Landmark XV-F, Patch II Frozen-Object Completions for the Joint-Graph Source Package: Residual Fiber, Six-Entry Presentations, Coefficient Instantiation, and the Correction Class

XV-F-patch-ii completes XV-F-patch at the level of frozen objects. It asks whether the four gaps recorded in the patch's F0 verdict—the residual codomain, the generators and module action of the presentation package, the ingredients of the six operational entries, and the parametrization and presentation-level form of the correction class—can be closed under the original F0 rule, which passes a core item only when its frozen member is identified, never when a parameter family is displayed.

Five gates are run. Q1: a residual bundle $E_{\mathrm{res}}$ of rank ten, with the metric slot at $z$ and the scalar slot at $y$, its density, and its duality pairing are specified as a convention adopted here, no sheet defining them; the index bookkeeping of the displayed entries supports the choice, and it is proved that smooth residual testing erases every covector with nonzero $z$-component. Q2: the six entries are enumerated; the $K_5$ placements have XIV-I's chart module with its reconstruction rule but without reproduced generators; for the supported sector it is proved that, for every completion of the undefined data obeying XIV-H's bound $|\alpha |+|\beta |\le 2$, $K_6+K_8+K_9$ is a normal jet of order at most three along the null incidence $\{q_{+}=0,r_{+}=0\}$, which reproduces XIV-J's normal order exactly, and that ${\partial }_{\alpha }\mathcal{B}=\frac{1}{2}{x}_{\alpha }R$, so $K_8$ is carried by the null incidence, correcting a remark of the patch; $K_{47}$ has no presentation space in any sheet, and Q2 fails there. Q3: the construction of $\mathsf{D}$ from ${\dot{a}}_2$ is displayed in XIV-I and identified; $\mathsf{A}$, ${\sigma }_{\epsilon }$, $f$, $a_2$, ${q^{L,I,J}}_{\alpha \beta }$, $\mathsf{P}$, $\mathfrak{b}$, $g$, ${E^{\mathrm{A}}}_m$ are open, each reduced to a named sheet and sentence. Q4: XIII-C's three roles remain roles; the correction class as maps on presentations is not defined by the sheets, and the lift of the collision sets is refused. Q5: the list of cross-line products proved on $U_{\mathrm{can}}$ with an operational domain is stated to be empty. The verdict is Unresolved at F0, at Q2; F1 is not entered; no lift, null-defect test, characteristic prescription, or admission is asserted.

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

Figure fig:scope-XV-F-patch-ii records that separation. The drawing adds no hypothesis.

XV-F-patch XVFpatch ended at F0 with Proposition prop:F0, cited in the commissioning plan as prop:F0patch. Its blocking list has four frozen gaps:

1. the residual codomain $E_{\mathrm{res}}$ of the target space ${\mathcal{D}}^{\prime }(W;E_{\mathrm{res}})$ is undefined (D0.5);

2. the generators, coefficient class, and module action of the presentation package ${\mathcal{P}}_{\mathrm{int}}$ are not reproduced, so $\ker \mathsf{Rec}$ is not computable (D0.1–D0.4);

3. the ingredients of the six operational entries—$\mathsf{A}$, ${\sigma }_{\epsilon }$, $f$, ${\dot{a}}_2$, $\mathsf{D}$, ${q^{L,I,J}}_{\alpha \beta }$, $\mathsf{P}$, $\mathfrak{b}$, $g$, ${E^{\mathrm{A}}}_m$—are not fixed (D0.13–D0.14);

4. the correction class is a class of contact values whose parametrization ($D_{\Gamma }$, $w_{\Gamma }$, $S_{\Gamma /I}$) and presentation-level form $\mathcal{A}\subseteq \mathrm{Hom}({\mathcal{P}}_{\mathrm{int}},Y)$ are not supplied (D0.17);

together with the domain item (D0.8): $O=U_{\mathrm{can}}$ is a named domain whose equality with $W\setminus \overline{{\mathcal{N}}_{\mathrm{att}}}$ is refused at three located places. This manuscript asks whether (a)–(d) and the domain item can be closed under the original F0 rule.

F0 rule. A core item passes when its frozen member is identified in a direct sheet, or when a convention is adopted here and explicitly offered as the frozen choice. A displayed parameter family is an open datum. A locator, a role, or a search term is not a member.

Stop rule, as applied. The F0 verdict is fixed by the first failing gate, in the order Q1–Q5. “Stop” means that F1 is not entered and that no later gate can change the verdict. The remaining patches are nevertheless executed, since the plan allocates them and since the successor needs the complete blocking list; their outcomes are recorded as documentary and cannot contribute to a pass.

