Landmark XV-D Instantiating the Frozen Joint Graph: Source Recovery, the Null Defect of Extension Lifts, and the Boundary of the First Full Calculation

XV-D continues the supplied XV-C manuscript at its source-completeness stop. Its task is to recover the frozen data for the six-branch joint characteristic target, define the preliminary full lift, compute its reconstruction-null defect, and test the inherited correction and analytic requirements; or, if the data cannot be recovered, to report precisely why the graph-level calculation cannot yet be performed.

The second alternative applies. A twenty-item provenance audit of the supplied record (the full XV-A, XV-B, and XV-C texts and the retrieved XIV-H, XIV-I, and XIV-J excerpts) shows that the presentation space, the full reconstruction map, the grouped cross-line operators, the canonical contraction, the frozen candidate and correction classes, the coefficient distributions $v_{\alpha }$ and $c_{D,\beta }$, and the output definitions $Q_H$, ${\mathcal{E}}_H$ are named but not defined in that record. No preliminary lift for the actual graph is therefore definable, and no graph-specific null defect, obstruction, or estimate is asserted. The graph-level outcome is Unresolved at D0.

The new mathematical content is the one null-defect calculation that the record does make source-complete. For the finite normal-jet module with reconstruction to a punctured base, we define the extension lifts ${F_0}^T={\mathsf{Rec}}_T$ associated with the distributional extensions $T$ of the punctured coefficient, compute their reconstruction-null defect exactly, prove that it is always supported on the puncture, that it depends on the extension, that it induces an injection of the quotient of the two reconstruction kernels, and that the inherited identity $\mathsf{Rec}((qi\theta +1)M_{\mathrm{R}})=0$ never contributes to it. An explicit nonzero defect is exhibited and cancelled by a change of extension. A conditional reduction of XV-C's coefficient estimates to extendability of the cross-line traces is proved under an explicit smoothness hypothesis on $v_{\alpha }$. None of these results is transferred to the inherited graph; they fix what must be computed once the missing definitions are supplied.

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

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

XV-C XVC proves local reconstruction and extension tests and stops at the gate it calls C0: the data defining the full joint characteristic operation for the frozen six-branch graph are not in the supplied record. XV-D asks the question that XV-C left open:

Given the exact frozen data for the six-branch graph, can one define its full characteristic operation, prove that it descends under reconstruction, and meet the inherited correction and analytic requirements?

Three endings are permitted for the graph-level problem.

1. Constructed: an admissible, presentation-independent full joint map with the required bounds.

2. Obstructed in a specified exhaustive class: a graph-specific proof that no allowed map exists.

3. Unresolved at a named gate: the exact missing definition, calculation, or estimate is identified.

“Not constructed” is not “does not exist.” None of the three endings is assumed in advance.

We use five descriptive gates for this manuscript.

D0 stop rule. If any definition needed to specify the full operation or its allowed corrections is unavailable, this is stated plainly; D1–D4 are then not entered for the actual graph, and no graph-specific construction, obstruction, or null-defect value is claimed. XV-D may then be a source audit together with any local calculation that is source-complete.

The direct inputs are:

1. the supplied full texts of XV-A XVA, XV-B XVB, and XV-C XVC;

2. the retrieved XIV-H, XIV-I, and XIV-J excerpts Hkernel,Hverdict,Ipole,Iamplitude,Ijoint,Irelease, Jdescent,Jnull,Jreference,Jconditional,Jverdict,Jrelease,Jadmission;

3. the retrieved XIV-G release statement, used only for its dimension bookkeeping Grelease;

4. the proposed XV-D plan, used as a statement of objectives and acceptance criteria, not as a source of theorems.

The coverage account record supplies documentary context only. A retrieved locator, a predecessor citation, or a sentence reporting that a predecessor proved something is not treated as the missing definition. Nothing from the relative classical branch (XIV-A, XIV-B) or the XIII-D/XIII-E routes is imported. Distributional arguments are local in Euclidean charts and extend componentwise to finite-dimensional residual fibers.

The technical-continuation convention of XV-B and XV-C is retained: XV-B, XV-C, XV-D study the unresolved construction before admission. The admission success criterion remains

a prerequisite for downstream quantum use, not for writing this continuation. It is not asserted here.

