Landmark XV-E Source Recovery for the Frozen Joint Graph, Descending Extensions in the Punctured Normal-Jet Module, and the Boundary of the Full Lift

XV-E continues XV-D at its gate D0. Its first task is a source recovery pass: to obtain, from the supplied record, the definitions that XV-D found missing for the frozen six-branch joint target, and, if they are obtained, to define the preliminary full lift, evaluate its reconstruction-null defect, and test cancellation in the inherited correction class.

The recovery pass finds one change. Excerpts of XII-A and XII-D now in the record display tensor and kernel formulas that are candidates for the undefined symbols $\mathsf{A}$, $L$, and $R$ in the internal kernels. The record does not identify them with those symbols, and XIV-H's defining equation for $L$ is still not reproduced, so they are recorded as candidates and not used. All other items remain unavailable. No lift for the actual graph is definable, and no graph-specific defect, obstruction, or estimate is asserted. The graph-level outcome is Unresolved at E0.

The new mathematics concerns the one source-complete module, the finite normal-jet module with punctured reconstruction. We prove that the null defect of an extension lift is governed by the annihilator ideals of the punctured and extended coefficients, with $H/H_T\cong (\mathrm{Ann}{T}^{\circ }/\mathrm{Ann}T{)}^{M_r}$; that null relations divide into universal, flat-annihilator, and jet-detected relations, of which only the last produce a defect, and that the defect is locally a functional of a finite jet along the puncture. For a point puncture we prove that a descending extension exists whenever any extension exists, with an explicit correction by finitely many derivatives of the delta at the puncture; in one dimension every extension descends. For a general puncture we prove that a descending extension exists if and only if a stated relative estimate holds, and we leave open whether that estimate can fail. Finally we correct an overstatement in XV-D's coefficient proposition and show by examples exactly which implications between extendability of a trace and of its smooth multiple hold. None of these results is transferred to the inherited graph.

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

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

XV-D XVD audited the supplied record item by item and found that the data defining the full joint characteristic operation for the frozen six-branch graph were not in it. XV-E asks:

Can the frozen six-branch operation be defined on the specified domain, extended to tests meeting the characteristic locus $N$, and made independent of presentation within the inherited correction class?

Exactly one of three endings is permitted for the graph-level problem: Constructed through descent; Obstructed in the frozen class, with the class proved exhaustive; or Unresolved at a named gate. Failure to recover a source is not an analytic obstruction. Successful descent would not be quantum admission.

E0 stop rule. If an item needed to state the lift or its allowed corrections remains unavailable, this is said plainly; E1–E4 are not entered for the actual graph; no graph formula with placeholders is written and called a lift.

Direct inputs: the full texts of XV-A, XV-B, XV-C, XV-D XVA,XVB,XVC,XVD; the retrieved XIV-H, XIV-I, XIV-J excerpts Hkernel,Hverdict,Ipole,Iamplitude,Ijoint,Irelease,Jdescent, Jnull,Jreference,Jconditional,Jverdict,Jrelease,Jadmission; the retrieved XIV-C and XIV-G release statements Crelease,Grelease; the retrieved XII-A and XII-D passages XIIA,XIID; and the proposed XV-E plan, used as a statement of objectives only. The coverage account record is documentary context. The record itself states that similar words in distinct manuscripts do not identify their models, response maps, or sign conventions; this rule is applied to every candidate identification below. Nothing from XIV-A, XIV-B, XIII-D, or XIII-E is imported.

The technical-continuation convention of XV-B–XV-D is retained. The admission criterion

is a later project gate, not a prerequisite for this manuscript, and is not asserted here.

The pass consists of a complete re-reading of the record as supplied with the present request, item by item against XV-D's Definition D0.1–D0.20. The record is the same retrieval corpus as before record; the passages newly present in it, relative to the record audited by XV-D, are the XIV-C release Crelease, the XII-A composite definitions XIIA, and the XII-D root and transpose passage XIID. No external registry is consulted, and no claim about documents outside the record is made.

Status codes: S supplied (formula or definition reproduced); T transcribed from a reproduced excerpt; C candidate (a formula in the record for a differently named object, whose identification with the required symbol is not supplied); R reported only (name, locator, or sentence); U unavailable.

