Abstract

We give a unified construction and extension ledger for a massless scalar field on the regular conical product prescribed in I-B. The spatial operator, its self-adjoint extension, the state prescription, the curvature coupling, and the flat-bundle holonomy are explicit inputs. The extension class consists of transverse self-adjoint extensions tensored with the ordinary longitudinal Laplacian; it includes rotation-breaking finite-dimensional boundary conditions but excludes general non-product extensions.

For the Friedrichs extension, an explicit heat-kernel estimate proves infrared finiteness of the massless spectral covariance on compact tests. Direct Euclidean subtraction leaves a jointly real-analytic bisolution at every regular diagonal point, including image-boundary parameters. Analytic continuation gives the local Hadamard form used for point splitting.

For one complex field, with $\Phi =({\varphi }_1+\mathrm{i}{\varphi }_2)/\sqrt{2}$, $T=T[{\varphi }_1]+T[{\varphi }_2]$, signature $(-,+,+,+)$, and mixed components ordered as $(t,r,\phi ,z)$, define

The Friedrichs results are

The neutral real tensor is half the untwisted complex tensor. The current uses the complex first jet of the regular two-point function; replacing that function by its real part would erase it.

For every nonnegative extension in the declared product class, a finite-rank Krein formula gives an infrared-finite spectral state and an analytic regular-part correction. We supply the formula also on boundary charts of the extension space. Positivity is equivalent to positivity of the finite boundary relation. Negative transverse eigenvalues produce unstable longitudinal bands; a stationary finite-second-moment state is then excluded by a commutator-growth argument. Two projected alternatives are recorded, together with their nonlocality and Hadamard defects.

Scale-invariant irregular modes admit closed forms expressed through analytic branches of the Friedrichs polynomials at shifted arguments. At half holonomy an additional ${\mathbb{CP}}^1$ family has explicit angular-dependent observables. Scale-dependent nonnegative extensions have convergent integral representations and proved asymptotics. Generic rotation-breaking extensions with unequal deficient orders can carry a radial current.

The familiar Friedrichs coefficients are not claimed as new. Historical formula comparisons and all inherited numerical residuals are retained with their evidence grades. No numerical execution, apex renormalization, backreaction, interacting construction, or downstream programme closure is asserted.