Unified extension and state ledger

Throughout this table $m=0$, unless a row says otherwise. “Constructed” refers to the declared spectral prescription, not universal uniqueness of algebraic states. “Evaluated by integrals” means a proved convergent formula, not a claim of elementary closed form.

{}1.16 {@{}L{.29}L{.18}L{.46}@{}} Extension or construction & Status & Formula, proof, or limitation
Extension or construction & Status & Formula, proof, or limitation

Friedrichs, $\beta =0$ & Constructed; closed forms & Infrared Lemma lem:infrared; Theorem thm:friedrichs. Only nonnegative extension in the untwisted massless product ledger, not uniqueness of every algebraic state.

Friedrichs, $0<\beta <1$ & Constructed; closed forms & Theorems thm:friedrichs and thm:current; all cone angles, including image-boundary parameters.

No deficient modes & Friedrichs only & The transverse minimal operator is essentially self-adjoint.

${\mathsf{L}}_L$, every finite $L>0$ & Excluded as stationary finite-moment state & One negative transverse eigenvalue and an unstable full band; Theorems thm:spectrum and thm:nogo.

Entire transverse bound channel removed & Projected construction; not local Hadamard & Extra logarithmic singularity and divergent subtracted square; Proposition prop:projections(a). Projected CCR.

Only negative full spectrum removed & Projected Wick square evaluated; stress prescription not supplied & Principal-value Wick square; nonsmooth direction-dependent remainder and boost breaking; Proposition prop:projections(b). Not the original local field.

${\mathsf{E}}_0$ in one deficient mode & Constructed; closed forms & Theorem thm:shift, with the legitimate shifted branch. Replaces predecessor “Open” status.

$({\mathsf{E}}_0,{\mathsf{E}}_0)$ & Constructed; closed forms & Additive formula in Theorem thm:shift.

Diagonal ${\lambda }_j\in [0,\infty ]$ & Constructed; convergent integrals & Theorems thm:nonnegative-state, thm:diagonal–thm:large. Finite-$\lambda$ endpoint and asymptotic qualifications apply.

Any diagonal combination with a negative ${\lambda }_j$ & Excluded as stationary finite-moment state & Theorems thm:spectrum and thm:nogo.

Half-holonomy ${\mathbb{CP}}^1$ line family & Constructed; closed forms & Theorem thm:line. Rotation breaking is allowed; radial current vanishes.

Full finite $U(2)$ chart, $H\ge 0$ & Constructed; integral evaluation & Theorems thm:krein, thm:nonnegative-state; Propositions prop:matrix, prop:matrix-asymptotics. Large-radius formula needs $H>0$.

Nonnegative boundary charts & Constructed; integral evaluation & Compressed coefficient (eq:CW). The full finite-chart two-mode ultraviolet limit is not imposed on these relations.