A source-complete instantiation of the frozen joint target consists of the following items, each given by an actual definition.

1. the presentation space ${\mathcal{P}}_{\mathrm{int}}$;

2. the reconstruction map $\mathsf{Rec}:{\mathcal{P}}_{\mathrm{int}}\to {\mathcal{K}}_{\mathrm{int}}$ on all of ${\mathcal{P}}_{\mathrm{int}}$;

3. the carrier ${\mathcal{K}}_{\mathrm{int}}$ and the topology required of the joint map;

4. a complete description, or complete method of handling, of $\ker \mathsf{Rec}$;

5. the attachment domain $W$ and its coordinates;

6. the characteristic locus $N$ and its lift to $W$;

7. the response incidence $S_{\mathrm{resp}}$;

8. the canonical open domain $O=U_{\mathrm{can}}$;

9. the collision strata and observable diagonals;

10. the canonical contraction $m^{\circ }$ and its proved domain;

11. the grouped cross-line operators ${\mathcal{C}}_{\mathrm{cross}}$;

12. the mixed vertex Hessian and vertex weights;

13. the tensor maps ${\mathsf{A}}^{\alpha \beta ,I}$, ${\mathsf{D}}^{\alpha \mu ,I,J}$, the kernels $L^{\mu ,J}$, $R$, and the weights $f$, ${\sigma }_{\epsilon }$;

14. the explicit internal maps of $K_8$ and $K_9$;

15. the normalization of the pushed-forward coefficient;

16. the frozen candidate class of joint maps;

17. the frozen correction (ambiguity) class;

18. the coefficient distributions $v_{\alpha }$ and $c_{D,\beta }$;

19. the complete-graph target wavefront cone and uniform estimates, including the operator $\mathcal{L}(\cdot )$ of the XIV-J insertion wavefront lemma;

20. the output operations, $Q_H$, and ${\mathcal{E}}_H$.

Status codes: S = directly supplied in the record (formula or definition reproduced); T = faithfully transcribed from a named source excerpt, with the excerpt reproduced; R = reported only (a locator, a name, or a sentence stating that a predecessor defined or proved it); U = unavailable (no definition in the record).

{}{@{}p{0.07} p{0.32}p{0.08}X@{}} Item & What the record contains & Status & What is missing
D0.1 & The phrase “finite retarded-pole module of Definition def:amodule” Iamplitude; XV-B's abstract interface. & R & The definition of the module and its elements.
D0.2 & Two values: $\mathsf{Rec}(M_{\mathrm{R}})=\delta (q)R(r)$, $\mathsf{Rec}(i\theta M_{\mathrm{R}})={\delta }^{\prime }(q)R(r)$ Jnull. & T (two values) & $\mathsf{Rec}$ on all of ${\mathcal{P}}_{\mathrm{int}}$; the chart in which $q$, $\theta$, $r$ are defined.
D0.3 & “${\mathcal{K}}_{\mathrm{int}}=\mathrm{im}\mathsf{Rec}$” as an interface convention XVB. & R & The inherited carrier topology.
D0.4 & One null identity (eq:inheritednull) Jnull. & T (one relation) & Whether it generates $\ker \mathsf{Rec}$; any other relation.
D0.5 & Name $W$; internal variables $(a,b,z,y)$ Hkernel; “dimension sixteen” Grelease. & R & Defining coordinates and the residual bundle $E_{\mathrm{res}}$.
D0.6 & Name $N={\mathcal{N}}_{\mathrm{att}}$; “dimension eight” Grelease; “codimension eight” record. & R & Defining equations; the lift to attachment variables.
D0.7 & Name $S_{\mathrm{resp}}$; “codimension five” record; normal coordinates $q_1,q_2$ appear in (eq:referenceWF) Jreference. & R & Defining equations.
D0.8 & Name $U_{\mathrm{can}}$; the statement that $U_{\mathrm{can}}=W\setminus \overline{{\mathcal{N}}_{\mathrm{att}}}$ does not follow Ijoint. & R & The definition of the microlocally admissible domain.
D0.9 & Names only record. & U & Equations of the collision strata and observable diagonals.
D0.10 & Name; “canonical grouped expression eq:target” Ijoint. & R & The expression and its proved domain.
D0.11 & Name ${\mathcal{C}}_{\mathrm{cross}}$; existence of a traceability lemma lem:traces Jreference. & R & The operators, their wavefront sets, their normal traces.
D0.12 & Names; XIV-D equation labels record. & U & The Hessian and the weights.
D0.13 & $\mathsf{D}$ fixed by ${\dot{a_2}}^{\alpha \mu }[G;h]={\mathsf{D}}^{\alpha \mu ,I,J}{h}_I{G}_J$ Iamplitude; pole chart of $R$ (eq:pole) Ipole; names $\mathsf{A}$, $L$, $f$, ${\sigma }_{\epsilon }$. & T/R & ${\dot{a}}_2$; cone formula eq:R; eq:L; $\mathsf{A}$; $f$; ${\sigma }_{\epsilon }$.
D0.14 & “Exact kernel maps eq:K8reduced and eq:K9” Hverdict. & R & The formulas.
D0.15 & “Normalization eq:Rconvention” Jadmission. & R & The convention.
D0.16 & “The class fixed in Definition def:mapclass” Jverdict; “Definition def:class” Ijoint. & R & The definitions.
D0.17 & “XIII-C's partia

l-diagonal ambiguity class” record. & R & Its definition and parametrization.
D0.18 & Symbols in (eq:reference); the proof sentence “multiplication by $v_{\alpha }$ adds no wavefront” Jreference. & R & Formulas for $v_{\alpha }$, $c_{D,\beta }$.
D0.19 & Supported-sector bound (eq:referenceWF) Jreference; “$\mathrm{WF}(t_D)\subseteq \bigcup \mathcal{L}(b_{D,\gamma })$” Jconditional. & T (partial) & The full target cone; the operator $\mathcal{L}$; the uniform estimates.
D0.20 & The line $Q_H{R}_1^{(1)}\in {\mathcal{E}}_H$ Jadmission. & R & $Q_H$, ${\mathcal{E}}_H$, the output operations.

The supplied record does not contain definitions for items D0.1, D0.2 (beyond two values), D0.4 (beyond one relation), D0.5–D0.12, D0.14–D0.18, and D0.20 of Definition def:package. In particular it does not define the domain, codomain, or value of a preliminary full lift $F_0:{\mathcal{P}}_{\mathrm{int}}\to {\mathcal{D}}^{\prime }(W;E_{\mathrm{res}})$, nor the correction space $\mathcal{A}$ against which a null defect would be tested.

Proof. Each row of the provenance table lists the literal content of the record for the item. For the items named, that content is a name, a locator, a defining relation in terms of a further undefined object ($\mathsf{D}$ via ${\dot{a}}_2$), or a sentence reporting a predecessor statement. The full texts of XV-A, XV-B, and XV-C themselves record that the source definitions of the cross-line factors, the frozen map class, $Q_H$, and ${\mathcal{E}}_H$ were not reproduced XVA,XVB,XVC; the present audit confirms that the retrieved excerpts do not add them.

A preliminary lift needs a domain (${\mathcal{P}}_{\mathrm{int}}$, D0.1), a codomain ($W$, $E_{\mathrm{res}}$, D0.5), and a defining operation on each presentation (the products of the internal kernels with ${\mathcal{C}}_{\mathrm{cross}}$ and the vertex data, D0.10–D0.13). The correction class needs D0.16–D0.17. Each is in status R or U. This is a statement about the record, not about the existence of the definitions elsewhere.

Under the stop rule, Gates D1–D4 are not entered for the actual graph. Sections sec:branches and sec:lift record what the freeze and the lift definition would require; Sections sec:nulljet and sec:coefficients contain the calculations that are source-complete.

The retained kernel and supported sector are

$K_{47}$ groups the original branches $B_4+B_7$; the two $K_5$ placements belong to the original branch $B_5$. The six operational entries are

The supplied formulas are Hkernel,Iamplitude

Both $K_5$ placements are finite differential expressions in the tensor-product pullbacks of $R(b-a)R(a-z)$ and $R(b-z)R(z-a)$, which exist in the independent variables $(a,b,z)$ including at $a=z$ Hkernel,XVB. The internal pole chart is

with $(\lambda -i0{)}^{-1}=\mathrm{PV}\frac{1}{\lambda }+i\pi \delta (\lambda )$ Ipole. The delta part is fixed by the retarded line and is not adjusted.

This table is a freeze check. It records that the decomposition is carried forward unchanged; it is not evidence that the full operation exists. The propagator, retarded pole, regulator, carrier, normalization, and ambiguity class are not altered in this manuscript. No new model is introduced.

A preliminary lift $F_0:{\mathcal{P}}_{\mathrm{int}}\to {\mathcal{D}}^{\prime }(W;E_{\mathrm{res}})$ with $r_O{F}_0=m^{\circ }\mathsf{Rec}$ requires three distinct operations:

1. reconstruction of the internal distributions (defined in the named module, D0.1–D0.2);

2. canonical cross-line composition on $O=U_{\mathrm{can}}$ (D0.8, D0.10, D0.11);

3. the full operation on tests meeting $N$, including the products of the internal distributions with the singular cross-lines in the attachment variables.

The internal product calculations for the $K_5$ placements settle only (i). Operation (iii) is exactly the object left unresolved at I3 and J2 Ijoint,Jverdict.

No linear map $F_0$ with domain ${\mathcal{P}}_{\mathrm{int}}$ and codomain ${\mathcal{D}}^{\prime }(W;E_{\mathrm{res}})$ is specified by the supplied record, and no such map can be specified from it without supplying at least the items D0.1, D0.2, D0.5, D0.10, D0.11, and D0.13.

Proof. A linear map is specified by its domain, codomain, and value on each element of the domain. The domain ${\mathcal{P}}_{\mathrm{int}}$ is not defined (D0.1), so “for each element of the domain” has no content. The codomain is not defined (D0.5). Even granting a domain and codomain, the value on a presentation of any of the six entries must be the composed distribution of the entry with the cross-line operators and vertex data (D0.10–D0.13), none of which is defined. Writing a formula with the undefined symbols would be a plausible-looking operator, not a definition.

Consequently the equality $r_O{F}_0=m^{\circ }\mathsf{Rec}$ has been proved nowhere in this record and is not asserted here for any map. The null defect of the actual graph is not an evaluable object in this release. This proves the plan's requirement that the calculation be reported blocked rather than filled.

The only module in the record for which presentations, reconstruction, canonical domain, and target are all defined is XV-C's finite normal-jet module. XV-C computed its full reconstruction kernel. XV-D now carries out, for this module, the full D1–D2 programme: define the preliminary lifts, compute their null defects, locate the defects, and test cancellation. Everything below is for this 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

For $b\in {\mathcal{D}}^{\prime }(X)$ and $\varphi \in \mathcal{D}(U)$ the insertion is

The same formula with $X^{\circ }$, $U^{\circ }$ in place of $X$, $U$ defines ${{\mathsf{E}}^{\circ }}_{\gamma }:{\mathcal{D}}^{\prime }(X^{\circ })\to {\mathcal{D}}^{\prime }(U^{\circ })$.

Fix $r\ge 0$ and the presentation space

For $a\in {{\mathcal{P}}_r}^{\mathrm{jet}}$ and $|\gamma |\le r$ define the smooth coefficient

Fix $T^{\circ }\in {\mathcal{D}}^{\prime }(X^{\circ })$. The punctured reconstruction is

For an extension $T\in {\mathcal{D}}^{\prime }(X)$ of $T^{\circ }$ (that is, $T{|}_{X^{\circ }}=T^{\circ }$), the full reconstruction is

Write $H=\ker {\mathsf{Rec}}^{\circ }$ and $H_T=\ker {\mathsf{Rec}}_T$.

In the language of XV-B's interface XVB, this module is the instance

where $r:{\mathcal{D}}^{\prime }(U)\to {\mathcal{D}}^{\prime }(U^{\circ })$ is restriction. The canonical contraction is the identity because the punctured reconstruction already is the composed object on the canonical domain. The role of the characteristic set is played by $Z\times V_q$; the role of the response incidence by $S$.

For every $T\in {\mathcal{D}}^{\prime }(X)$ and $a\in {{\mathcal{P}}_r}^{\mathrm{jet}}$,

The same identity holds for ${\mathsf{Rec}}^{\circ }$ with $T^{\circ }$, ${{\mathsf{E}}^{\circ }}_{\gamma }$, and ${\beta }_{\gamma }(a){|}_{X^{\circ }}$.

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

by the Leibniz rule. Since $(-1{)}^{|\alpha |}=(-1{)}^{|\beta |}(-1{)}^{|\alpha -\beta |}$, each term is $(-1{)}^{|\beta |}(\genfrac{}{}{0ex}{}{\alpha }{\beta })⟨{\mathsf{E}}_{\alpha -\beta }(({{\partial }_q}^{\beta }{a}_{\alpha })(\cdot ,0)T),\varphi ⟩$. Substituting $\gamma =\alpha -\beta$ and summing over $\alpha$ gives (eq:normalform) with the coefficient (eq:beta). The punctured case is the same computation on $U^{\circ }$.

Let $b_{\gamma }\in {\mathcal{D}}^{\prime }(X)$ for $|\gamma |\le r$ and $t={\sum }_{|\gamma |\le r}{\mathsf{E}}_{\gamma }{b}_{\gamma }$. Then

1. $t=0$ if and only if $b_{\gamma }=0$ for all $|\gamma |\le r$;

2. $\mathrm{supp}t=\left({\bigcup }_{|\gamma |\le r}\mathrm{supp}{b}_{\gamma }\right)\times \{0\}$;

3. $t{|}_{U^{\circ }}={\sum }_{|\gamma |\le r}{{\mathsf{E}}^{\circ }}_{\gamma }(b_{\gamma }{|}_{X^{\circ }})$.

Proof. Choose $\chi \in \mathcal{D}(V_q)$ equal to one near $0$ and set ${\mathsf{J}}_{\gamma }\psi (x,q)=\frac{(-1{)}^{|\gamma |}}{\gamma !}{q}^{\gamma }\chi (q)\psi (x)$ for $\psi \in \mathcal{D}(X)$. Since $\chi$ is constant near $0$, $(-1{)}^{|\beta |}{{\partial }_q}^{\beta }{\mathsf{J}}_{\gamma }\psi (x,0)={\delta }_{\beta \gamma }\psi (x)$ for all $\beta$, so $⟨t,{\mathsf{J}}_{\gamma }\psi ⟩=⟨b_{\gamma },\psi ⟩$. This proves (i) and the inclusion “$\supseteq$” in (ii): if $t$ vanishes near $(x_0,0)$, choose $\chi$ supported in the normal neighborhood and $\psi$ supported near $x_0$ to see that every $b_{\gamma }$ vanishes near $x_0$. The inclusion “$\subseteq$” in (ii) is immediate from (eq:insert): a test vanishing near $(\bigcup \mathrm{supp}{b}_{\gamma })\times \{0\}$ has normal jets vanishing on $\bigcup \mathrm{supp}{b}_{\gamma }$. For (iii), a test $\varphi \in \mathcal{D}(U^{\circ })$ has ${{\partial }_q}^{\gamma }\varphi (\cdot ,0)\in \mathcal{D}(X^{\circ })$, so $⟨{\mathsf{E}}_{\gamma }{b}_{\gamma },\varphi ⟩=(-1{)}^{|\gamma |}⟨b_{\gamma }{|}_{X^{\circ }},{{\partial }_q}^{\gamma }\varphi (\cdot ,0)⟩$.

Let $T$ be an extension of $T^{\circ }$. Then

and $H_T\subseteq H$. Moreover $r\circ {\mathsf{Rec}}_T={\mathsf{Rec}}^{\circ }$, so every ${\mathsf{Rec}}_T$ is a preliminary lift for the punctured reconstruction.

Proof. (eq:Hpunct) and (eq:Hfull) follow from Lemma lem:normalform and Lemma lem:extraction(i) on $X^{\circ }$ and on $X$ respectively. For the compatibility, apply Lemma lem:extraction(iii) to (eq:normalform): ${\mathsf{Rec}}_T(a){|}_{U^{\circ }}=\sum {{\mathsf{E}}^{\circ }}_{\gamma }({\beta }_{\gamma }(a){|}_{X^{\circ }}T{|}_{X^{\circ }})={\mathsf{Rec}}^{\circ }(a)$. If ${\mathsf{Rec}}_T(a)=0$ then its restriction ${\mathsf{Rec}}^{\circ }(a)=0$, giving $H_T\subseteq H$.

For an extension $T$ of $T^{\circ }$, the extension lift is ${F_0}^T={\mathsf{Rec}}_T:{{\mathcal{P}}_r}^{\mathrm{jet}}\to {\mathcal{D}}^{\prime }(U)$. Its null defect is

Let $T$ extend $T^{\circ }$ and let $a\in H$. Then:

1. ${d_0}^T(a)={\sum }_{|\gamma |\le r}{\mathsf{E}}_{\gamma }({\beta }_{\gamma }(a)T)$, and every coefficient ${\beta }_{\gamma }(a)T$ is supported in $Z$;

2. $\mathrm{supp}{d_0}^T(a)\subseteq Z\times \{0\}$; in particular ${d_0}^T(a)\in Y$ and ${d_0}^T(a){|}_{U^{\circ }}=0$;

3. $\ker {d_0}^T=H_T$; hence ${d_0}^T$ induces an injective linear map $H/H_T\to Y$, and ${d_0}^T=0$ if and only if $H_T=H$;

4. ${d_0}^T(a)=0$ on all of $U$ if and only if ${\beta }_{\gamma }(a)T=0$ in ${\mathcal{D}}^{\prime }(X)$ for all $|\gamma |\le r$; otherwise the defect is a nonzero distribution supported in $Z\times \{0\}$, and there is no open $G\subset U\setminus (Z\times \{0\})$ with ${d_0}^T(a){|}_G\ne 0$.

Proof. (i) is (eq:normalform). For $a\in H$, $({\beta }_{\gamma }(a)T){|}_{X^{\circ }}={\beta }_{\gamma }(a){|}_{X^{\circ }}{T}^{\circ }=0$ by (eq:Hpunct), so $\mathrm{supp}({\beta }_{\gamma }(a)T)\subseteq Z$. (ii) follows from Lemma lem:extraction(ii) and the definition of $Y$. (iii): $\ker {d_0}^T=H\cap \ker {\mathsf{Rec}}_T=H\cap H_T=H_T$ by Theorem thm:kernels. (iv) is Lemma lem:extraction(i)–(ii).

For every tested null presentation $a\in H$ the outcome is never “defect on an open set away from the puncture.” Either the defect vanishes on all of $U$ (when $a\in H_T$) or it is a nonzero distribution carried by $Z\times \{0\}$. The second case is the genuine content of “vanishes only on the canonical domain”: it is produced not by the insertion structure but by tangential annihilator relations $\beta T^{\circ }=0$ that fail for $T$.

Let $d=2$, $X={\mathbb{R}}^2$ with coordinates $(x_1,x_2)$, $Z=\{0\}$, $m=1$, $r=0$. Let $T^{\circ }\in {\mathcal{D}}^{\prime }(X^{\circ })$ be the restriction to $X^{\circ }$ of $\delta (x_1)$, that is, $⟨T^{\circ },\psi ⟩=\int \psi (0,x_2)\mathrm{d}{x}_2$ for $\psi \in \mathcal{D}(X^{\circ })$. Consider the two extensions

and the presentation $a\in {{\mathcal{P}}_r}^{\mathrm{jet}}=C^{\infty }(U)$, $a(x,q)=x_1$. Then:

1. $H=\{a\in C^{\infty }(U):a(0,x_2,0)=0\text{for all }{x}_2\}$, and $a=x_1$ belongs to $H$;

2. $H_{T_1}=H$, so ${d_0}^{T_1}=0$;

3. $H_{T_2}=\{a\in H:({\partial }_{x_1}a)(0,0,0)=0\}$, the quotient $H/H_{T_2}$ is one-dimensional, and

   $${d_0}^{T_2}(x_1)={\mathsf{E}}_0\left(x_1{\partial }_{x_1}{\delta }_0\right)=-{\mathsf{E}}_0{\delta }_0\ne 0,\mathrm{supp}{d_0}^{T_2}(x_1)=\{(0,0,0)\}=Z\times \{0\}.$$

Proof. For $r=0$, ${\beta }_0(a)=a(\cdot ,0)$. (i): $a(\cdot ,0)T^{\circ }=0$ on $X^{\circ }$ means $\int a(0,x_2,0)\psi (0,x_2)\mathrm{d}{x}_2=0$ for all $\psi \in \mathcal{D}(X^{\circ })$, i.e. $a(0,x_2,0)=0$ for $x_2\ne 0$, hence for all $x_2$ by continuity. For $a=x_1$ this holds. (ii): the same computation on $X$ gives $H_{T_1}=H$. (iii): for $a\in H$,

so $a(\cdot ,0)T_2=-({\partial }_{x_1}a)(0,0,0){\delta }_0$. This vanishes if and only if $({\partial }_{x_1}a)(0,0,0)=0$, giving $H_{T_2}$; the linear functional $a\mapsto ({\partial }_{x_1}a)(0,0,0)$ is onto $\mathbb{C}$ on $H$ (take $a=x_1$), so $H/H_{T_2}$ is one-dimensional. For $a=x_1$ the coefficient is $-{\delta }_0$, and Lemma lem:extraction gives the nonvanishing and the support.

Let $T,T^{\prime }$ be extensions of $T^{\circ }$ and $s=T^{\prime }-T$, so $\mathrm{supp}s\subseteq Z$. Then

and ${d_0}^{T^{\prime }}={d_0}^T+{\mathsf{Rec}}_s{|}_H$. The defect of ${F_0}^T$ is cancelled by a change of extension if and only if there is an extension $T^{\prime }$ with $H_{T^{\prime }}=H$. In Proposition prop:example, $s=-{\partial }_{x_1}{\delta }_0$ cancels the defect of ${F_0}^{T_2}$.

Proof. Linearity of $T\mapsto {\mathsf{Rec}}_T$ gives ${F_0}^{T^{\prime }}-{F_0}^T={\mathsf{Rec}}_s$. By (eq:normalform), ${\mathsf{Rec}}_s(a)=\sum {\mathsf{E}}_{\gamma }({\beta }_{\gamma }(a)s)$ with each coefficient supported in $Z$, so Lemma lem:extraction(ii) gives the support statement. Cancellation means ${d_0}^{T^{\prime }}=0$, which by Theorem thm:defect(iii) is $H_{T^{\prime }}=H$. In the example, $T_2+s=T_1$ and $H_{T_1}=H$.

Let ${\mathcal{A}}_Z=\mathrm{Hom}({{\mathcal{P}}_r}^{\mathrm{jet}},V_{Z\times \{0\}})$. For every extension $T$, the relative obstruction class ${\mathrm{Ob}}_{{\mathcal{A}}_Z}({F_0}^T)=[{d_0}^T]\in \mathrm{Hom}(H,Y)/\{a{|}_H:a\in {\mathcal{A}}_Z\}$ vanishes. The cancelling correction constructed in the proof is algebraic; it is not asserted to be continuous, to be a change of extension, or to lie in any inherited class.

Proof. By Theorem thm:defect(ii), ${d_0}^T(H)\subseteq V_{Z\times \{0\}}$. Choose an algebraic complement ${{\mathcal{P}}_r}^{\mathrm{jet}}=H\oplus L$ with projection ${\pi }_H$ and set $a=-{d_0}^T{\pi }_H\in {\mathcal{A}}_Z$. Then $({F_0}^T+a){|}_H={d_0}^T-{d_0}^T=0$. This is the algebraic criterion of XV-B XVB instantiated with the actual spaces of this module.