The XII-A passage XIIA defines, on the XII sheet,

and the XII-D passage XIID fixes the sign ${G^{\mathrm{R}}}_{\mathrm{gr}}=-\mathfrak{P}{E_0}^{\mathrm{R}}$ and displays ${{\alpha }_{\mathrm{co}}}^{ab}=-{\sigma }_{\epsilon }{\mathfrak{a}}^{\lambda \mu ,ab}{\partial }_{\lambda }{E_0}^{\mathrm{A}}(f{\partial }_{\mu }\varphi )$. The coverage account states that the XIV analytic branch inherits XII-C's cubic action and XII-A's carrier record. The following identifications are therefore candidates:

None of the four identifications is supplied by the record. Each would be promoted to status S by reproducing XIV-H's equation eq:L and the definitions of $\mathsf{A}$ and $R$ in XIV-H or XIV-I, together with a statement in those manuscripts that their symbols are the XII-A objects, or by a transport proof meeting the four conditions of XV-A's domain-sensitive transfer criterion XVA. Even if all four were promoted, items D0.1, D0.2, D0.4, D0.8, D0.10–D0.12, D0.14, D0.16–D0.18, and D0.20 would remain in status R or U.

Proof. The XIV-H excerpt uses $L$ and refers to “(eq:L)” without reproducing it Hkernel; the XIV-I excerpt uses $L$, $R$, $\mathsf{D}$ without definitions Iamplitude. The XII-A passage names $\Lambda$ and ${E_0}^{\mathrm{R}}$ and does not mention XIV-H's symbols. The record's own reading rule forbids identifying objects across manuscripts by notation. The promotion conditions are read off from XV-A's transfer criterion. The last sentence is the manifest.

The record does not define ${\mathcal{P}}_{\mathrm{int}}$, $\mathsf{Rec}$ on ${\mathcal{P}}_{\mathrm{int}}$, $O$, $m^{\circ }$, ${\mathcal{C}}_{\mathrm{cross}}$, the vertex data, the candidate class, or the correction class. Hence no preliminary lift $F_0:{\mathcal{P}}_{\mathrm{int}}\to {\mathcal{D}}^{\prime }(W;E_{\mathrm{res}})$ with $r_O{F}_0=m^{\circ }\mathsf{Rec}$ can be stated, and no correction space $\mathcal{A}$ against which a null defect could be tested is available.

Proof. A linear map requires a domain, a codomain, and a value on each element of the domain. The domain ${\mathcal{P}}_{\mathrm{int}}$ (D0.1) and codomain (D0.5) are not defined. The value on any presentation of the six entries is the composition of the internal kernel with ${\mathcal{C}}_{\mathrm{cross}}$, the vertex Hessian, and weights (D0.10–D0.13), none of which is defined; the candidates of sec:candidates concern only the internal kernels. The correction class needs D0.16–D0.17. All are in status R, U, or C.

Gates E1–E4 are not entered for the actual graph. Section sec:branches records the freeze; Sections sec:jet and sec:erratum contain the source-complete calculations.

The retained kernel and supported sector are

$K_{47}$ groups $B_4+B_7$; both $K_5$ placements belong to $B_5$. The supplied internal expressions are Hkernel,Iamplitude

and the internal pole chart is

with $(\lambda -i0{)}^{-1}=\mathrm{PV}\frac{1}{\lambda }+i\pi \delta (\lambda )$ Ipole. The supported-sector reference and its wavefront bound are Jreference

This is a freeze check, not evidence that the full operation exists. Propagator, pole, regulator, carrier, normalization, and ambiguity class are unchanged.

The only module in the record with defined presentations, reconstruction, canonical domain, and target is the finite normal-jet module with punctured reconstruction of XV-D XVD. XV-D computed the null defect of every extension lift and exhibited a nonzero defect cancelled by a change of extension. XV-E now determines exactly which relations carry a defect and when a descending extension exists. Everything in this section concerns that module; it is not identified with ${\mathcal{P}}_{\mathrm{int}}$.

