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 β governing p(n) ~ n^β was identified in companion papers as the sole remaining structural parameter of the admissibility sub-programme: O3 constrains β^* ∈ (0.09, 0.13) from lepton masses, while O4 derives the quadratic upper bound p(n) Cn² 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(|Cl(G)|) = O(q), far below |G| = O(q³), through a direct consequence of the representation theory of G = PSL(2, Fq). 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 P¹(Fq) achieves the first q-structural saturation (|S^*| ~ q1.15, |S^*|/|G| → 0), but the pre-saturation window remains too short to extract a reliable effective exponent. We conclude that the smallness of β^* is not a representation-theoretic dimension effect but a genuinely dynamical redundancy phenomenon, and identify Hₘₐₜ = O(q²) 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.