Purpose, inputs, and the essential qualification

Cheung's I-B prescribes a regular conical spacetime and a separate finite operational interface CheungIB. I-B-G constructs intrinsic scalar Green operators after a spatial extension has been supplied CheungIBG. I-B-VP-v2 constructs the distinguished massless Friedrichs spectral ground state, proves its local Hadamard form, and evaluates its Wick square and stress tensor CheungIBVP. I-B-VP-0-v1 evaluates the charged circulating current CheungIBVPZero.

Those observables are smooth on the regular spacetime. Their singular radial behaviours do not, without an additional prescription, define distributions across the completed apex worldsheet. The present paper supplies one such prescription and classifies its finite freedom.

qualification that cannot be omitted. The metric completion is not a smooth Lorentzian manifold carrying the displayed conical metric at its apex. Before speaking of tensor distributions there, one must specify the test space, smooth structure, tensor bundle, density, and meaning of the identities to be preserved. In particular, an expression such as $\Gamma (g)T$, with an apex-supported $T$, is not automatically a product of distributions.

We therefore construct distributions on a declared smooth completion and preserve a declared smooth representative of the regular invariant Ward equations. We do not claim that this defines the Levi-Civita derivative of arbitrary tensor distributions for the singular metric. The distinction is consistent with the general cautions about concentrated sources and distributional geometry in Geroch–Traschen GerochTraschen.

and the non-extraction boundary. The philosophy of Enter distinguishes a destructive readout from a quantum-return map, a record from a decoder, and an analogy from an identification CheungPhilosophy. I-B and En09-A implement a finite destructive instrument returning a classical code and a surviving residual, not the measured clock register CheungIB,CheungEn09. That instrument does not supply the field, apex condition, state, continuum test space, or renormalization prescription used below.

The cone, worldsheet, Friedrichs extension, spectral state, and completion prescription remain declared inputs. No continuum distribution is extracted from a finite conditional density matrix. Conservation does not impose Einstein dynamics CheungIB (:not-source) CheungXAST (:noeinstein).

context and contribution. Extension by Taylor subtraction, scaling degree, and distributions supported on a submanifold are classical tools BF,Dang,Hormander. Almost-homogeneous extensions and their dilation residues are likewise established concepts NikolovStoraTodorov. Our contribution is their explicit application to the simultaneous one-point data, including the stress Ward repair and the complete contact space in the prescription fixed here.