Landmark XV-B Relative Lifting and Reconstruction Descent for the Full Joint Characteristic Target

We study the compatibility between distributional extension and reconstruction descent for linear constructions on presentations. Given a canonical map on a proved open domain, a preliminary lift, and a specified affine space of permitted corrections, we prove that a reconstruction-descending candidate exists exactly when the preliminary lift's null defect vanishes in the corresponding relative obstruction quotient. Contact-supported corrections cannot cancel a null defect detected away from their support.

For a specified finite normal-jet module, we obtain an explicit normal form and a complete distributional null-relation criterion. Supported extension preserving the normal-order bound is equivalent to extension of its unique coefficients; the complete unrestricted supported ambiguity is also determined. We give a general closed-set extension criterion, distinguish universal and flat-annihilator relations from potentially jet-detected defects, and construct a simultaneous annihilator-preserving correction at a point puncture by finitely many derivatives of a delta distribution. The point-puncture result is not asserted for a positive-dimensional characteristic locus.

The application retains the full six-branch target $K_{47}+K_5+K_6+K_8+K_9$, including both placements of $K_5$. The interface results do not construct its missing characteristic contraction, establish its complete-graph estimates, or admit a quantum coefficient.

Figure fig:split-XV-B records one view of the same separation. Figure fig:chain-XV-B records one view of the same separation. Figure fig:pair-XV-B records one view of the same separation. Figure fig:gate-XV-B records one view of the same separation. The drawings add no hypothesis.

Figure fig:scope-XV-B records that separation. The drawing adds no hypothesis.

A distributional construction on a presentation space need not depend only on the reconstructed distribution. This distinction is particularly important when different presentation degrees can cancel after reconstruction. Extending each displayed term separately does not, by itself, preserve those cancellations.

The problem has three logically distinct parts:

1. extend prescribed canonical values from their proved domain;

2. annihilate every reconstruction-null presentation;

3. satisfy the specified support, normalization, continuity, and microlocal requirements.

The first two parts are already different before any complete-graph wavefront theorem is considered.

Cheung's XIV-J makes the descent condition explicit and records a null relation mixing presentation degrees CheungXIVJ. The present paper develops an exact relative criterion for correcting such defects. It also incorporates the finite normal-jet reconstruction test supplied in XV-C CheungXVC and develops the annihilator and point-puncture interfaces discussed in the supplied account of XV-E CheungXVE. All abstract statements used below are proved here; no unavailable proof from a companion manuscript is required.

The distributional background is standard Hormander. Extension with prescribed scaling or wavefront control requires further hypotheses, as in the frameworks of Brunetti–Fredenhagen and Dang BF,Dang. Our algebraic lifting criterion is not a substitute for those analytic hypotheses.

The principal results are:

1. the exact affine relative-lifting criterion (Theorem thm:lifting);

2. the contact-support obstruction (Theorem thm:contact);

3. the complete null criterion for a specified finite normal-jet module (Theorem thm:jetnull);

4. the supported coefficient-extension equivalence and complete unrestricted supported ambiguity (Theorem thm:jetextension);

5. the local uniform finite-order criterion for extension across a general relatively closed set (Theorem thm:closedextension);

6. the finite-jet localization of natural null defects and the simultaneous point-puncture correction (Lemma lem:jetdefect and Theorem thm:pointcorrection).

The normal-jet module is explicitly defined. It is not identified with the inherited physical presentation space. Likewise, unrestricted supported ambiguities are not identified with the frozen partial-diagonal correction class.

The retained application asks for a map

for the full joint characteristic target inherited from XIV-I and XIV-J CheungXIVI,CheungXIVJ. Here $W$ is the attachment domain and $E_{\mathrm{res}}$ is the residual-species bundle.

A successful construction must agree with canonical composition on its proved domain, act on tests meeting the characteristic locus, retain every branch and prescribed tensor operation, respect reconstruction, and satisfy the complete-graph microlocal bounds.

In this paper, XV-B denotes the technical continuation studying these obligations. The predecessor phrase “XV-B remains closed” concerns its earlier downstream admission contract, not the nonexistence of a technical continuation. The admission condition

is not established here. Changing the manuscript convention neither proves (eq:admission) nor supplies a free quantum BV carrier. In particular, the model-and-domain audit cited as XV-A below is not identified with the differently described free-carrier prerequisite appearing under that label in older passages CheungXVA,CheungXIVI,CheungXIVJ.