Let $X\subset {\mathbb{R}}^d$ and $V_q\subset {\mathbb{R}}^m$ be open with $0\in V_q$, let $Z\subset X$ be relatively closed, and put $X^{\circ }=X\setminus Z$, $U=X\times V_q$, $U^{\circ }=X^{\circ }\times V_q$. For $b\in {\mathcal{D}}^{\prime }(X)$, $\varphi \in \mathcal{D}(U)$,

and ${{\mathsf{E}}^{\circ }}_{\gamma }:{\mathcal{D}}^{\prime }(X^{\circ })\to {\mathcal{D}}^{\prime }(U^{\circ })$ is defined by the same formula. Fix $r\ge 0$, let $M_r=\#\{\gamma \in {\mathbb{N}}^m:|\gamma |\le r\}$, and set ${{\mathcal{P}}_r}^{\mathrm{jet}}={⨁}_{|\alpha |\le r}{C}^{\infty }(U)$. For $a\in {{\mathcal{P}}_r}^{\mathrm{jet}}$ put

Fix $T^{\circ }\in {\mathcal{D}}^{\prime }(X^{\circ })$. The punctured reconstruction and, for an extension $T\in {\mathcal{D}}^{\prime }(X)$ of $T^{\circ }$, the full reconstruction are

with kernels $H=\ker {\mathsf{Rec}}^{\circ }$, $H_T=\ker {\mathsf{Rec}}_T$. The extension lift is ${F_0}^T={\mathsf{Rec}}_T$ and its null defect is ${d_0}^T={F_0}^T{|}_H$. In XV-B's interface, $O=U^{\circ }$, $m^{\circ }=\mathrm{id}$, $r$ is restriction, and $Y=\{t\in {\mathcal{D}}^{\prime }(U):\mathrm{supp}t\subset Z\times V_q\}$.

For every $T\in {\mathcal{D}}^{\prime }(X)$ and $a\in {{\mathcal{P}}_r}^{\mathrm{jet}}$, ${\mathsf{Rec}}_T(a)={\sum }_{|\gamma |\le r}{\mathsf{E}}_{\gamma }({\beta }_{\gamma }(a)T)$, and ${\sum }_{\gamma }{\mathsf{E}}_{\gamma }{b}_{\gamma }=0$ iff all $b_{\gamma }=0$. Consequently

$r{F}_0^T={\mathsf{Rec}}^{\circ }$, ${d_0}^T(a)={\sum }_{\gamma }{\mathsf{E}}_{\gamma }({\beta }_{\gamma }(a)T)$ with every coefficient supported in $Z$, $\ker {d_0}^T=H_T$, and ${d_0}^T=0$ iff $H_T=H$.

Proof. The normal form is the Leibniz rule applied to ${{\partial }_q}^{\alpha }(a_{\alpha }\varphi )(\cdot ,0)$ in (eq:insert). Extraction: with $\chi \in \mathcal{D}(V_q)$ equal to one near $0$ and ${\mathsf{J}}_{\gamma }\psi (x,q)=\frac{(-1{)}^{|\gamma |}}{\gamma !}{q}^{\gamma }\chi (q)\psi (x)$, one has $(-1{)}^{|\beta |}{{\partial }_q}^{\beta }{\mathsf{J}}_{\gamma }\psi (x,0)={\delta }_{\beta \gamma }\psi (x)$, so testing on ${\mathsf{J}}_{\gamma }\psi$ returns $⟨b_{\gamma },\psi ⟩$. The rest follows as in XVD: restriction of the normal form to $U^{\circ }$ gives $r{F}_0^T={\mathsf{Rec}}^{\circ }$ and $H_T\subseteq H$; for $a\in H$ each ${\beta }_{\gamma }(a)T$ restricts to zero on $X^{\circ }$.

$\mathrm{Ann}(T^{\circ })=\{g\in C^{\infty }(X):g{|}_{X^{\circ }}{T}^{\circ }=0\}$ and, for $T\in {\mathcal{D}}^{\prime }(X)$, $\mathrm{Ann}(T)=\{g\in C^{\infty }(X):gT=0\}$. Both are ideals of $C^{\infty }(X)$, and $\mathrm{Ann}(T)\subseteq \mathrm{Ann}(T^{\circ })$ when $T$ extends $T^{\circ }$.

The map $\sigma :C^{\infty }(X{)}^{M_r}\to {{\mathcal{P}}_r}^{\mathrm{jet}}$, $\sigma (g{)}_{\alpha }(x,q)=g_{\alpha }(x)$, satisfies $\beta \circ \sigma =\mathrm{id}$. In particular $\beta$ is surjective.

Proof. If $a_{\alpha }$ is independent of $q$ then ${{\partial }_q}^{\alpha -\gamma }{a}_{\alpha }=0$ unless $\alpha =\gamma$, so (eq:beta) gives ${\beta }_{\gamma }(\sigma (g))=g_{\gamma }$.

