grammar and dynamical-systems theory at the service of generativity.