Image-Sum Stress-Energy and Multi-Horizon Structure on the Flat Helical Zn Spacetime

We study a free scalar field on flat 4-dimensional Minkowski spacetime with a Zn helical identification (t, r, ϕ) ∼ (t − l/n, r, ϕ + 2π/n), the minimal exactly solvable model carrying the topological structure of the Zn helical identification program of Refs. [1–3]. Using the method of images we obtain ⟨Tμν⟩ren in closed form. Three results are established. (i) Multi-horizon structure (principal result). For n ≥ 3 the identification produces n − 1 distinct chronology horizons at rc,j = jl/(2n sin(πj/n)), j = 1, . . . , n − 1, and we prove analytically that rc,1 < rc,2 < · · · < rc,n−1, so the horizons are separated in radius and do not accumulate. This has no analogue in single-horizon Misner space. (ii) Invariant-sector weight. Near each rc,j the renormalized stress-energy diverges as σ−2 j ; in the Zn-invariant sector each divergence enters with the projector weight 1/n, and at leading UV order the real invariant-sector energy density equals 1/n times the unrestricted density on the same torus, because the n−1 twisted Fourier sums are purely imaginary [Li−2(e2πij/n) ∈ iR, proved inline]. (iii) The divergence is not removed. The 1/n is a weight on a still-divergent quantity. Consistently with the Kay–Radzikowski–Wald theorem [5], the construction redistributes and re-weights the divergence across n − 1 horizons but does not evade chronology protection. We note that Cramer–Kay [9] proved singularity persists for automorphic fields (all modes present, acquiring phase factors eiθ under the identification); the Zn-projected sector studied here (modes with k ̸≡ m (mod n) absent from the Hilbert space entirely) is a mathematically distinct object, and Cramer– Kay does not apply to it directly. The flat model fixes the topological weight in a fully controlled QFTCS setting; the curved Ori background, with frame-dragging β(r, θ) = −Ω1r2 sinh2 θ solved in the companion paper, carries additional O(β) corrections that are not addressed here. Version note (v4). The mode constraint is corrected to k ≡ m (mod n) (the n = 2 case reduces to k + m ≡ 0 (mod 2)); the 1/n claim is restated as an invariant-sector weight rather than a suppression of the divergence; the Li−2 identity is proved inline; and the corrected Table I values of v2 are retained.