Abstract

We construct distributional extensions across the prescribed apex worldsheet of the massless Friedrichs ground-state one-point observables computed in I-B-VP and I-B-VP-0. The construction fixes a smooth structure on the topological completion, the conical volume density, mixed stress-tensor components, and an explicit extension of the regular radial Ward identities. These choices are necessary: multiplication of apex-supported distributions by the singular conical metric or its connection is not silently assumed.

A single transverse finite part of $r^{-2}$ generates simultaneous extensions of the Wick square, stress tensor, and circulating current. The regulated distributions converge strongly, with fixed-support seminorm estimates. Direct componentwise finite parts are also constructed. Their stress divergence and trace defects are evaluated exactly and repaired by explicit second derivatives of the worldsheet Dirac distribution.

The principal result is a complete ambiguity theorem in the declared stationary, boost-invariant, rotation-covariant, charge-conjugation-compatible, almost-homogeneous class. For a generic charged holonomy sector, the simultaneous Ward-compatible extension has three real contact parameters; the real neutral problem has two. Nonzero normal-dilation residues cannot be removed in this class. A current contraction identity exhibits explicitly why extending a formula and preserving all its differential presentations are different requirements.

The contribution is this prescribed extension and compatibility analysis, not new regular cosmic-string coefficients or a canonical singular-metric renormalization. It partially resolves the Friedrichs one-point extension question V1 in a stated distributional structure. It does not extend the quantum state or field algebra to the apex, select a defect action, construct backreaction, or derive Einstein dynamics. Finite Enter reporting remains an interface for installed descriptions, not acquisition of a continuum distribution.