The inherited formulas and historical statuses in Section sec:application are attributed to the supplied manuscript excerpts. Bibliographic references to those manuscripts use the supplied author, version, and statement labels; no journal publication, independent corpus verification, or public availability of a complete manuscript is asserted.

The additional record contains definitions and formulas absent from the original XV-B excerpt, including the coordinate-response line, the characteristic attachment locus, the grouped cross-line expression, and the partial-diagonal class. Their recovery changes the available evidence. It does not construct the full characteristic operation.

The reported XII-A and XII-D candidate identifications are not used to assign values to unspecified inherited symbols. In particular, a displayed candidate tensor or response formula is not an identification of the complete package required in Section sec:obligations. The independently supplied XII-C and XII-D classical passages are not premises of the abstract theorems below.

All vector spaces are over $\mathbb{C}$; the real versions are identical. $\mathrm{Hom}$ denotes algebraic linear maps unless continuity is explicitly required. Manifolds are smooth, Hausdorff, and second countable. Distributions on manifolds act on compactly supported smooth densities; in Euclidean charts we use the fixed coordinate density and write tests as functions.

Bundle-valued assertions are interpreted in local trivializations of a finite-rank smooth bundle and proved componentwise. Multi-index inequalities are componentwise, and

For $K⋐X\subset {\mathbb{R}}^d$ and $M\in \mathbb{N}$, write

The notation $K⋐X$ means that $K$ is compact and contained in the open set $X$.

Let ${\mathcal{P}}_{\mathrm{int}}$ be a vector space and let

be a surjective linear map. Set

Let $O\subset W$ be open, and put

Let $r:V\to V_O$ be restriction, and let

be a prescribed linear canonical map. Define

A preliminary lift is a linear map $F_0:{\mathcal{P}}_{\mathrm{int}}\to V$ satisfying

No surjectivity of $r$ is assumed. A distribution on an open subset need not extend across its boundary; an explicit example appears in Proposition prop:nonextendable.

For distributions,

Indeed, zero restriction is equivalent to vanishing on all tests supported in $O$.

A linear map $F:{\mathcal{P}}_{\mathrm{int}}\to V$ has a unique factorization

if and only if

If $rF=c$, the induced map satisfies $r\mathsf{M}=m^{\circ }$.

Proof. A factorization annihilates $\ker \mathsf{Rec}$.

Conversely, suppose $F{|}_H=0$ and define

If $\mathsf{Rec}A=\mathsf{Rec}B$, then $A-B\in H$, so $F(A)=F(B)$. Thus $\mathsf{M}$ is well defined and linear. Surjectivity of $\mathsf{Rec}$ gives uniqueness. Finally,

This is the elementary factorization underlying the corresponding XIV-J and XV-A statements CheungXIVJ,CheungXVA.

If a larger carrier is also denoted by ${\mathcal{K}}_{\mathrm{int}}$, uniqueness holds only on $\mathrm{im}\mathsf{Rec}$ until a further extension is specified.

If ${\mathcal{P}}_{\mathrm{int}}$ is a topological vector space, ${\mathcal{K}}_{\mathrm{int}}$ carries the quotient topology, and $F$ is continuous and annihilates $H$, the induced $\mathsf{M}$ is continuous. This follows from the defining property of the quotient topology. It does not identify that topology with an inherited physical carrier topology.

Fix a preliminary lift $F_0$ and a vector subspace

The candidate family is $F_0+\mathcal{A}$. Define

Since $c{|}_H=0$, the null defect

takes values in $Y$. Its relative obstruction is

The symbol $\partial$ here denotes restriction of a map, not a distributional derivative.

For Definition def:obstruction:

1. The class ${\mathrm{Ob}}_{\mathcal{A}}(F_0)$ is unchanged if $F_0$ is replaced by another element of $F_0+\mathcal{A}$.

2. A reconstruction-descending member of $F_0+\mathcal{A}$ exists if and only if

   $${\mathrm{Ob}}_{\mathcal{A}}(F_0)=0.$$

3. An individual candidate $F_0+a$ descends if and only if

   $$d_0+\partial a=0.$$

4. If $F_{\ast }$ is one descending candidate, all descending candidates are exactly

   $$F_{\ast }+{\mathcal{A}}_0,{\mathcal{A}}_0=\{a\in \mathcal{A}:a{|}_H=0\}.$$

5. Every descending candidate induces a unique map $\mathsf{M}:{\mathcal{K}}_{\mathrm{int}}\to V$ agreeing with $m^{\circ }$ after restriction to $O$. The induced ambiguity consists precisely of those maps $\bar{a}:{\mathcal{K}}_{\mathrm{int}}\to Y$ for which $\bar{a}\mathsf{Rec}\in {\mathcal{A}}_0$.

