I-B-VP-1: Prescribed apex finite parts on a conical Enter spacetime Wick square, stress tensor, and charged current: strong extension, Ward identities, and classified contact freedom

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.

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.

VII-D's strong finite-part construction and VII-H's explicitly restricted counterterm space are methodological comparisons CheungVIID,CheungVIIH. Their numerical kernels and contact classes are not imported. Similarly, the relative-lifting and normal-jet distinctions in XV-B motivate a compatibility check, not an identification with its characteristic graph CheungXVB.

dependencies. The displayed regular coefficients in Section sec:data are the input identified with the Friedrichs state by the cited companions. Every assertion about their distributional extension, cutoff limit, Ward repair, scaling, and contact classification is proved below. No missing interacting product or characteristic composition is used.

Fix

The regular host is

Its metric completion adds

a worldsheet, not a worldline CheungIB,CheungIBG.

Identify the topological completion with

Equip it with this Cartesian smooth structure and the auxiliary smooth metric

The pairing density is

It equals the host volume density on $M$.

All tensor distributions below are sections of the ordinary smooth tensor bundles of $\overline{M}$. Stress is extended as a mixed tensor $T^{\mu }{}_{\nu }$; the current is extended first as a contravariant vector $J^{\mu }$.

These $x$-coordinates are not the developing coordinates $(r\cos (\alpha \phi ),r\sin (\alpha \phi ))$. Assumption ass:completion is additional data. For a quotient angle $\alpha =1/N$, it is not automatically the orbifold test-function structure of the covering plane.

Write

Thus, as a generalized function relative to $\mathrm{d}\Omega$, ${\delta }_{\mathcal{W}}={\alpha }^{-1}{{\delta }^{(2)}}_{\mathrm{Leb}}(x)$. Derivatives are transposed with respect to the constant density (eq:density). Put

Indices $A,B$ range over $t,z$, and $i,j$ over $1,2$.

Let $ϵ$ be the positive quarter-turn matrix,

The smooth vector $R$ is ${\partial }_{\phi }$ on $M$.

The charged field obeys

in a lifted parallel frame. Its spatial extension and state are the massless Friedrichs operator and distinguished spectral ground state of I-B-VP-v2 CheungIBVP (:state, :infrared, :hadamard). For the real field take $\beta =0$.

Introduce a multiplicity

Define

The regular Wick square is

meaning $⟨{\varphi }^2{⟩}_{\mathrm{ren}}$ for the real field and $⟨{\Phi }^{†}\Phi {⟩}_{\mathrm{ren}}$ for the complex field. The effective field square in the stress trace is $n{F}^{\circ }$.

In the smooth coordinates of Assumption ass:completion, the regular mixed stress is

These are precisely the mixed components $r^{-4}\mathrm{diag}(a,b,-3b,a)$ in order $(t,r,\phi ,z)$ CheungIBVP (:phi2, :T-real, :T-complex).

For the charged field put

With

I-B-VP-0-v1 gives

The current uses the complex first jet of the regular covariance, not the real part used for stress CheungIBVPZero (:current, :current). Set $j=0$ for the real field.

For the stationary distributions considered here, use

For a stationary scalar distribution $G$, define the smooth reference improvement

In the present translation-invariant sector,

For the regular tensors (eq:regularstress),

For every smooth radial scalar $G(r)$, the mixed improvement $(g{\square }_g-{\nabla }^g{\nabla }^g{)}^{\mu }{}_{\nu }G$ equals $\mathcal{I}[G{]}^{\mu }{}_{\nu }$. Consequently,

Proof. In polar coordinates the only nonzero difference between the transverse Christoffel symbols of $g$ and $\bar{g}$ is

Its contraction with the diagonal mixed tensor in (eq:regularstress) vanishes in the mixed divergence. The two volume densities differ by the constant $\alpha$, so vector divergences agree as well.

For a radial scalar, both mixed Hessians have radial entry $G^{\prime \prime }$ and angular entry $G^{\prime }/r$; both wave operators give $G^{\prime \prime }+G^{\prime }/r$. This proves the improvement assertion. Finally,

on $r>0$, which gives the trace and improvement identities.

