Purpose, background, and corrections to the proposed route

Cheung's I-B prescribes a regular conical spacetime together with a separate finite operational interface CheungIB. I-B-G constructs intrinsic scalar propagation after a spatial self-adjoint extension has been supplied CheungIBG. I-B-VP-v2 constructs the distinguished massless Friedrichs spectral ground state and establishes its local Hadamard form CheungIBVP. I-B-VP-0 evaluates its charged current CheungIBVPZero; I-B-VP-1 constructs prescribed extensions of selected one-point distributions across the apex in an additional smooth completion CheungIBVPOne.

The present object is different: the global covariance on the regular spacetime $M$, with both endpoints satisfying $r,r^{\prime }>0$. No smooth structure at the apex and no apex-supported one-point counterterm enters the proof.

analytical background. Sommerfeld-type cone kernels and cosmic-string Euclidean Green functions have a substantial history Dowker,Linet. Geometric and diffractive propagation on cones and edges are treated by Cheeger–Taylor, Melrose–Wunsch, and Melrose–Vasy–Wunsch CheegerTaylorI,CheegerTaylorII,MelroseWunsch,MelroseVasyWunsch. The magnetic Aharonov–Bohm diffraction problem provides another relevant model Yang. We use these works for context, not as substitutes for checking the exact holonomy, state, and transition formulas below.

The global statement is proved directly from an explicitly summed kernel. The standard tools needed are distributional boundary values, the wavefront pullback and product theorems, microlocal elliptic regularity, and propagation of singularities for a smooth real-principal-type operator on the regular host HormanderI,HormanderIV.