Let $T$ extend $T^{\circ }$. Then $H={\beta }^{-1}(\mathrm{Ann}(T^{\circ }{)}^{M_r})$, $H_T={\beta }^{-1}(\mathrm{Ann}(T{)}^{M_r})$, and $\beta$ induces a linear isomorphism

In particular ${d_0}^T=0$ if and only if $\mathrm{Ann}(T)=\mathrm{Ann}(T^{\circ })$.

Proof. The descriptions of $H$ and $H_T$ are (eq:H)–(eq:HT) read through $\beta$. Since $\ker \beta \subseteq H_T\subseteq H$, $\beta$ induces a well-defined linear map $H/H_T\to \mathrm{Ann}(T^{\circ }{)}^{M_r}/\mathrm{Ann}(T{)}^{M_r}$. It is injective: if $\beta (a)\in \mathrm{Ann}(T{)}^{M_r}$ then $a\in H_T$. It is surjective by Lemma lem:surj: for $g\in \mathrm{Ann}(T^{\circ }{)}^{M_r}$, $\sigma (g)\in H$ and $\beta (\sigma (g))=g$. Finally $\mathrm{Ann}(T^{\circ }{)}^{M_r}/\mathrm{Ann}(T{)}^{M_r}\cong (\mathrm{Ann}(T^{\circ })/\mathrm{Ann}(T){)}^{M_r}$. The last sentence is Proposition prop:recalled.

Let $a\in H$. Call $a$

1. universal if ${\beta }_{\gamma }(a)=0$ in $C^{\infty }(X)$ for all $|\gamma |\le r$;

2. a flat-annihilator relation if each ${\beta }_{\gamma }(a)$ is flat on $Z$, i.e. all its derivatives vanish at every point of $Z$;

3. jet-detected otherwise.

Universal relations include the jet analogue of the inherited null identity $\mathsf{Rec}((qi\theta +1)M_{\mathrm{R}})=0$ Jnull,XVD. XV-D proved that universal relations carry no defect for any $T$. The next results show that flat-annihilator relations also carry no defect, and that jet-detected relations produce a defect that is locally a functional of a finite jet.

Let $Z\subset {\mathbb{R}}^d$ be any set and $\epsilon >0$. There is ${\theta }_{\epsilon }\in C^{\infty }({\mathbb{R}}^d)$ with $0\le {\theta }_{\epsilon }\le 1$, ${\theta }_{\epsilon }=0$ on $\{\mathrm{dist}(x,Z)<\epsilon \}$, ${\theta }_{\epsilon }=1$ on $\{\mathrm{dist}(x,Z)>3\epsilon \}$, and $|{\partial }^{\alpha }{\theta }_{\epsilon }|\le C_{\alpha }{\epsilon }^{-|\alpha |}$ with $C_{\alpha }$ independent of $\epsilon$ and $Z$.

Proof. Let $\rho \in \mathcal{D}({\mathbb{R}}^d)$ be nonnegative, supported in the unit ball, with integral one, and ${\rho }_{\epsilon }(x)={\epsilon }^{-d}\rho (x/\epsilon )$. Put ${\theta }_{\epsilon }=1_{\{\mathrm{dist}(\cdot ,Z)\ge 2\epsilon \}}\ast {\rho }_{\epsilon }$. It vanishes where $\mathrm{dist}<\epsilon$, equals one where $\mathrm{dist}>3\epsilon$, and ${\partial }^{\alpha }{\theta }_{\epsilon }=1\ast {\partial }^{\alpha }{\rho }_{\epsilon }$ is bounded by $\Vert {\partial }^{\alpha }{\rho }_{\epsilon }{\Vert }_{L^1}={\epsilon }^{-|\alpha |}\Vert {\partial }^{\alpha }\rho {\Vert }_{L^1}$.