Proof. If $F_1=F_0+a_1$, then

which proves (i).

A candidate annihilates $H$ precisely when (eq:cancellation) holds. Such an $a$ exists exactly when $d_0\in \partial \mathcal{A}$, because $\partial \mathcal{A}$ is a vector subspace. This proves (ii) and (iii) by Proposition prop:descent.

If $F_{\ast }$ annihilates $H$, then $F_{\ast }+a$ annihilates $H$ exactly when $a{|}_H=0$, proving (iv).

Every correction takes values in $\ker r$, so

Proposition prop:descent gives the claimed agreement. The same proposition factors every $a\in {\mathcal{A}}_0$ uniquely through $\mathsf{Rec}$, and conversely any stated $\bar{a}$ gives such a correction. This proves (v).

The same criterion holds when $\mathcal{A}$ is a specified vector space of continuous corrections and the candidate maps are continuous. Vanishing means membership in the actual image $\partial \mathcal{A}$, not merely its closure.

Proof. The proof uses the exact equation $d_0+\partial a=0$ in the specified correction space. It does not replace the image of $\partial$ by a closure or completion.

Equation (eq:vanishing) asserts existence of a descending candidate; it does not assert that every candidate descends.

An application to a frozen analytic construction must establish that the selected family really is affine, that its members meet the required constraints, and, for an exclusion theorem, that it exhausts the class being excluded. Defining (eq:obstruction) does not evaluate that class for a particular graph.

The following are equivalent:

1. a preliminary linear lift exists;

2. every $m^{\circ }(k)$, $k\in {\mathcal{K}}_{\mathrm{int}}$, has some extension in $V$;

3. there exists a linear $\mathsf{M}:{\mathcal{K}}_{\mathrm{int}}\to V$ with $r\mathsf{M}=m^{\circ }$.

This equivalence imposes no continuity, locality, normalization, or wavefront condition.

Proof. For (i)$\Rightarrow$(ii), choose $A$ with $\mathsf{Rec}A=k$ and use $F_0(A)$.

For (ii)$\Rightarrow$(iii), choose a Hamel basis of ${\mathcal{K}}_{\mathrm{int}}$, choose an extension for the canonical value of each basis element, and extend linearly. Every vector is a finite linear combination, so the resulting assignment is a linear map with the required restriction.

For (iii)$\Rightarrow$(i), take $F_0=\mathsf{M}\mathsf{Rec}$.

If a preliminary lift exists and $\mathcal{A}=\mathrm{Hom}({\mathcal{P}}_{\mathrm{int}},Y)$, then ${\mathrm{Ob}}_{\mathcal{A}}(F_0)=0$.

Proof. Choose an algebraic complement ${\mathcal{P}}_{\mathrm{int}}=H\oplus L$ and its projection ${\pi }_H:{\mathcal{P}}_{\mathrm{int}}\to H$. The correction

takes values in $Y$ and satisfies $a{|}_H=-d_0$. Apply Theorem thm:lifting.

These basis and complement choices are not graph prescriptions. In particular, they need not preserve a topology, differential operator structure, covariance, contact restrictions, or microlocal bounds.

Let $C\subset W$ be closed and set

Suppose every $a\in \mathcal{A}$ takes values in $V_C$. If there exist $A\in H$ and an open set $G\subset W\setminus C$ such that

then no member of $F_0+\mathcal{A}$ descends. Equivalently, ${\mathrm{Ob}}_{\mathcal{A}}(F_0)\ne 0$.

Proof. Every permitted correction restricts to zero on $G$. Therefore

for every $a\in \mathcal{A}$. No candidate annihilates $H$. Theorem thm:lifting proves the obstruction statement.

If $C\subset W\setminus O$ and $\mathcal{A}=\mathrm{Hom}({\mathcal{P}}_{\mathrm{int}},V_C)$, a descending candidate exists if and only if $F_0(H)\subset V_C$.

