Theory of Quantum Groupoid Fields: Geometric Submanifold of SU(n), Character Variety of SU(3), Exact Lower Frontier bmin(a, c) and Diagonal Algebraicity

Starting from Burnside's theorem, which identifies the effective rank r_eff(n) = 2n^2 as the real dimension of M_n(C), notes R33-R35 establish that the geometric submanifold M_geom is strictly contained in the truncated correlation body C(N, 2n^2) for n >= 3, with exact character minimum chi_min(n) = cos(2pi/n). R36-R37 show that the interval-and-PSD conditions are necessary but not sufficient, owing to the Procesi-Lawton cubic identity in the character variety of SL(3,C)^2. R38 reduces the problem to optimal assignment on the Weyl torus via the Birkhoff polytope, and R39 derives the perturbative expansion b_min(a,c) ~ a+c-1-2*sqrt((1-a)(1-c)) near (1,1). R40 identifies a two-region decomposition A u I and proves that {det(J-hat) = 0} coincides with the frontier only on the symmetric orbit O_sym; two claims from R40 are corrected in R41: b_min is not concave (explicit counterexample, Delta ~ 0.21), and the transcendence argument is logically invalid. The main result of R41 is an exact closed-form formula on the diagonal, b_min(a,a) = 6a^2 + 10a + 3 - 2(2a+1)*sqrt(6a+3), proved via a KKT analysis showing the symmetric orbit is the unique global minimiser; this places b_min(a,a) in Q(sqrt(6a+3)), algebraic of degree 2 over Q(a), with exact threshold a* = 1/6 for the active set.