Let $K\subset X$ be compact, and let $\varrho >0$ be such that the closed $\varrho$-neighborhood $K^{\prime }$ of $K$ is contained in $X$. Let $g\in C^{\infty }(X)$ have vanishing $k$-jet at every point of $Z$. Then for $x\in K$ with $\mathrm{dist}(x,Z)<\varrho$ and $|\alpha |\le k$,

Proof. Put $\delta =\mathrm{dist}(x,Z)<\varrho$ and choose $\delta <{\varrho }^{\prime }<\varrho$. The closed ball $\bar{B}(x,{\varrho }^{\prime })$ lies in $K^{\prime }\subset X$, so $Z\cap \bar{B}(x,{\varrho }^{\prime })$ is closed in $\bar{B}(x,{\varrho }^{\prime })$, hence compact and nonempty; a nearest point $z\in Z$ with $|x-z|=\delta$ exists, and the segment $[z,x]\subset \bar{B}(x,\delta )\subset K^{\prime }$. Taylor's formula for ${\partial }^{\alpha }g$ at $z$ to order $k-|\alpha |$, all of whose Taylor coefficients vanish, gives the integral-remainder bound stated, with the derivatives of order $k+1$ estimated on $K^{\prime }$.

Let $T$ extend $T^{\circ }$. Let $K$, $\varrho$, $K^{\prime }$ be as in Lemma lem:taylor, and let $k\ge 0$, $C_T$ be such that $|⟨T,\psi ⟩|\le C_T{p}_k(\psi )$ for all $\psi \in \mathcal{D}(X)$ with $\mathrm{supp}\psi \subset K^{\prime }$, where $p_k(\psi )={\max }_{|\alpha |\le k}\Vert {\partial }^{\alpha }\psi {\Vert }_{\infty }$. If $g\in \mathrm{Ann}(T^{\circ })$ has vanishing $k$-jet at every point of $Z$, then $⟨gT,\varphi ⟩=0$ for every $\varphi \in \mathcal{D}(X)$ with $\mathrm{supp}\varphi \subset K$. Consequently, if $g\in \mathrm{Ann}(T^{\circ })$ is flat on $Z$, then $gT=0$, i.e. $g\in \mathrm{Ann}(T)$.

Proof. Put $\psi =g\varphi$; its $k$-jet vanishes on $Z$ and $\mathrm{supp}\psi \subset K$. Let ${\theta }_{\epsilon }$ be as in Lemma lem:cutoff with $3\epsilon <\varrho$. Then ${\theta }_{\epsilon }\psi \in \mathcal{D}(X^{\circ })$, so

Moreover $(1-{\theta }_{\epsilon })\psi$ is supported in $K\cap \{\mathrm{dist}(\cdot ,Z)\le 3\epsilon \}$, and for $|\alpha |\le k$, by the Leibniz rule, Lemma lem:cutoff, and Lemma lem:taylor applied to $\psi$,

Hence $p_k((1-{\theta }_{\epsilon })\psi )\to 0$, and the order-$k$ estimate for $T$ on $K^{\prime }$ gives $⟨T,\psi ⟩={\lim }_{\epsilon \to 0}⟨T,{\theta }_{\epsilon }\psi ⟩=0$. For flat $g$ every $k$ is admissible and every $\varphi \in \mathcal{D}(X)$ has support in some compact $K$, so $gT=0$.

Let $T$ extend $T^{\circ }$, and let $K$, $\varrho$, $K^{\prime }$, $k$ be as in Lemma lem:flat. For $a\in H$, the restriction of ${d_0}^T(a)$ to $\mathrm{int}(K)\times V_q$ depends only on the $k$-jets along $Z$ of the functions ${\beta }_{\gamma }(a)$, $|\gamma |\le r$. In particular:

1. flat-annihilator relations carry no defect for any extension $T$;

2. a nonzero defect is produced only by jet-detected relations.

Proof. Let $a,a^{\prime }\in H$ have ${\beta }_{\gamma }(a)-{\beta }_{\gamma }(a^{\prime })$ with vanishing $k$-jet on $Z$ for all $\gamma$. Since ${\beta }_{\gamma }(a-a^{\prime })\in \mathrm{Ann}(T^{\circ })$, Lemma lem:flat gives $⟨{\beta }_{\gamma }(a-a^{\prime })T,\psi ⟩=0$ for $\psi \in \mathcal{D}(X)$ with $\mathrm{supp}\psi \subset K$. For $\varphi \in \mathcal{D}(U)$ supported in $\mathrm{int}(K)\times V_q$, each ${{\partial }_q}^{\gamma }\varphi (\cdot ,0)$ is such a $\psi$, so $⟨{d_0}^T(a)-{d_0}^T(a^{\prime }),\varphi ⟩={\sum }_{\gamma }(-1{)}^{|\gamma |}⟨{\beta }_{\gamma }(a-a^{\prime })T,{{\partial }_q}^{\gamma }\varphi (\cdot ,0)⟩=0$. Taking $a^{\prime }=0$ with a flat $a$ gives (i); (ii) is (i) together with XV-D's result on universal relations.