Proof. If $F_0+a$ annihilates $H$, then $F_0(A)=-a(A)\in V_C$ for $A\in H$.

Conversely, assume $F_0(H)\subset V_C$ and choose the algebraic projection ${\pi }_H$ used above. Then

is contact-valued and cancels the defect.

Theorem thm:contact is the precise support-theoretic version of the observation that collision contacts cannot alter an off-collision prescription. The supplied XIII-E release records that observation CheungXIIIE. This comparison imports no regulator, characteristic limit, selected finite part, or convergence theorem from XIII-E.

If $r{F}_0=c$, then $r{F}_0(A)=0$ for every $A\in H$. This does not in general imply $F_0(A)=0$ on $W$.

Proof. The first assertion follows from $c{|}_H=0$. If $p\in W\setminus O$ and the residual bundle has a nonzero fiber there, a nonzero delta distribution at $p$, with a chosen fiber coefficient, restricts to zero on $O$. Thus restriction need not detect a defect supported outside $O$.

The counterexample concerns injectivity of restriction; it does not assert that an arbitrary delta term belongs to the frozen analytic ambiguity class. This is the domain qualification in XIV-J and XV-A CheungXIVJ,CheungXVA.

For $i=1,2$, let ${\mathsf{Rec}}_i:P_i\to K_i$ be linear and let $D_i\subset P_i$ be vector subspaces. Suppose $F_1:D_1\to V_1$ annihilates $D_1\cap \ker {\mathsf{Rec}}_1$. Let

be linear maps satisfying

Then

annihilates $D_2\cap \ker {\mathsf{Rec}}_2$ and descends onto ${\mathsf{Rec}}_2(D_2)$.

No assertion is made outside $D_2$. If all maps are continuous and the reconstructed image carries the appropriate quotient topology, the corresponding induced map is continuous.

Proof. For $A\in D_2\cap \ker {\mathsf{Rec}}_2$,

Since $\alpha A\in D_1$, the assumed null condition gives $F_1(\alpha A)=0$, hence $F_2(A)=0$. Apply Proposition prop:descent to ${\mathsf{Rec}}_2{|}_{D_2}:D_2\to {\mathsf{Rec}}_2(D_2)$. The continuity assertion is the quotient-topology statement.

This lemma makes the needed identifications explicit. A common label, an analogous formula, or an internal Fourier representation does not supply $\alpha$, $\beta$, $\tau$, or the domain inclusion $\alpha (D_2)\subset D_1$. In particular, it does not enlarge a canonical composition domain to a characteristic domain.

Fix a product chart

where $X\subset {\mathbb{R}}^d$ and $V_q\subset {\mathbb{R}}^m$ are open and $0\in V_q$. The notation and representation are tied to these fixed normal coordinates.

For $\gamma \in {\mathbb{N}}^m$ and $b\in {\mathcal{D}}^{\prime }(X)$, define

Thus ${\mathsf{E}}_{\gamma }b=b\otimes {{\partial }_q}^{\gamma }{\delta }_0$ in the chosen product coordinates.

Choose $\chi \in \mathcal{D}(V_q)$ equal to one near zero and put

These are continuous maps between the corresponding test-function spaces, and

For a finite family $b_{\gamma }\in {\mathcal{D}}^{\prime }(X)$,

Moreover, testing the sum against ${\mathsf{J}}_{\gamma }\psi$ extracts $⟨b_{\gamma },\psi ⟩$.

Proof. Equation (eq:extractjets) and the insertion convention give

The forward implication follows for all $\psi$; the reverse is immediate.

A distribution $t\in {\mathcal{D}}^{\prime }(U)$ has supported normal order at most $r$ along $S$ if $\mathrm{supp}t\subset S$ and it annihilates every test $\varphi$ satisfying

On an open subset of $U$, this condition is understood locally in product neighborhoods of the points of $S$.

A distribution has supported normal order at most $r$ along $S$ if and only if it has a unique representation

Proof. Suppose $t$ has the stated normal order. Define

Continuity of ${\mathsf{J}}_{\gamma }$ makes $b_{\gamma }$ a distribution.

For $\varphi \in \mathcal{D}(U)$, set

Each $a_{\gamma }$ is a test on $X$, and

has zero normal jets through order $r$. Consequently,

which is (eq:normal).

Conversely, every insertion is supported on $S$ and depends only on its indicated jet. The finite sum therefore has the required normal order. Lemma lem:extraction proves uniqueness.

