Topological Protection of Z9 Sector Charges in the PTRH Framework: Frobenius Rigidity, Cohomological Classification, and Spectral Gap --- Closure of Gap G2

Gap G2 of the PTRH/TMRB framework called for a proof that the Z9 sector charges of the horizon C-field are topologically protected, complementing the dynamical Zeno suppression established in Paper 16 (Gap G8). We provide three independent results, each at status [V]. (i) Frobenius eigenvalue rigidity: the Z9 eigenspace decomposition H = direct-sum H_n is invariant under any Z9-equivariant deformation of the PTRH spacetime, because the Frobenius eigenvalues zeta_9^n are algebraic integers belonging to a finite discrete set and cannot vary continuously under deformation. (ii) Cohomological charge classification: sector n corresponds to the character chi_n in H^1(BZ9, U(1)) = Hom(Z9, U(1)) = Z9, a homotopy invariant of the Z9-equivariant structure; the sector label is therefore a topological invariant. The isomorphism is established via K(G,1) group cohomology: H^1_gp(Z9, U(1)) = Hom(Z9, U(1)) (1-cocycles with trivial action are group homomorphisms; 1-coboundaries vanish). (iii) Spectral gap: the minimum separation between distinct Frobenius eigenvalues is Delta_min = 2 sin(pi/9) approximately 0.684. For any bounded perturbation delta_Phi with norm less than (1/2) Delta_min, the eigenvalues of Phi' = Phi + delta_Phi cluster into nine pairwise disjoint disks D_n, each containing exactly dim(H_n) eigenvalues (algebraic multiplicity), by the Bauer-Fike theorem for normal matrices (each sector H_n is finite-dimensional per the holographic bound). Additionally, the Z9-graded horizon Hilbert space is identified with the Hilbert space of untwisted Dijkgraaf-Witten topological gauge theory for gauge group Z9, evaluated on the boundary circle of a torsion cell. The DW basis state |n> corresponds to the flat connection with holonomy g^n in Z9, recovering the torsion-phase matching zeta_9^n = ei theta_n of Paper 14 Theorem 2. Combined with Paper 16, the Z9 charges are doubly protected: topologically (this paper) and dynamically (Zeno suppression). Gap G2 is upgraded from [O] to [V].