Topological Mode Suppression and the Chronology Protection Conjecture: The Zn Stress-Energy Decomposition, Polylogarithm Vanishing, and Unconditional 1/n KRW Suppression at Leading UV Order

The Zn helical identification program imposes the mode constraint k ≡ m (mod n) (diagonal sublattice Dn) on scalar field modes in the interior of rotating spacetimes with backward v-shift, and the constraint kz + m ≡ 0 (mod n) (anti-diagonal sublattice An) in the Chronotunnel Toroid with forward z-shift. Both sublattices have density 1/n in Z 2 . We provide a single unified treatment establishing three results that stand independently of the companion preprint series. (i) Exact algebraic decomposition. Using the discrete Fourier representation of the sublattice indicator function, the renormalized stress-energy tensor in the method-of-images construction decomposes exactly as ⟨Tµν⟩ (n) ren = 1 n ⟨Tµν⟩ (1) ren + 1 n nX−1 j=1 T µν j , where T µν j are twisted image sums carrying the phase e 2πij(k−m)/n (or e 2πij(kz+m)/n for the antidiagonal case). This decomposition is exact and model-independent, and the 1/n prefactor of the direct term follows identically from the method-of-images structure of the two-point function. (ii) Unconditional leading-UV vanishing. For all j ̸≡ 0 (mod n), the leading UV contribution to the twisted image sums involves the polylogarithm Li−2(e iθ) evaluated at θ = 2πj/n. We prove that Li−2(e iθ) is purely imaginary for all θ ∈ (0, 2π), so Re Li−2(e iθ) = 0 exactly. Consequently, in zeta-function regularization and at leading UV order, Re⟨Tµν⟩ (n) ren = (1/n) Re⟨Tµν⟩ (1) ren +O(β) with no hypothesis on the twisted sums. This is an unconditional result at leading UV order; whether subleading O(β) corrections are bounded or divergent near the horizon is identified as the central remaining open computation. (iii) Verification in the exactly solvable flat model. The flat helical Zn spacetime (Ref. [8]) provides a fully controlled, closed-form check: the method-of-images Green’s function there yields ⟨Tvv⟩ (n) ren ∼ C (n) /(r−rc,1) 4 with C (n) = C (1)/n exactly, confirming that the image-sum decomposition correctly tracks the coefficient of the near-horizon divergence, not merely its UV mode density. The three mathematical results above are self-contained. As an application to near-extremal supermassive Kerr (drawing on companion preprints), the 1/n suppression combined with geodesic blueshift suppression yields a residual stress-energy ∼ 10−230ρPl and a horizon destruction timescale ∼ 10300 yr, rendering the primary quantum mechanism for Hawking’s chronology protection conjecture inoperative on any cosmological timescale in this class of spacetime