What question did this study set out to answer?

This research aims to demonstrate the topological protection of Z9 sector charges within the PTRH framework, addressing Gap G2.

May 26, 2026Open Access

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

Puntos clave

This research aims to demonstrate the topological protection of Z9 sector charges within the PTRH framework, addressing Gap G2.
Analyzed Frobenius eigenvalue rigidity in Z9-equivariant deformations.
Established a cohomological classification of Z9 sectors as topological invariants.
Computed the minimum spectral gap for perturbations in Frobenius eigenvalues.
Frobenius eigenvalues are invariant under Z9-equivariant deformation, maintaining stability.
Z9 sector classification corresponds to the homotopy invariant, confirming its topological nature.
The minimum spectral gap is approximately 0.684, indicating robust eigenvalue clustering for bounded perturbations.

Resumen

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.

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