Subdiffusive Valence Growth under Bounded Relational Flux: Structural Derivation of the Cascade Exponent β

The cascade exponent $\beta$, governing the power-law growth $p(n) \sim n^\beta$ of the effective relational valence along the Cosmochrony relaxation cascade, was identified in a companion paper as the sole remaining structural postulate of the admissibility sub-programme. We show that the bounded-flux constraint $|\partial_t \chi_v| \leq c_\chi$, inherited from Born–Infeld saturation, renders super-quadratic valence growth structurally impossible. Under a natural closure hypothesis relating valence to the cumulative exploration of the relational configuration space $\Omega_\chi$, the Cheeger isoperimetric inequality for Ramanujan–LPS graphs implies equation* p(n) c_ p(n), equation* from which it follows that $p(n) \lesssim C n^2$, i.e.\ any admissible power-law exponent must satisfy $\beta \leq 2$. By convention, the saturated growth rate $p(n) \sim n^2$ is identified with the maximal admissible regime; sub-saturated growth laws satisfying $p(n) = o(n^2)$ correspond to $\beta < 2$, and the physically relevant regime is $\beta \in (0,1]$ (see Remark rem:beta-convention). This structural upper bound is derived without invoking phenomenological mass data and is therefore independent of the lepton or quark sector. The sharper confinement to the phenomenologically identified window $\beta^* \in (0.09, 0.13)$ is not established here; we identify the additional geometric input required and formulate the residual problem precisely. The present result removes the last unconstrained degree of freedom of the O-series from the class of super-quadratic growth laws, consolidating the structural foundation of the programme.