From Recursive Depth to Dynamical Systems - A Formalization of Eglash's Recursive-Depth Measure of Generativity 【（Preliminary）Draft】

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

Author: Wanhong Huang · June 2026 · (working draft)

Abstract

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. 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 ρ of the resulting fertility operator gives a dynamical criterion separating regeneration (ρ > 1) from extraction (ρ < 1). A consequence invisible to the static score is that recursive depth is not regeneration: an acyclic nesting has ρ = 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.

Keywords: generative justice; recursive depth; weighted context-sensitive grammar; semiring parsing; multitype branching process; spectral radius; regeneration.

At a glance

Let G be a weighted context-sensitive grammar with parent–child edge weight w_ij = W_j d_j P_j^(−1) E_j.

• Claim 1 (parsing). Eglash’s recursive depth R is realized as the root inside score of G, which admits the root-to-leaf path-sum closed form R = ⟨S⟩ = Σ over leaf paths of the product of edge weights w_ij. The dependence of the weight on context through E pushes the parse to the (weighted) context-sensitive level.
• Claim 2 (dynamics). Reading w_ij as an expected number of offspring turns G into a multitype branching process with fertility matrix M = (w_ij). The same R is then its expected total progeny, R = e_S (I − M)^(−1) 1 (finite iff ρ < 1, where ρ := ρ(M)), and the spectral radius ρ is the asymptotic growth rate: self-sustaining ρ > 1, critical ρ = 1, declining ρ < 1.

A consequence invisible to the static recursive-depth score: recursive depth is not regeneration. An acyclic nesting has ρ = 0; genuine self-maintenance requires the loop network to close into a cycle whose net gain exceeds one.