Definition def:Ward extends a regular, already-cancelled invariant equation by a smooth differential operator. It does not define products of the conical connection with delta derivatives. Likewise, the trace is the smooth mixed trace; we do not first lower or raise distributional indices with the singular metric at $\mathcal{W}$. All claims of Ward compatibility below refer to this definition.

Fix a length $L>0$ and a smooth profile

Put

Define

Its limit will be denoted by $U_L$. The first term is interpreted as a locally integrable function with the density $\mathrm{d}\Omega$.

The limit

exists strongly in ${\mathcal{D}}^{\prime }(\overline{M})$, is independent of $\rho$, and restricts to $r^{-2}$ on $M$. It is rotation invariant, translation invariant in $t,z$, and real.

For every compact test support there is a constant $C$ such that

where $p_2$ is a fixed-support smooth seminorm. In particular, all fixed distributional derivatives converge strongly, with the corresponding extra derivatives in the seminorm.

Proof. Let

Taylor's theorem and angular averaging give

uniformly on the relevant compact set.

For the sharp cutoff, the expression

has a limit: subtract $f(y,0)$ in a fixed small radial neighbourhood and integrate the remainder with measure ${\sigma }_{\alpha }\mathrm{d}r/r$. The omitted remainder on $0<r<\epsilon$ is $O({\epsilon }^2)$.

The smooth cutoff differs from the sharp cutoff only on $\epsilon <r<2\epsilon$. Its constant Taylor term contributes

which is exactly removed in (eq:Uepsilon). Its remaining contribution is $O({\epsilon }^2{p}_2(f))$. This proves existence, profile independence, and (eq:Urate).

The estimates are uniform on bounded sets of tests with fixed compact support, and hence give strong distributional convergence. The symmetries and reality follow directly from the construction. Derivative convergence follows by transposition.

The following elementary extension lemma will also be used to audit direct cutoff prescriptions.

For $p\in \{2,3,4\}$ and a smooth angular coefficient $A$, define

For a test $f$, put

When $k<p-2$, set

The limit

exists and is independent of the profile. The sum is empty when $p=2$. It defines a distribution of transverse scaling degree at most $p$, equal to ${f_A}^{\circ }$ off $\mathcal{W}$.

For the scalar, stress, and vector kernels below, parity improves the fixed-support convergence remainder to $O({\epsilon }^2)$.

Proof. Expand the test transversely through degree $p-2$. Radial integration of the $k$-th term gives the power ${\epsilon }^{k+2-p}$ when $k<p-2$, and a logarithm when $k=p-2$. These are precisely the subtractions in (eq:rawfinitepart). The Taylor remainder is integrable.

Comparing two profiles on their transition annuli gives the same moment contributions, so the normalized limit is profile independent. Taylor remainder estimates prove continuity and the stated scaling-degree bound.

For $r^{-2}$, angular odd terms vanish. For the stress kernels the angular coefficient is even, so the cubic Taylor term vanishes. For $x_i/r^4$, the angular coefficient is odd, so the quadratic Taylor term vanishes. In these cases the first unremoved angular term yields an $O({\epsilon }^2)$ remainder, bounded respectively by test seminorms of orders $2,4,3$.

The codimension is two. Thus $r^{-4}$ produces an ${\epsilon }^{-2}$ divergence and a logarithmic second-jet subtraction, not an ${\epsilon }^{-4}$ divergence. For the circulating vector, the prospective ${\epsilon }^{-1}$ term has zero angular moment; only a logarithmic first-jet subtraction remains.

We shall write

The superscript $\mathrm{c}$ denotes componentwise cutoff finite parts, not a conserved prescription.

Define

and

The distributions (eq:Fhat)–(eq:Jhat) are real extensions of the regular data (eq:regularstress)–(eq:regularcurrent). They have transverse scaling degrees at most $2,4,3$, respectively. They satisfy

The reference-lowered stress ${\bar{g}}_{\lambda \mu }({\hat{T}}_L{)}^{\mu }{}_{\nu }$ is symmetric.

