Purpose, attribution, and logical scope

The spacetime companions X-A-ST, X-B-ST, X-C-ST, X-D-ST, V-B, IX-A-ST, IX-B-ST, IX-C-ST, and III-B use a supported developing presentation of the prescribed I-B host CheungIB,CheungXAST,CheungXBST,CheungXCST,CheungXDST, CheungVB,CheungIXAST,CheungIXBST,CheungIXCST,CheungIIIB. Their global free scalar and Green operators belong to a separately declared Minkowski completion. A developing chart does not itself supply an intrinsic conical inverse.

This paper supplies such an inverse after an apex extension has been specified. It does not change the hypotheses or conclusions of those companions. The connection to their supported scalar calculations is a separate theorem (Theorem thm:transfer and Corollary cor:expressions), not a reinterpretation of their original global analytical domains.

The analytical ingredients are classical: self-adjoint spectral theory; Bessel transforms; the Sommerfeld and Carslaw construction; Dowker's cone and flux calculations; and the geometric diffraction framework of Cheeger and Taylor ReedSimonII,Watson,Sommerfeld1896,Carslaw1899,Carslaw1920, Dowker1977,Dowker1987,CheegerTaylor1,CheegerTaylor2. Kay and Studer discuss the apex extension problem for cones and idealised cosmic strings KayStuder; the related radial inverse-square operators are treated explicitly by Dereziński and Richard DerezinskiRichard. These references provide mathematical context, not a derivation of the prescribed Enter geometry.

The Green-operator terminology is consistent with Bär–Ginoux–Pfäffle BGP. Their globally hyperbolic existence theorem is not invoked on the regular cone. Here existence comes from a declared spatial extension, and the exact sequence is proved on a declared spectral domain.

“open” means here. An item marked Open is not resolved in this paper. This is not a claim that the corresponding problem is unknown in the literature. In particular, no assertion about the absence of results in an entire manuscript corpus is needed for any proof.

-identifications. The geometry, the apex condition, and any independent Aharonov–Bohm phase are inputs. The construction gives no Einstein dynamics, spacetime emergence, conical quantum state, Hadamard covariance, time-ordered product, or interacting algebra. The finite reporting interface introduced below does not measure an unknown continuum field.