Let $Z=\{z_0\}$ and suppose $T^{\circ }$ has some extension $\tilde{T}\in {\mathcal{D}}^{\prime }(X)$. Let $\bar{B}=\bar{B}(z_0,{\varrho }_0)\subset X$ be a closed ball and $k$ an order of $\tilde{T}$ on $\bar{B}$. Then there is a distribution

such that $T=\tilde{T}-s$ extends $T^{\circ }$ and $\mathrm{Ann}(T)=\mathrm{Ann}(T^{\circ })$. Consequently $H_T=H$, ${d_0}^T=0$, and ${F_0}^T={F_0}^{\tilde{T}}-{\mathsf{Rec}}_s$ is a descending lift in the class of extension changes.

Proof. Let $B=B(z_0,{\varrho }_0)$ be the open ball and $A_B=\{g\in \mathrm{Ann}(T^{\circ }):\mathrm{supp}g\subset B\}$. Define $\lambda :A_B\to \mathbb{C}$, $\lambda (g)=⟨\tilde{T},g⟩$.

{Step 1: $\lambda$ factors through $k$-jets.} Let $g\in A_B$ have vanishing $k$-jet at $z_0$. Choose $\chi \in \mathcal{D}(B)$ with $\chi =1$ on $\mathrm{supp}g$. Apply Lemma lem:flat with $K=\mathrm{supp}\chi$, $\varrho$ so small that $K^{\prime }\subset \bar{B}$, and the order-$k$ bound of $\tilde{T}$ on $\bar{B}$: $⟨\tilde{T},g⟩=⟨g\tilde{T},\chi ⟩=0$. Hence $\lambda (g)$ depends only on ${j^k}_{z_0}g=({\partial }^{\mu }g(z_0){)}_{|\mu |\le k}$, i.e. $\lambda ={\Lambda }_0\circ {j^k}_{z_0}$ for a linear functional ${\Lambda }_0$ on the subspace ${j^k}_{z_0}(A_B)$ of the finite-dimensional jet space $J^k={\mathbb{C}}^{\#\{|\mu |\le k\}}$.

Step 2: the correction. Extend ${\Lambda }_0$ linearly to $\Lambda$ on $J^k$ and write $\Lambda (j)={\sum }_{|\mu |\le k}{c}_{\mu }{j}_{\mu }$. With $s$ as displayed, $⟨s,g⟩={\sum }_{\mu }{c}_{\mu }{\partial }^{\mu }g(z_0)=\Lambda ({j^k}_{z_0}g)$ for all $g\in \mathcal{D}(X)$. Set $T=\tilde{T}-s$. Since $\mathrm{supp}s=\{z_0\}\subset Z$, $T$ extends $T^{\circ }$. For $g\in A_B$, $⟨T,g⟩=\lambda (g)-\Lambda ({j^k}_{z_0}g)=0$.

{Step 3: $\mathrm{Ann}(T)=\mathrm{Ann}(T^{\circ })$.} The inclusion $\mathrm{Ann}(T)\subseteq \mathrm{Ann}(T^{\circ })$ holds for every extension. Let $g\in \mathrm{Ann}(T^{\circ })$ and $\varphi \in \mathcal{D}(X)$; fix $\chi \in \mathcal{D}(B)$ with $\chi =1$ near $z_0$. Then $(1-\chi )g\varphi \in \mathcal{D}(X^{\circ })$ and $⟨T,(1-\chi )g\varphi ⟩=⟨T^{\circ },(1-\chi )g\varphi ⟩=⟨g{|}_{X^{\circ }}{T}^{\circ },(1-\chi )\varphi ⟩=0$; and $\chi g\varphi \in A_B$, so $⟨T,\chi g\varphi ⟩=0$ by Step 2. Hence $⟨gT,\varphi ⟩=0$ for all $\varphi$, i.e. $g\in \mathrm{Ann}(T)$.

The consequences follow from Theorem thm:ann and linearity of $T\mapsto {\mathsf{Rec}}_T$.

