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 |∂ₜχᵥ|≤ c_χ induces a hierarchy of maximal admissible mode amplitudes across irreducible spectral sectors. In the linearised regime, the admissibility bound takes the universal form \[ A_nᵐᵃˣ={c_χ}{√λ_n}, \] where λₙ 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 Aₙᵐᵃˣ=c_χ/√λₙ 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 chain \[ Q_8 ⊂ 2I ⊂ SU(2). \] The analysis is further extended to the SU(3) chain, following the exact parallel structure. On the minimal SU(3)-type group Δ(27) = Heis₃(F₃), the tracelessness mechanism produces μ₃ = 0 for the Weil representation, establishing the structural anchor of the SU(3) hierarchy. On the group Σ(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 \[ λ₃ = λ3̄ = 28 \;<\; λ₆ = λ₈ = 42 \;<\; λ₇ = 48, \] with 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 χ₃(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.