Replacing $U_L$ by ${U_{L,\epsilon }}^{\rho }$ gives strongly convergent regulated extensions. The errors are bounded by $C{\epsilon }^2{p}_2$, $C{\epsilon }^2{p}_4$, and $C{\epsilon }^2{p}_3$ for field square, stress, and current.

Proof. On $r>0$,

This proves exact regular-part matching.

All derivatives in (eq:That)–(eq:Jhat) are derivatives of a defined distribution. Since

conservation follows without a boundary-flux omission. The mixed trace is

The changes of $a/4$ and $b/2$ under a change from conformal coupling are $nqs$ and $-nqs$, respectively, giving (eq:extendedimprovement). Symmetry is immediate from the block expressions. The scaling-degree and strong-convergence claims follow from Theorem thm:U.

This is a construction of c-number one-point distributions. It is not an extension of the two-point function, quantum state, or field algebra to $\mathcal{W}$.

The direct finite parts satisfy

In particular,

Proof. Profile independence permits the sharp cutoff in this comparison. Twice integrating by parts on $r>\epsilon$, and expanding the boundary test jets, gives

The boundary computations are detailed in Appendix app:boundary. Their trace gives (eq:convertV); the first identity gives (eq:convertQ); subtracting the traced second identity gives (eq:convertS). Taking a divergence of (eq:convertS) proves the first identity in (eq:rawdefects). The second follows either by tracing the regular angular kernel before applying the linear finite-part operation, or directly from the moment subtractions.

Define the preliminary componentwise lift by

The preliminary lift has

The corrections

give exactly Theorem thm:extension.

Proof. The divergence follows from (eq:rawdefects). The raw trace is

Equation (eq:convertV) makes $3q\Delta (n{\hat{F}}_L)=12k{V}_L^{\mathrm{c}}-3k{\sigma }_{\alpha }\Delta {\delta }_{\mathcal{W}}$, proving (eq:tracedefect). The vector in (eq:convertQ) differs from a gradient by a delta derivative, so its rotation by $ϵ$ is conserved. The corrections are the three conversions multiplied by their regular coefficients.

The raw finite part exists, but is not automatically a Ward-compatible extension. Corollary cor:repair evaluates the defect and supplies an actual correction in an explicit supported space. This is the relevant compatibility calculation; merely naming a relative-lifting criterion would not supply it CheungVIID,CheungXVB,CheungEmOneA,CheungEmOneD. Nothing here defines a product of singular observables.

A finite-dimensional ambiguity requires more than a bound on normal derivative order. We now fix the full class.

For each fixed $(\alpha ,\beta ,\xi )$, consider real distributional extensions $(F,T,J)$ of the displayed regular data satisfying:

1. Translation invariance along $\mathcal{W}$, covariance under longitudinal Lorentz transformations, and transverse rotations. The reference-lowered stress is symmetric.

2. Charge conjugation sends $\beta$ to $1-\beta$: $F,T$ are even and $J$ is odd. Transverse reflection reverses the meridian and therefore also sends $\beta$ to $1-\beta$. These requirements apply to the family of sectors.

3. The componentwise transverse scaling degrees are at most $2,4,3$ for $F,T,J^i$, respectively, and the extensions are almost homogeneous of logarithmic order at most one:

   $$(E+2{)}^{2}F=0,(E+4{)}^2{T}^{\mu }{}_{\nu }=0,(E+3{)}^2{J}^{\mu }=0.$$

   These are equations for the stated Cartesian component distributions.

4. The Ward representative is Definition def:Ward, and the required identities are

   $$\mathfrak{D}T=0,\mathfrak{d}J=0,\mathfrak{t}T=3q\Delta (nF).$$

For a coupling-dependent family, $F,J$ are required to be independent of $\xi$, and the improvement identity is required with this same $F$.

The almost-homogeneity condition excludes new power-law defect scales such as $L^{-2}{\delta }_{\mathcal{W}}$ in the stress. It does not require strict homogeneity, which will generally be impossible. The class is a prescription, not a classification of every possible singular-background renormalization.