This fixed-order statement is a special case of the standard local structure of distributions supported on a smooth submanifold Hormander,Dang. A general supported distribution need only have a locally finite normal-order bound; a single global $r$ is an additional hypothesis.

For $c\in C^{\infty }(U)$ and $v\in {\mathcal{D}}^{\prime }(X)$,

Proof. Pair the left side with $\varphi$. The Leibniz rule gives

The insertion on the right contributes the sign $(-1{)}^{|\beta |}(-1{)}^{|\alpha -\beta |}=(-1{)}^{|\alpha |}$, giving the formula.

The formula concerns smooth multipliers. Applying an analogous formula to singular cross-lines requires separately proved traces and products.

Fix $T\in {\mathcal{D}}^{\prime }(X)$ and $r\in \mathbb{N}$. Let

and define

All products are multiplication by smooth functions.

For $a=(a_{\alpha })\in {{\mathcal{P}}_r}^{\mathrm{jet}}(T)$, define the smooth functions

and the distributions

Then

and

Thus (eq:completekernel) is a complete distributional null-relation test.

Proof. Apply Lemma lem:leibniz to each summand $a_{\alpha }{\mathsf{E}}_{\alpha }T$ and collect the coefficient of ${\mathsf{E}}_{\gamma }$. This gives (eq:fcoeff)–(eq:jetnormal). Lemma lem:extraction then gives (eq:completekernel).

This is the explicit local criterion in the supplied XV-C theorem CheungXVC, now proved with the insertion and extraction conventions fixed in the present paper.

The equations $f_{\gamma }(a)T=0$ are distributional equations. They need not imply $f_{\gamma }(a)=0$ pointwise. For example, $x{\delta }_0=0$ although $x$ is not the zero smooth function. Replacing (eq:completekernel) by pointwise coefficient vanishing would lose genuine null relations.

In one normal variable,

Suppose an adapted reconstruction convention satisfies

and multiplication by $q$ is reconstructed as multiplication by the same normal coordinate. Then

If $R\ne 0$, the degree-zero and degree-one summands reconstruct separately to nonzero opposite distributions.

Proof. For $\psi \in \mathcal{D}(\mathbb{R})$,

Tensoring $q{\delta }^{\prime }=-\delta$ with $R$ gives the identity. The tensor $\delta \otimes R$ is nonzero when $R\ne 0$, as is seen by choosing separated tests with nonzero pairings.

In the convention of Lemma lem:null, the same nonzero reconstructed kernel has representatives

of different highest polynomial degrees. Consequently, highest degree of a chosen presentation is not an invariant of the reconstructed kernel.

Likewise, projection to the degree-one summand does not descend: it sends the null presentation in (eq:null) to a summand with nonzero reconstruction.

Proof. Both displayed representatives reconstruct to $\delta (q)\otimes R$. The degree-one projection of the null presentation reconstructs to $-\delta (q)\otimes R\ne 0$. Apply Proposition prop:descent.

This does not rule out a filtration defined by reconstructed images of filtered presentation spaces, nor does it invalidate XIV-I's estimates on its named module. It rules out promoting a representative-dependent highest term or splitting to an intrinsic joint symbol without a descent theorem and a compatible joint calculus CheungXIVI,CheungXIVJ.

Let $A\subset U$ be relatively closed and put

Then $Z$ is relatively closed in $X$.

Suppose $t^{\circ }\in {\mathcal{D}}^{\prime }(U\setminus A)$ is supported on $S\setminus A$ and locally has supported normal order at most $r$. Its unique normal-jet coefficients glue to distributions

The following are equivalent:

1. there exists $t\in {\mathcal{D}}^{\prime }(U)$ supported on $S$, of supported normal order at most $r$, with

   $$t{|}_{U\setminus A}=t^{\circ };$$

2. every ${b_{\gamma }}^{\circ }$ extends to some $b_{\gamma }\in {\mathcal{D}}^{\prime }(X)$.

For any chosen coefficient extensions,

is such a realization. If realizations exist, all of them form an affine space over

Proof. For every $x\in X\setminus Z$, choose a product neighborhood of $(x,0)$ disjoint from $A$. Theorem thm:normal gives its coefficients there. Uniqueness makes these coefficients agree on overlaps, so they glue on $X\setminus Z$.