The theorem extends verbatim to finite $Z$, with one correction $s_i$ per point. For XV-D's example ($X={\mathbb{R}}^2$, $Z=\{0\}$, $T^{\circ }=\delta (x_1){|}_{X^{\circ }}$, $\tilde{T}=\delta (x_1)+{\partial }_1{\delta }_0$, $k=1$) one finds, for $g\in A_B$, $\lambda (g)=\int g(0,x_2)\mathrm{d}{x}_2-{\partial }_{1}g(0)=-{\partial }_{1}g(0)$, so $\Lambda (j)=-j_{e_1}$, $c_{e_1}=-1$, $s={\partial }_1{\delta }_0$, and $T=\delta (x_1)$, in agreement with XVD.

Let $d=1$, $Z=\{z_0\}$, and suppose $z_0\in \overline{\mathrm{supp}{T}^{\circ }}$ (closure in $X$). Then every $g\in \mathrm{Ann}(T^{\circ })$ is flat at $z_0$, and consequently $\mathrm{Ann}(T)=\mathrm{Ann}(T^{\circ })$ and ${d_0}^T=0$ for every extension $T$ of $T^{\circ }$.

Proof. If $g\in \mathrm{Ann}(T^{\circ })$ and $g(p)\ne 0$ for some $p\in \mathrm{supp}{T}^{\circ }$, then $g$ is invertible near $p$ and $T^{\circ }=g^{-1}(g{T}^{\circ })=0$ near $p$, a contradiction; so $g=0$ on $\mathrm{supp}{T}^{\circ }$. By hypothesis there are points $x_n\in \mathrm{supp}{T}^{\circ }$, $x_n\to z_0$, $x_n\ne z_0$; passing to a subsequence on one side of $z_0$, $g$ has infinitely many zeros accumulating at $z_0$, so $g(z_0)=0$ and, by Rolle's theorem applied inductively, $g^{(j)}(z_0)=0$ for all $j$. Lemma lem:flat gives $gT=0$ for every extension $T$.

Let $Z\subset X$ be relatively closed and $T^{\circ }\in {\mathcal{D}}^{\prime }(X^{\circ })$. The following are equivalent.

1. There is an extension $T\in {\mathcal{D}}^{\prime }(X)$ of $T^{\circ }$ with $\mathrm{Ann}(T)=\mathrm{Ann}(T^{\circ })$ (equivalently, by Theorem thm:ann, ${d_0}^T=0$).

2. For every compact $K\subset X$ there are $k_K\ge 0$ and $C_K<\infty$ such that

   $$|⟨T^{\circ },\varphi ⟩|\le C_K{p}_{k_K}(\varphi +g)$$

   for all $\varphi \in \mathcal{D}(X^{\circ })$ and $g\in \mathrm{Ann}(T^{\circ })$ with $\mathrm{supp}\varphi ,\mathrm{supp}g\subset K$.

Proof. (i)$\Rightarrow$(ii). Let $T$ be as in (i) and $K$ compact; let $k,C$ be an order and constant for $T$ on tests supported in $K$. For $\varphi ,g$ as in (ii), $⟨T,g⟩=⟨gT,\chi ⟩=0$ with $\chi \in \mathcal{D}(X)$ equal to one on $\mathrm{supp}g$, so $|⟨T^{\circ },\varphi ⟩|=|⟨T,\varphi +g⟩-⟨T,g⟩|=|⟨T,\varphi +g⟩|\le C{p}_k(\varphi +g)$.

(ii)$\Rightarrow$(i). Fix a compact $K$ and let ${\mathcal{D}}_K=\{\psi \in \mathcal{D}(X):\mathrm{supp}\psi \subset K\}$, $S_K=\{\varphi +g\}\subset {\mathcal{D}}_K$ with $\varphi ,g$ as in (ii). Define $\ell (\varphi +g)=⟨T^{\circ },\varphi ⟩$. This is well defined: if $\varphi +g={\varphi }^{\prime }+g^{\prime }$ then $h=\varphi -{\varphi }^{\prime }=g^{\prime }-g\in \mathcal{D}(X^{\circ })\cap \mathrm{Ann}(T^{\circ })$, and $⟨T^{\circ },h⟩=⟨h{T}^{\circ },\chi ⟩=0$ for $\chi \in \mathcal{D}(X^{\circ })$ equal to one on $\mathrm{supp}h$. By (eq:relest), $|\ell |\le C_K{p}_{k_K}$ on $S_K$. The Hahn–Banach theorem for functionals dominated by a seminorm extends $\ell$ to a linear $B_K$ on ${\mathcal{D}}_K$ with $|B_K|\le C_K{p}_{k_K}$.