Define the conserved, traceless mixed contact tensor

Indeed, $\mathfrak{D}\mathcal{H}=0$ and $\mathfrak{t}\mathcal{H}=2\Delta {\delta }_{\mathcal{W}}-2\Delta {\delta }_{\mathcal{W}}=0$.

Every extension in Assumption ass:class is uniquely of the form

with real $c,\tau ,w$.

Conversely, these formulas give all the required extensions. For coupling-dependent families, $c,\tau ,w$ can be taken independent of $\xi$, and then the improvement identity holds.

The coefficients $c,\tau$ are even in $\beta$, and $w$ is odd. Consequently:

1. for a generic charged sector $\beta \notin \{0,1/2\}$, the contact space has dimension $3$;

2. at a self-conjugate charged sector $\beta =0$ or $1/2$, $w=0$, and the dimension is $2$;

3. for the real neutral field, there is no charged current and the dimension is $2$.

The dimension is for fixed background and field parameters, not for arbitrary functions of those parameters.

Proof. The difference of two extensions is supported on $\mathcal{W}$. The structure theorem for distributions supported on a submanifold expresses it as a finite sum of transverse delta derivatives with coefficient distributions on $\mathcal{W}$ Dang (Section 8) Hormander (Theorem 2.3.5). Translation invariance forces those coefficient distributions to be constants times the longitudinal Lebesgue distribution.

A transverse derivative of order $m$ obeys

The scaling-degree bounds and (eq:AHclass) therefore allow only order $0$ for the field-square difference, order $2$ for stress, and order $1$ for current. For example, a stress term proportional to ${\delta }_{\mathcal{W}}$ has $(E+4{)}^2{\delta }_{\mathcal{W}}=4{\delta }_{\mathcal{W}}$, and is excluded.

Rotation covariance and the reflection/charge-conjugation conditions give

with no mixed blocks. The parity-odd symmetric transverse tensor is excluded by the even charge-conjugation behaviour of stress combined with meridian reflection.

For the current, longitudinal Lorentz covariance excludes a constant longitudinal vector. Rotation covariance allows ${\partial }_i{\delta }_{\mathcal{W}}$ and ${ϵ}_i{}^k{\partial }_k{\delta }_{\mathcal{W}}$. The radial term is even under meridian reflection, whereas the current must reverse under charge conjugation. The simultaneous requirements exclude the radial term. Thus the current difference is precisely the term in (eq:allJ).

Stress conservation gives $v+z=0$, so write the transverse block as $v^{\prime }{K}_{ij}{\delta }_{\mathcal{W}}$. Its trace is

The trace equation in (eq:classWard) requires

Put $\tau =u-nqc$. Then $v^{\prime }=2\tau -nqc$, which is exactly (eq:allT) by (eq:Iblocks) and (eq:Hcontact). The parameters are unique because the listed contact generators are linearly independent.

Conversely, the contact tensors in (eq:allF)–(eq:allJ) satisfy all the symmetries, scaling conditions, and Ward equations. The improvement identity follows by subtracting the same formula at $q=0$. At a self-conjugate sector, oddness forces $w=0$.

Without longitudinal translation invariance, the coefficients of the normal delta derivatives could themselves be distributions on $\mathcal{W}$, giving an infinite-dimensional ambiguity. Without (eq:AHclass), the scaling-degree bound also permits lower-order contacts, including a longitudinal stress proportional to $L^{-2}{\delta }_{\mathcal{W}}$. Without the reflection and charge-conjugation requirements, additional oriented tensor structures can occur. The dimension in Theorem thm:classification is not a universal dimension for every apex prescription.

The Friedrichs condition fixes the spatial wave operator on the regular host. It does not fix $c,\tau ,w$. A specified defect action, core model, variational prescription, or independent apex normalization could impose additional conditions. None is introduced here.

The generating finite part obeys

and hence

For the transverse dilation $x\mapsto \lambda x$,

as a pullback identity of generalized functions. No extension of $r^{-2}$ differing from $U_L$ by a multiple of ${\delta }_{\mathcal{W}}$ can be strictly homogeneous.

