What question did this study set out to answer?

This work aims to bridge the gap in understanding the cascade exponent and its implications for spectral theory.

July 13, 2026Open Access

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

Key Points

This work aims to bridge the gap in understanding the cascade exponent and its implications for spectral theory.
Introduced the notion of admissible frontier and its complete saturation at scale O(|Cl(G)|)
Applied representation theory to derive saturation behavior in vertex-based and transition-based frameworks
Evaluated the impact of character-based and matrix-level refinements on the saturation threshold.
Identified that the smallness of β* is a dynamical redundancy, not solely a representation-theoretic effect
Achieved a form of structural saturation that indicates a new direction in the admissible subspace
Determined that fixed-dimensional matrix proxies do not serve as effective structural mechanisms.

Abstract

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 governing p (n) n^ was identified in companion papers as the sole remaining structural parameter of the admissibility sub-programme: O3 constrains ^* (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^*| q^1. 15, |S^*|/|G| 0), but the pre-saturation window remains too short to extract a reliable effective exponent. We conclude that the smallness of ^* is not a representation-theoretic dimension effect but a genuinely dynamical redundancy phenomenon, and identify H₌₀ₓ = 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.

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