Enter, I-B-VP-1-v4, 2026, pp. 1-71
I-B-VP-1: Prescribed apex finite parts on a conical Enter spacetime
Ngo Cheung
October 2026
Abstract
We construct distributional extensions across the apex worldsheet 𝒲 of a conical Enter spacetime for the one-point functions of a scalar field (Wick square, mixed stress tensor, charged current), on a declared smooth completion with Ward prescription. Version 1 classified the contact freedom for the Friedrichs state. Version 2 extended this to non-Friedrichs, thermal and massive states, and proved that no canonical dilation-invariant point exists. Version 3 adds six results. (1) For general non-diagonal U(2) extensions with H ≥ 0, the leading apex coefficients are those of the ultraviolet Lagrangian of H. The off-diagonal sector has closed-form leading asymptotics ∝ κ₀₁(H) r^{−1/α}, with an angle-dependent stress tensor and a radial current Jᵣ ≠ 0; it has a unique Ward-compatible extension with no residue and no contact. (2) Explicit flower-pot cores show that the zero-thickness limit of the apex contact is not universal and is decided by the curvature coupling. For ξ = 0 the limit is finite and model-dependent. For ξ > 0 the contact diverges like −1/(4π)log log (1/R). For ξ < 0 there is no vacuum. The (ξ − 1/6)R δ term moves the contact off the dilation orbit. (3) The restriction φ|_{𝒲} exists for β = 0 (a dimension-1 generalized free field on 𝒲) and vanishes for β ≠ 0. Boundary values aⱼ of deficient modes are worldsheet generalized free fields of dimension 1 − νⱼ; for E_{λ} they flow to dimension 1 + νⱼ. The normal-ordered Wick square smears on 𝒲, but its covariant c-number part does not. (4) For the massless Dirac field with flux we give closed forms for T^{μ}_{ν} and J^{φ} (⟨ψ̅ψ⟩ = 0), the residues, and the two-parameter contact space. These agree with Linet and with Bezerra de Mello–Saharian. A disagreement with Spinelly–Bezerra de Mello is disclosed. (5) The first-order λφ⁴ tadpole turns the free contact c into a physical r⁻² term; beyond first order there is a precise obstruction. (6) A re-audit of version 2 confirms its theorems and corrects the physical interpretation of its core theorem. All checks are symbolic or numerical, and no formal certificate is asserted.
Bibliographic record
- Title
- I-B-VP-1: Prescribed apex finite parts on a conical Enter spacetime
- Author
- Cheung, Ngo
- Publication date
- 2026/10/01
- Institution
- Enter
- Report number
- I-B-VP-1-v4
- Publisher
- Cheung Ngo Medical Limited
- First page
- 1
- Last page
- 71
- Language
- en