Restriction of the TCFQ Functional to the Geometric Submanifold of the Strong Groupoid: Global Convexity and Absence of Bifurcation

We study the restriction of the TCFQ functional S[Jˆ] = Tr(Jˆ2)+c3 Tr(Jˆ3)+c4 Tr(Jˆ4) to the submanifold Mgeom ⊂ Sym4(R) of matrices realizable by the strong groupoidi.e., matrices of the form Jˆ = cc⊤ + ss⊤ arising from the coherence law φik = φij +φjk. We prove three structural results: (T1) the restriction S|M_geom is strictly convex; (T2) it has a unique global minimum; (T3) it admits no cusp-type bifurcation. As a corollary we establish that the critical transition of R25/R26 ‒which requires a degenerate interior critical point‒ is a phenomenon of the extended spectral layer, not of Mgeom. The two layers are mathematically compatible and structurally distinct.