Spectral Admissibility under Bounded Flux: Non-Abelian Mode Selection on Cayley Graphs of SU(2) and SU(3) Subgroups

We present the first analytically tractable instance of the spectral admissibility programme proposed in the Cosmochrony framework. On the undirected Cayley graph of the quaternion group Q₈, with generators S=\ i, j, k\, a local bounded–flux constraint |ₜᵥ| c_ induces a hierarchy of maximal admissible mode amplitudes across irreducible spectral sectors. In the linearised regime, the admissibility bound takes the universal formₙ^=c䂸, ₙ are the Laplacian eigenvalues of the graph. This produces a spectral hierarchy in which the non-abelian sector admits a larger maximal effective amplitude than the nontrivial abelian sectors by a factor 4/3. We then perform a weakly nonlinear modal reduction of the discrete Dirac–Born–Infeld dynamics and show that the first nonlinear corrections preserve this hierarchy. Solving the one-mode DBI dynamics exactly on the Q₈ graph, we further show that the same envelope Aₙ^=c_/ₙ coincides with the exact saturation boundary, demonstrating structural stability beyond the perturbative regime. The analysis is extended to the binary icosahedral group 2I, the largest finite subgroup of SU (2), where the same admissibility envelope generates a four-level hierarchy across spin sectors with contrast 14/9. Together these results provide explicit realizations of spectral admissibility along the chain₈ 2I SU (2). \ The analysis is further extended to the SU (3) chain, following the exact parallel structure. On the minimal SU (3) -type group (27) = Heis₃ (F₃), the tracelessness mechanism produces ₃ = 0 for the Weil representation, establishing the structural anchor of the SU (3) hierarchy. On the group (168) = PSL (2, 7) of order~168, the analogue of 2I for SU (3), a complete character-theoretic computation for the canonical generating set S = C₄ (the 42 elements of order~4) yields\₃ = ₃ = 28\;<\;₆ = ₈ = 42\;<\;₇ = 48, admissibility contrast 12/7 1. 309. This establishes that the fundamental and anti-fundamental representations of SU (3) are co-admissible and strictly dominate all other non-trivial sectors, in exact analogy with the SU (2) result on 2I. The algebraic origin is the character identity ₃ (g) = 1 for all g C₄, the SU (3) counterpart of the golden-ratio identity on 2I. More generally, the results show that bounded substrate capacity can act as a purely spectral filter favouring low-lying non-abelian sectors of relational dynamics, and that this mechanism operates uniformly across the SU (2) and SU (3) chains.