Admissible Frontier Saturation and the Cascade Exponent: A Representation-Theoretic Obstruction and its Matrix-Level Refinement

The Cosmochrony relaxation cascade generates an inter-generational mass hierarchy through the contraction of the Kesten–McKay spectral support as the effective valence $p(n)$ grows. The cascade exponent $\beta$ governing $p(n) \sim n^\beta$ was identified in companion papers as the sole remaining structural parameter of the admissibility sub-programme: O3 constrains ${\beta^*} \in (0.09, 0.13)$ from lepton masses, while O4 derives the quadratic upper bound $p(n) \lesssim Cn^2$ from bounded-flux and Cheeger arguments, leaving a factor of $10$–$20$ unexplained. The present paper addresses this gap by introducing the notion of admissible frontier: the subset of the raw BFS boundary whose elements introduce genuinely new directions in the admissible spectral subspace. We prove that the vertex-based admissible frontier saturates completely at scale $O(|{\mathrm{Cl}}(G)|) = O(q)$, far below $|G| = O(q^3)$, through a direct consequence of the representation theory of $G = {\mathrm{PSL}}(2, {\mathbb{F}}_q)$. This Proposition is a theorem, independent of any phenomenological input. We then examine transition-based refinements. Character-based transition fingerprints collapse to the same saturation behaviour on LPS graphs, where all generators lie in a single conjugacy class. Fixed-dimensional matrix proxies produce a clear pre-saturation decay of the admissible frontier fraction but with a $q$-independent saturation threshold, disqualifying them as structural mechanisms. A Steinberg-based fingerprint using the permutation action on $\mathbb{P}^1({\mathbb{F}}_q)$ achieves the first $q$-structural saturation ($|S^*| \sim q^{1.15}$, $|S^*|/|G| \to 0$), but the pre-saturation window remains too short to extract a reliable effective exponent. We conclude that the smallness of ${\beta^*}$ is not a representation-theoretic dimension effect but a genuinely dynamical redundancy phenomenon, and identify $\dim \mathcal{H}_{\mathrm{mat}} = O(q^2)$ as the minimal ambient dimension for a structurally complete theory. The correct class of mechanisms is isolated; quantitative extraction of the exponent is deferred to the next paper.