Direct full texts. XV-F-patch XVFpatch, as the record being completed; its documentary ledgers are inputs, its F0 refusal is binding, and its verified lemmas are cited by label. XIV-G Grelease,Gresidual,GG34, for $W$, $N={\tilde{\mathcal{N}}}_{\mathrm{att}}$, and the residual test. XIV-J Jdescent,Jnull,Jreference,Jconditional,JJ2, Jrelease,Jadmission,Jmapclass,Jtarget,Jcross,Jhessian,Jrefuse, for the presentation interface, the retained sum, def:mapclass, and the descent criterion. XIV-I Ipole,IeqR,Ichart,Iamodule, II2,II3,Irelease,Iresponse,Irefuse, for the chart module, the source-internal identification of $R$ as the kernel of ${E_0}^{\mathrm{R}}$, and the copied $K_8$, $K_9$ displays. XIV-H Hkernel,HeqL,Hq,Hrefuse,Hfiber,HH2,Hrelease,HK9, for the coefficient input statements. XIII-C XIIICcoll,XIIICpart, XIIICvertex,XIIICU,XIIICT0, for ${\Delta }_I$, def:Cpart, ${\mathcal{U}}_{\Gamma }$, and $T_0$.

Copied premises, usable only where the display is already quoted inside one of the six direct sheets: XII-E XIIEa2 for $a_2$ (and $a_1$, $b_1$ to the extent XIV-H's input statement input:EC HinputEC quotes them); XII-C XIICrelease for the undifferentiated cubic; XII-D XIIDfiber,XIIDweight for $f$ and the observable; XIII-B XIIIB for the time-ordered line convention copied into XIII-C.

Non-inputs. XI-B, XII-A, XII-B, XIV-D, XIV-E, and the quantum coefficient ${R_1}^{(1)}$. A citation inside a handoff does not import them. The equality $U_{\mathrm{can}}=W\setminus \overline{{\mathcal{N}}_{\mathrm{att}}}$ remains refused and is recorded once, in (eq:refused).

Reading rule. Similar words in distinct manuscripts do not identify their models, response maps, or sign conventions record. Within the six direct sheets, a symbol defined in one and used in another with an explicit input statement is one symbol; without such a statement it is two.

Status codes. identified (frozen member displayed in a direct sheet, with locator); convention (adopted here, offered as the frozen choice); open (role, family, locator, or unreturned quotation); documentary (a gate whose pass condition is a statement about the record).

Standard distribution theory is used with reference to Horm. Throughout, $\eta =\mathrm{diag}(+,-,-,-)$, $\square ={{\partial }_t}^2-\Delta$, $J^{+}=\{t\ge |\mathbf{x}|\}$, $\partial J^{+}=\{t=|\mathbf{x}|\}$, $x_{\alpha }={\eta }_{\alpha \nu }{x}^{\nu }$, $x^2={\eta }_{\mu \nu }{x}^{\mu }{x}^{\nu }$; $R=\frac{\Theta (t)}{2\pi }\delta (x^2)$ and $\mathcal{B}=R\ast R=\frac{1}{8\pi }\Theta (t)\Theta (x^2)$ are XIV-I's displays IeqR, verified in XVFpatch (Lemmas lem:R, lem:RR).

{(G1) The support of $K_8$.} The patch's Proposition prop:support placed $\mathrm{supp}{K}_8$ in the solid cone $S_{J^{+}}=\{a=z,b-a\in J^{+}\}$, and its Remark rem:conormal argued that $K_8$ is singular at points with $(b-a{)}^2>0$, where $\mathcal{B}(b-a)$ is locally constant, so that the conormal statement of XIV-H could not be checked. The derivative ${\partial }_{b^{\alpha }}$ inside the bracket of (eq:K8) acts on $\mathcal{B}(b-v)$, and ${\partial }_{\alpha }\mathcal{B}=\frac{1}{2}{x}_{\alpha }R$ (Lemma lem:dB). Hence $\mathrm{supp}{K}_8\subseteq S_{\partial }=\{a=z,b-a\in \partial J^{+}\}$ (Corollary cor:K8supp); the premise of the Remark fails, and the check it declared not performable passes at the level of supports. Item (g) of the patch's Proposition prop:checks is moved to the performed list.

{(G2) The construction of $\mathsf{D}$.} The patch stated that the construction of ${\mathsf{D}}^{\alpha \mu ,I,J}$ from ${\dot{a}}_2$ is XIV-H's and not reproduced. XIV-I displays it II2: $\mathsf{D}$ is the constant bilinear fiber map determined by ${\dot{a_2}}^{\alpha \mu }[G;h]={\mathsf{D}}^{\alpha \mu ,I,J}{h}_I{G}_J$ ((eq:Dfrom) below). The construction is identified; the value of $\mathsf{D}$ depends on $a_2$, which remains open (sec:patch3).

