Automorphic Polylogarithmic Vacuum on a Helical Quotient of a Toroidal Chronology Boundary: Horizon Relocation and an Adapted-Phase Stress-Tensor Cancellation

We study a free massless scalar field on a helical quotient of the exact hyperextreme two-soliton (Kerr) vortex of Paper 1, in which the azimuthal circle closes only through an orientable screw identification Γ : (σ, ϕ) (σ+2π, ϕ+2π/n) on the covering space Rσ Rϕ. This is a toroidal mem-ber of the Grant (generalized-Misner) family. Three results follow. (i) Horizon relocation: the screw does not remove the null closed geodesics that drive the Kay–Radzikowski–Wald (KRW) divergence; on the model world function it moves the chronology horizon from the classical boundary gϕϕ = 0 inward to the polarized surface gϕϕ = n²g¯σσ. (ii) Correct polylogarithmic weights and bulk finiteness: by direct image summation the vacuum polarization and stress ten- 2 iα iα sor carry the twisted-Casimir weights ⟨Φ ⟩ ∝ ℜLi₂(e ) and ⟨Tµν⟩ ∝ ℜLi₄(e ) (two point-split derivatives raise the polylog index by two; the earlier Li₁, Li₀ assignment is corrected). Outside 2 iα 4 2 the relocated horizon we obtain the closed form ⟨Φ ⟩ = ℜ[Li₂(e )]/[8π (g¯σσ +gϕϕ/n )], finite in the bulk and diverging as the horizon is approached. (iii) Adapted ph√a ses: the leading horizon 2 iα divergence of ⟨Φ ⟩ is can celled at ℜLi₂(e ) = 0, i.e. α = π(1 ∓ 1/ 3), and that of ⟨Tµν⟩ at ℜLi₄(e ) = 0, i.e. α = π 1 ± J1 − 2√30/15 ≈ 1.510, 4.773. These phases are distinct : the phase that zeroes Tµν leaves Φ² finite but nonzero, so it diverges at the horizon. This is the toroidal-screw realization of the automorphic self-consistent vacua of Sushkov and Li–Gott, and reproduces the Cassidy–Hawking obstruction: the cancellation is phase-specific, removes only the leading (conformal) divergence, and does not extend to Φ² . We claim no topologi-cal evasion of KRW. All polylog weights, closed forms, and adapted phases are verified three independent ways.