If (i) holds, extract the coefficients of $t$ by Theorem thm:normal. On every such smaller product neighborhood, uniqueness identifies their restrictions with the coefficients of $t^{\circ }$. This proves (ii).

If (ii) holds, (eq:jetrealization) is a distribution supported on $S$ with the stated normal order. Near every point of $S\setminus A$, its coefficients agree with those of $t^{\circ }$, so the distributions agree by Theorem thm:normal. Away from $S$ both distributions vanish. These local identities establish equality on $U\setminus A$, proving (i).

Finally, two extensions of a coefficient differ by a distribution supported on $Z$. Conversely, any such difference changes no restriction on $U\setminus A$. Coefficient uniqueness proves that (eq:jetambiguity) is the full ambiguity under the stated support and normal-order conditions.

For the supported-sector reference in XIV-J, the intended local substitution is

in a specified response-normal chart in which $N_U$ is relatively closed. The copied reference and its collected coefficients are recorded in Section sec:reference CheungXIVJ.

The support hypothesis is essential. An arbitrary extension may also contain terms supported in $A$ outside $S$; such terms are not described by the response-normal insertions.

The space (eq:jetambiguity) is the unrestricted supported ambiguity with the fixed response-normal bound. It is not automatically the partial-diagonal class, a normalized contact class, or a space of admissible maps on presentations.

A finite normal-order bound does not make this ambiguity finite-dimensional. For example, when $d\ge 1$ and $Z=\{0\}$, already at $r=0$ the distributions

give an infinite linearly independent family. A further bound on the tangential order is needed for a finite-dimensional space.

Let $X\subset {\mathbb{R}}^d$ be open, let $Z\subset X$ be relatively closed, and let $b^{\circ }\in {\mathcal{D}}^{\prime }(X\setminus Z)$. Then $b^{\circ }$ extends to a distribution in ${\mathcal{D}}^{\prime }(X)$ if and only if, for every $K⋐X$, there exist $C_K>0$ and $M_K\in \mathbb{N}$ such that

for all $\varphi \in \mathcal{D}(X\setminus Z)$ with $\mathrm{supp}\varphi \subset K$.

The constants and order in (eq:closedestimate) are uniform as these supports approach $Z$. The criterion imposes no wavefront or prescribed scaling-degree bound.

Proof. Necessity follows from the finite-order estimate of any extension on the fixed compact set $K$.

For sufficiency, choose a locally finite cover of $X$ by relatively compact open sets $U_i$ whose closures $K_i$ are contained in $X$. On the subspace $\mathcal{D}(U_i\setminus Z)\subset \mathcal{D}(U_i)$, the functional $b^{\circ }$ is bounded by

where tests are extended by zero when needed. The complex Hahn–Banach theorem extends it to a linear functional $B_i$ on $\mathcal{D}(U_i)$ with the same bound. This bound proves that $B_i$ is a distribution on $U_i$.

Choose a smooth locally finite partition of unity $({\eta }_i)$ subordinate to the cover and define

Only finitely many terms occur for any fixed compact support, and their estimates show distributional continuity. If $\varphi$ is supported in $X\setminus Z$, every term agrees with the original distribution, giving

Thus $b$ is an extension.

The extension in Theorem thm:jetextension exists if and only if every ${b_{\gamma }}^{\circ }$ satisfies (eq:closedestimate) on $X\setminus Z$. Since there are finitely many $\gamma$, the orders and constants may be chosen common to all coefficients on each fixed compact set.

Proof. Apply Theorem thm:closedextension to each coefficient and then Theorem thm:jetextension. Take the maximum of the finitely many orders and enlarge the finitely many constants.

The criterion is applicable to a general closed locus, not only a finite puncture. It does not select an admissible characteristic prescription or provide microlocal control. Those are separate tasks; compare the additional scaling and wavefront hypotheses in BF,Dang.

For $b\in {\mathcal{D}}^{\prime }(X)$,

Proof. The insertion is $b\otimes {{\partial }_q}^{\gamma }{\delta }_0$. The tensor-product wavefront bound, with zero covectors adjoined on the supports of the factors, gives (eq:insertWF). Differentiation does not enlarge that bound Hormander (Section 8.2).

An extension of $b^{\circ }$ may introduce tangential wavefront directions at $Z$. Therefore coefficient extendability alone is not a complete-graph wavefront theorem, and an insertion bound is not an output bound after the missing characteristic composition and pushforward.