Proof. For a test $f$,

The subtraction term vanishes because $Ef(y,0)=0$. Radial integration gives

after the longitudinal integration. This proves the first identity. Differentiating the defining subtraction proves the second, and integration in $L$ gives (eq:scaleshift). Integrating the Euler equation along dilations gives the pullback formula. Since $(E+2){\delta }_{\mathcal{W}}=0$, adding a scalar contact cannot remove the residue.

For the canonical prescription,

These residues are unchanged by the contacts in Theorem thm:classification. The $L$-derivative of each canonical distribution is the negative of its displayed residue.

Proof. Use $[E,{\partial }_i]=-{\partial }_i$ in (eq:EulerU), then apply the defining differential operators. Each permitted contact is exactly homogeneous at the relevant degree and is therefore killed by the corresponding Euler operator.

The residues are extension scaling defects. They must not be identified without further analysis with a curvature-dependent quantum trace anomaly. For example, at $\xi =1/6$ the mixed trace of the constructed stress is exactly zero, even when its normal-dilation residue is nonzero. Changing $L$ changes a contact convention and does not change any regular-part value.

The smooth contraction defining an angular scalar extension is

On $M$, this is the actual covariant angular component, because $g_{ij}{R}^i={\alpha }^2{R}_j$. Equation (eq:angularcontraction) uses a smooth polynomial multiplier on the completion; it does not multiply by the singular metric there.

For the derivative-compatible current,

For the general current (eq:allJ),

In contrast, the direct componentwise current obeys

Thus a contact can preserve this latter contraction identity, but not simultaneously the unmodified derivative identity (eq:Jhat) when $j\ne 0$.

Proof. Since $R_i{ϵ}^i{}_k=x_k$,

Equation (eq:EulerU) gives (eq:contractiondefect). Also $x^i{\partial }_i{\delta }_{\mathcal{W}}=-2{\delta }_{\mathcal{W}}$, proving (eq:generalangular). Equation (eq:currentrepair) then gives the direct finite-part contraction. Choosing $w=-j{\sigma }_{\alpha }/(4{\alpha }^2)$ removes the contraction defect, but changes (eq:Jhat) by that nonzero contact.

This is an explicit presentation-compatibility calculation. It does not rely on checking finitely many null relations and then assuming coverage of all relations. The distinction between a distributional extension and a descending realization of several presentations is also emphasized in the supplied XV-B material CheungXVB.

Every test supported in $\{r\ge r_0>0\}$ sees exactly the regular one-point functions, for all $c,\tau ,w,L$. No extension parameter alters the Friedrichs wave operator, the regular spectral state, or its local Hadamard subtraction.

The supported-transfer theorem of I-B-G concerns scalar retarded and advanced operators, with both the additional-arrival window and the independent retained-shadow reserve CheungIBG (:transfer, :transfer-contract). The X-A-ST and X-B-ST supported-chart contracts remain separate CheungXAST,CheungXBST. Neither Green-operator agreement nor a local developing chart transfers a vacuum covariance or selects apex contacts. No comparison with a flat-completion state is used in the proofs above.

For a static observer on $M$, the measured energy density is $-a/r^4$. Its cutoff transverse integral per unit longitudinal length is

It generally diverges. Likewise,

A finite distributional action on smooth compact tests is not a proof of a finite unrenormalized energy per unit length.

If $s>0$, no positive distribution on the completion can restrict to $s/r^2$ with this density: a positive distribution is a locally finite measure, whereas every neighbourhood of the apex has infinite regular $s/r^2$ mass. The constructed finite part is therefore not a positive measure. This does not contradict positivity of the ground-state two-point function on regular tests.

The conical metric in the chosen smooth coordinates is

It has no direction-independent value at $x=0$ when $\alpha \ne 1$. Products such as $g_{ij}{\partial }_k{\delta }_{\mathcal{W}}$, a distributional Einstein tensor built from unspecified products, or a singular-connection divergence of arbitrary contacts are not defined by the finite parts above.

