Degree Extensions, Universal Integral Criterion, and Effective Rank for SU(n)

We present three consecutive advances in the TCFQ programme for G = SU(2) and its extension to G = SU(n). R30 extends the functional to degree 6 and computes the boundary energy difference ∆(6) frontera exactly; it is shown that the structural threshold c∗4(N, r, c6) decreases strictly with c6, and that threshold elimination requires c3 < −1/(6ξ + 3). R31 replaces the degree-dependent analysis by a universal integral representation: ∆frontera = R1A Wr(t) f′′(t) dt, where Wr is a sign-changing triangular kernel whose positive lobe over [2ξ + 1, rξ + 1] carries all the structural information; the renormalised weights are νk(r) = (r2k−1(2k − r) + 22k)/(r − 2)2, positive if and only if r ≤ 2k. R32 determines the effective rank for SU(n): the quaternionic accident SU(2) =∼ S3 that sets reff = 4 for n = 2 does not generalise; by Burnside’s theorem the real span of SU(n) in Mn(C)R has dimension 2n2, giving reff(n) = 2n2 for n ≥ 3 and the positivity threshold k∗(n) = n2. All results are exact algebraic consequences of the groupoid structure; no large-N limit is invoked.