Retain $U=X\times V_q$, let $Z\subset X$ be relatively closed, and set

Fix $T^{\circ }\in {\mathcal{D}}^{\prime }(X^{\circ })$ and use the presentation space

Thus the coefficients are smooth across $Z$; we are not using the larger space of arbitrary smooth coefficients on $U^{\circ }$.

Define

The complete criterion of Theorem thm:jetnull reads

If $T_0\in {\mathcal{D}}^{\prime }(X)$ extends $T^{\circ }$, the natural preliminary lift is

For a null presentation,

Consequently the natural null defect is supported on $Z\times \{0\}$.

The formula (eq:naturaldefect) concerns the generator-compatible lift (eq:naturallift). An arbitrary linear preliminary lift may assign nonzero values even to universal presentation identities. The relation classification below is therefore not a theorem about every possible preliminary lift without additional compatibility assumptions.

Suppose $Z$ is locally a smooth embedded submanifold of $X$. In a product coordinate neighborhood $(s,h)$ with $Z=\{h=0\}$, let $T_0$ have distributional order at most $M$ on a fixed compact coordinate neighborhood.

If $f\in C^{\infty }(X)$ satisfies $f{T}^{\circ }=0$ off $Z$ and

then $f{T}_0=0$ on the interior of that neighborhood. Thus, among annihilator functions, $f{T}_0$ depends locally only on the transverse jet of $f$ through order $M$. In particular, an annihilator flat along $Z$ produces no natural defect.

Proof. Let $\varphi$ be a test supported in a smaller compact coordinate neighborhood. The distribution $u=f{T}_0$ is supported on $Z$. Choose $\rho \in \mathcal{D}({\mathbb{R}}^{\dim h})$ equal to one near zero and write ${\rho }_{ϵ}(h)=\rho (h/ϵ)$. Since $u$ is supported on $Z$,

Taylor's theorem and (eq:vanishingjet) imply, uniformly on the fixed compact set,

The derivatives of ${\rho }_{ϵ}$ contribute the corresponding powers of ${ϵ}^{-1}$. The Leibniz rule therefore gives

for sufficiently small $ϵ$. The order-$M$ estimate for $T_0$ implies $⟨u,\varphi ⟩=0$ on letting $ϵ\downarrow 0$.

If two annihilator functions have the same transverse jet through order $M$, their difference satisfies (eq:vanishingjet), so their defects agree. Flatness implies vanishing of every finite transverse jet.

In the preceding smooth-locus setting, put

and

Then

The successive categories are called universal relations, non-universal flat-annihilator relations, and potentially jet-detected relations.

The null defect of $F_{T_0}$ vanishes on $H_{\mathrm{flat}}$, and hence factors algebraically through $H^{\circ }/H_{\mathrm{flat}}$. Locally, its coefficient defects depend on finite transverse jets of the functions $f_{\gamma }(a)$.

These categories form a filtration, not a canonical direct-sum decomposition. A potentially jet-detected relation need not have a nonzero defect.

Proof. Universal relations vanish for every generator-compatible lift by (eq:naturaldefect). Flat-annihilator relations vanish locally by Lemma lem:jetdefect, hence globally by locality. The factorization follows. The same lemma proves finite-jet dependence. No argument asserts nonvanishing for every remaining relation.

For a positive-dimensional $Z$, the transverse jets are functions along $Z$, and supported coefficient defects can have distributional tangential dependence. They are not generally finite-dimensional data.

This formulation makes precise, for the stated module and natural lifts, the relation distinctions discussed in the supplied XV-E account CheungXVE. It does not evaluate the inherited graph's relative obstruction.

Let $X\subset {\mathbb{R}}^d$ be open with $0\in X$, and let $T^{\circ }\in {\mathcal{D}}^{\prime }(X\setminus \{0\})$ have a distributional extension $T_0\in {\mathcal{D}}^{\prime }(X)$. Define its smooth annihilator ideal

Then there exists

for some finite $M$ such that

still extends $T^{\circ }$ and satisfies

One may take $M$ to be a distributional order bound for $T_0$ on a compact neighborhood of zero.

Proof. Fix such an order bound $M$. For every $f\in I$, the distribution $f{T}_0$ is supported at zero. If ${j_0}^{M}f=0$, the point case of Lemma lem:jetdefect gives $f{T}_0=0$.