Our mixed tensors are ordinary tensor distributions on the auxiliary smooth completion. Their regular restrictions are the physical mixed tensors, and their extended Ward identities are those fixed in Definition def:Ward. A core regularization or nonlinear generalized-function framework would require a separate construction and comparison GerochTraschen. No such comparison is asserted.

The finite supplied-report instrument can install words containing

For a calibrated clock basis $\{|{\tau }_j⟩\}$, set

The recorded instrument is

Its Kraus completeness and discard identities are

including every untouched-reference extension CheungIB (:instrument) CheungEn09,CheungOOneST. It returns the surviving $A$, not a quantum $C$-register. The words are installed descriptions, not measurements acquiring an unknown continuum distribution.

is class relative. Every regular-part test annihilates all contacts in Theorem thm:classification. Thus even the collection of all tests supported away from $\mathcal{W}$ cannot select $c,\tau ,w$.

Within the declared three-dimensional contact space, a finite family of appropriately chosen apex-jet tests can be injective: the three generators are independent distributions, and linear algebra selects separating test actions. That conclusion uses the proved classification. It does not hold for an unrestricted space of worldsheet-supported distributions. This is the finite-coverage distinction of Oai282-A CheungOaiTwoEightTwo.

classes. We retain the reporting discipline used in XVI-O1-ST CheungOOneST:

ANALYTIC

A statement established by the displayed argument and its identified analytical inputs.

SUPPLIED

An inherited computational report, not independently rerun.

UNCERTIFIED

A numerical enclosure or verification claim without the artifacts and total error bounds required for that interpretation.

The results of this paper are analytical. No numerical apex runs or raw logs were supplied or are claimed. The numerical residuals in the regular-part companion papers are not relabeled as checks of these extension identities.

A future regulated numerical report must retain the test, density, smooth structure, component convention, cutoff profile, input encodings, raw subtraction terms, precision, residuals, and remainder bounds. Input error, cutoff error, and quadrature error are separate. A successful regression that retains a known discrepancy does not remove it. No successful pdflatex run, numerical execution, proof-assistant verification, or independent rebuild is asserted.

{}1.12

precise V1 advance. The Friedrichs one-point extension question is partially resolved: a concrete completion prescription exists, its direct-cutoff defects are repaired, and its residual contact space is fully classified. The original demand for a uniquely normalized or intrinsically defined apex observable is not thereby resolved. V2, V3, and V5–V8 of I-B-VP retain their stated scopes; I-B-VP-0's current evaluation remains its separate regular-part achievement CheungIBVP,CheungIBVPZero.

-sourcing and unchanged programme boundaries. A prescribed distributional extension is not a field equation. No Maxwell equation, semiclassical Einstein equation, backreaction solution, spacetime emergence, or interacting Ward theorem is constructed. The non-sourcing distinctions of I-B and X-A-ST remain unchanged CheungIB,CheungXAST.

No external OpenAI catalogue entry is a premise. No characteristic graph coefficient is constructed by this one-point extension. In particular, the existence of technical XV-B manuscripts is not the opening of its distinct downstream quantum-admission exit CheungXVB,CheungEn09. O.1 remains NOT_CLOSED{} wherever it was not independently closed; XVI-N is not discharged; bulk_ symmetry_ restored{} is not promoted CheungOOneST,CheungEn09. These restrictions do not deny the free-field distributions proved here.

. A declared smooth completion and Ward representative turn the regular $r^{-2}$, $r^{-4}$, and circulating-current formulas into a precise extension problem. One generating finite part solves it simultaneously. The raw finite-part defects, their repair, the remaining contact parameters, and the nonremovable dilation residues are explicit. The construction is therefore more than a proposal, but deliberately less than a canonical singular-background quantum theory or a derived source law.

We give the local transverse calculation; longitudinal integration is passive. Write $⟨\cdot {⟩}_{\phi }$ for an unnormalized angular integral, and use the constant density factor $\alpha$. All remainder estimates are uniform in the required compact longitudinal set.

For a smooth test,

The angular identities used below are

For $u=r^{-2}$, integration by parts on $r>\epsilon$ uses the inner outward normal $-\omega$: