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 Hₙ is invariant under any Z9-equivariant deformation of the PTRH spacetime, because the Frobenius eigenvalues zeta₉ⁿ 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 chiₙ in H¹ (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: H¹gp (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 Deltaₘin = 2 sin (pi/9) approximately 0. 684. For any bounded perturbation deltaPhi with norm less than (1/2) Deltaₘin, the eigenvalues of Phi' = Phi + deltaPhi cluster into nine pairwise disjoint disks Dₙ, each containing exactly dim (Hₙ) eigenvalues (algebraic multiplicity), by the Bauer-Fike theorem for normal matrices (each sector Hₙ 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 gⁿ in Z9, recovering the torsion-phase matching zeta₉ⁿ = e^i thetaₙ 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.
George H. Bressler (Sun,) studied this question.