Let

be the finite-dimensional algebra of jets through order $M$, and let $I_M={j_0}^M(I)$ be the image ideal in that algebra. Choose $\chi \in \mathcal{D}(X)$ equal to one near zero and define

This is well defined: representatives with the same jet differ by an element of $I$ with vanishing $M$-jet, whose product with $T_0$ is zero. The value is independent of the chosen $\chi$, since $f{T}_0$ is supported at zero.

Extend the linear functional $\ell$ from $I_M$ to a linear functional $\Lambda$ on the finite-dimensional vector space ${J_0}^M$. Define

This is a distribution supported at zero. In the jet basis $e_{\nu }=[x^{\nu }/\nu !]$, it has the explicit form

which is (eq:pointD).

For $f\in I$ and $\varphi \in \mathcal{D}(X)$, the product $f\varphi$ is also in $I$. Because $(f\varphi )T_0$ is supported at zero,

Thus $f(T_0-D)=0$ for every $f\in I$. Subtracting $D$ changes no restriction off zero.

In the module of Section sec:relations, with $Z=\{0\}$, if $T^{\circ }$ has any distributional extension, it has an extension $T$ for which $F_T$ annihilates all of $H^{\circ }$. Hence $F_T$ descends to a map on $\mathrm{im}{\mathsf{Rec}}^{\circ }$ extending every reconstructed value.

Relative to $F_{T_0}$, the correction is explicitly

where $D$ is given by (eq:pointD). Its values are supported at $(0,0)$, with response-normal order at most $r$ and tangential order at most $M$.

Proof. For $a\in H^{\circ }$, equation (eq:puncturednull) says that every $f_{\gamma }(a)$ belongs to $I$. By Theorem thm:pointcorrection, $f_{\gamma }(a)T=0$ for every $\gamma$. Theorem thm:jetnull therefore gives $F_T(a)=0$. Proposition prop:descent gives the factorization.

Equation (eq:pointmapcorrection) follows from $T=T_0-D$. The support and order bounds follow from the insertion formula and smooth multiplication.

The correction is simultaneous for the entire smooth annihilator ideal, not a separate choice for each null relation. It uses the finite-dimensional jet algebra of one point.

The result assumes extendability of $T^{\circ }$; it does not create an extension for a nonextendable coefficient. It does not prove preservation of a specified scaling degree, a frozen normalization, or a restricted physical contact class. The order $M$ depends on the chosen local extension; it is not bounded by the response-normal order $r$ alone.

The presentation maps in this local construction are continuous for the usual smooth coefficient topology, since they multiply fixed finite-order distributions by smooth coefficients. No identification with the physical carrier topology follows.

For any $a>0$, the smooth function on the punctured interval

defines a distribution on $(-1,1)\setminus \{0\}$ with no extension in ${\mathcal{D}}^{\prime }((-1,1))$.

Proof. Suppose an extension $B$ exists. On a fixed compact neighborhood of zero, there are $C>0$ and $M\in \mathbb{N}$ such that

for tests supported in that neighborhood.

Choose a nonnegative nonzero $\varphi \in \mathcal{D}((0,1))$ and set

Its support lies in $(\eta ,\eta +{\eta }^3)$ and

For small $\eta$, the support lies in the fixed compact neighborhood and satisfies $x\le 2\eta$. Since $B$ agrees there with ${b_a}^{\circ }$,

The lower bound exceeds every fixed power of ${\eta }^{-1}$ as $\eta \downarrow 0$, contradicting the distributional estimate.

A punctured supported distribution of normal order zero need not have a supported normal-order-preserving extension.

Proof. Take

and set $t^{\circ }={\mathsf{E}}_0{b_1}^{\circ }$ on $U\setminus A$. Theorem thm:jetextension would turn an extension of the required support and normal order into an extension of ${b_1}^{\circ }$, contradicting Proposition prop:nonextendable.

Define

Then $v$ is smooth and flat at zero, but

is nonextendable.

Proof. Every derivative of $e^{-2/x^2}$ is a finite sum of powers of $x^{-1}$ multiplied by $e^{-2/x^2}$, and tends to zero at zero. The same estimates after division by any fixed power of $x$ give the smooth flat extension. The product identity holds on the punctured interval, and Proposition prop:nonextendable applies.

The correct conclusion is that flatness alone does not guarantee extendability of a product. It is false that a flat multiplier can never repair nonextendability. For the same $v$,