Spectral Hierarchy, Critical Thresholds, and Universal Scaling in the TCFQ Programme for N > 4

We extend the TCFQ programme from N = 4 (established in R25R28) to general N, with G = SU(2) and eective rank r ≤ 4. For each pair (N, r), the spectral polytope ΛN is classied, its extreme vertices are identied, and the variational problem for the separable functional S(λ) = Pi f(λi), f(x) = x2 + c3x3 + c4x4, is solved exactly. The principal results are: (T1) a universal critical curve f′′(N/r) = 0, with a dichotomy between admissible (N/r ≤ 1) and inadmissible (N/r > 1) isospectrum; (T2) the dominant unstable mode at the rst bifurcation is always the balanced splitting (r/2, r/2), selected by vanishing of the cubic term; (T3) a hierarchical stability principle: each spectral level λk loses stability when f′′(λk) = 0; (T4) a structural threshold c4∗(N, r) below which the asymmetric vertex (N−r+1, 1, . . . , 1) competes with the balanced branch; (T5) a qualitative change of regime at N = 8, where the critical branch is born with the rank constraint active; and (T6) a universal asymptotic law c4∗(N, r) ∼ [(r − 2)/r] N−2, with a natural renormalisation scale c˜4 = c4N2. All results are exact algebraic consequences of the SU(2) groupoid structure; no large-N limit is invoked.