If a closed $C\subseteq U\setminus U^{\circ }$ contains $Z\times \{0\}$, then for every extension $T$ and every $a\in H$ there is no open $G\subset U\setminus C$ with ${F_0}^T(a){|}_G\ne 0$. The contact-support obstruction test of XV-B therefore cannot produce an obstruction for any extension lift of this module.

Proof. Theorem thm:defect(ii).

The record supplies one null identity for the actual graph Jnull:

Its analogue in the jet module is the presentation

for which ${\mathsf{Rec}}_T(a^{(j)})={\mathsf{E}}_{0}T+q_j{\mathsf{E}}_{e_j}T$.

For every $T\in {\mathcal{D}}^{\prime }(X)$, $a^{(j)}\in H_T$. Consequently $a^{(j)}\in H$ and ${d_0}^T(a^{(j)})=0$ for every extension $T$ of every $T^{\circ }$. More generally, if $a\in {{\mathcal{P}}_r}^{\mathrm{jet}}$ satisfies ${\beta }_{\gamma }(a)=0$ in $C^{\infty }(X)$ for all $|\gamma |\le r$, then $a\in H_T$ for every $T$ and ${d_0}^T(a)=0$.

Proof. From (eq:beta), ${\beta }_0(a^{(j)})={a^{(j)}}_0(x,0)-{\partial }_{q_j}{a^{(j)}}_{e_j}(x,0)=1-1=0$, ${\beta }_{e_j}(a^{(j)})={a^{(j)}}_{e_j}(x,0)=q_j{|}_{q=0}=0$, and all other ${\beta }_{\gamma }(a^{(j)})$ vanish because the corresponding $a_{\alpha }$ vanish. Thus ${\beta }_{\gamma }(a^{(j)})T=0$ for every $T$ and (eq:Hfull) gives $a^{(j)}\in H_T$. The general statement is the same argument.

In the jet module the null relations split into two kinds: universal relations, with all ${\beta }_{\gamma }(a)=0$ as smooth functions, which hold for every coefficient and never produce a defect; and annihilator relations, with some ${\beta }_{\gamma }(a)\ne 0$ but ${\beta }_{\gamma }(a)T^{\circ }=0$, which are the only source of a nonzero defect (Proposition prop:example). The supplied identity (eq:inheritednull) is of the first kind in this module. For the actual graph this classification is not available: $\mathsf{Rec}$ is defined on two elements only (D0.2), and the record does not say whether $\ker \mathsf{Rec}$ contains annihilator relations. The statement “the inherited identity carries no defect” is proved here for (eq:ajet) in the jet module and is not transferred to (eq:inheritednull) in ${\mathcal{P}}_{\mathrm{int}}$.

Let ${\mathcal{P}}_{\mathrm{int}}$, $\mathsf{Rec}$, $O$, $m^{\circ }$, $Y$ be as in XV-B's linear interface, let $n\in \ker \mathsf{Rec}$, and let $F_0$ be any preliminary lift. Then $F_0(n)\in Y$, and for a correction class $\mathcal{A}\subseteq \mathrm{Hom}({\mathcal{P}}_{\mathrm{int}},Y)$ the component of the obstruction class on $\mathrm{span}\{n\}$ is the class of $F_0(n)$ in $Y/\{a(n):a\in \mathcal{A}\}$. If every $a\in \mathcal{A}$ takes values in $V_C$ for a closed $C\subseteq W\setminus O$, then $\mathrm{supp}{F}_0(n)\subseteq C$ is necessary for descent in $F_0+\mathcal{A}$. For $n=(qi\theta +1)M_{\mathrm{R}}$ the descent equation is

Proof. $r_O{F}_0(n)=m^{\circ }\mathsf{Rec}(n)=0$, so $F_0(n)\in \ker r_O=Y$. A candidate $F_0+a$ annihilates $n$ iff $a(n)=-F_0(n)$, which is the quotient statement. If $a(n)\in V_C$ for all $a$, then $F_0(n)=-a(n)\in V_C$ is necessary. The last display is linearity.

For the actual graph, $F_0$ is undefined (Proposition prop:nolift), so (eq:nulltest) has no computed left-hand side in this release.

The supplied supported-sector reference is Jreference

with wavefront bound