(G3) XII-A and XII-B are non-inputs here. The patch used XII-B's trace reversal to state $\mathfrak{a}=-\text{PPcalnote}$. Under the present evidence contract that sheet is a non-input; the identity is restated as the stand-alone computation ${\mathfrak{a}}^{\alpha \mu ,ab}{r}_{ab}=\frac{1}{2}{\eta }^{\alpha \mu }{\mathrm{tr}}_{\eta }r-r^{\alpha \mu }$ (Lemma lem:fiber), with no second map named. XII-A's Hadamard blocks are not used; the identity of the blocks in ${\mathcal{C}}_{\mathrm{cross}}$ is carried from the patch as open and is not reassessed (sec:reassess).

Let $M={\mathbb{R}}^4$, $W={\mathbb{R}}^{16}$ with coordinates $(a,b,z,y)$, $a,b,z,y\in M$ Grelease,Ichart, and $Z={{\mathbb{R}}^4}_z$ the attachment variable. XIV-G's residual test is a section $G\in {C^{\infty }}_c(Z;{\mathrm{Sym}}^2{T}^{\ast }M)$, and XIV-G states that the residual pairing ${\mathrm{JD}}^{\prime }[G]$ has order at most two Gresidual. XIV-G does not define a bundle $E_{\mathrm{res}}$ over $W$.

Definition def:gtest types an {input direction on $Z$}: a compactly supported symmetric covariant two-tensor field on the four-dimensional attachment space. It is not a section of any bundle over the sixteen-dimensional $W$. The order statement is copied as a statement about XIV-G's object; no order bound is asserted for any object constructed in this manuscript.

The joint-map statements of XIV-I and XIV-J name a residual species with a metric slot at $z$ and a scalar slot at $y$ II3, Jmapclass; they do not display the bundle. The following convention is introduced here and offered as the frozen choice.

1. Fiber. $E_{\mathrm{res}}:={\mathrm{pr}}_z^{\ast }\left({\mathrm{Sym}}^{2}TM\right)\otimes {\mathrm{pr}}_y^{\ast }1$, where ${\mathrm{pr}}_z,{\mathrm{pr}}_y:W\to M$ are the coordinate projections and $1$ is the trivial line (the scalar slot at $y$). The fiber at $w=(a,b,z,y)$ is ${\mathrm{Sym}}^2{T}_{z}M$, of dimension $\left(\genfrac{}{}{0ex}{}{5}{2}\right)=10$; $E_{\mathrm{res}}$ is a trivial bundle of rank ten over $W$.

2. Density. The density bundle $|\Lambda {|}_W$ is trivialized by the coordinate density $|{\mathrm{d}}^{16}w|$ of $(a,b,z,y)$.

3. Tests. ${\Gamma }_c:={C^{\infty }}_c(W;{E_{\mathrm{res}}}^{\ast }\otimes |\Lambda {|}_W)\cong {C^{\infty }}_c(W)\otimes {\mathrm{Sym}}^2({\mathbb{R}}^4{)}^{\ast }$; an element is a symmetric array $\Phi =({\Phi }_{ab}{)}_{a,b=0}^3$ of test functions on $W$.

4. Pairing. ${\mathcal{D}}^{\prime }(W;E_{\mathrm{res}}):={\Gamma }_c^{\prime }$, with $⟨u,\Phi ⟩={{\sum }_{a,b=0}}^3⟨u^{ab},{\Phi }_{ab}⟩$ for $u=(u^{ab})$ a symmetric array of scalar distributions; the fiber pairing ${\mathrm{Sym}}^{2}V\times {\mathrm{Sym}}^2{V}^{\ast }\to \mathbb{C}$, $(u,\Phi )\mapsto u^{ab}{\Phi }_{ab}$, is the restriction of the duality $V^{\otimes 2}\times (V^{\ast }{)}^{\otimes 2}\to \mathbb{C}$ and is perfect.

For $u\in {\mathcal{D}}^{\prime }(W;E_{\mathrm{res}})$ and $G\in {C^{\infty }}_c(Z;{\mathrm{Sym}}^2{T}^{\ast }M)$, the prescription

defines $u[G]\in {\mathcal{D}}^{\prime }({\mathbb{R}}^{12})$; the map $G\mapsto u[G]$ is linear and continuous from ${C^{\infty }}_c(Z;{\mathrm{Sym}}^2{T}^{\ast }M)$ to ${\mathcal{D}}^{\prime }({\mathbb{R}}^{12})$.

Proof. $\varphi \mapsto G\otimes \varphi$ is linear and continuous $\mathcal{D}({\mathbb{R}}^{12})\to {\Gamma }_c$ (supports stay in ${\mathrm{pr}}_{(a,b,y)}^{-1}(\mathrm{supp}\varphi )\cap {\mathrm{pr}}_z^{-1}(\mathrm{supp}G)$, compact, and seminorms are multiplicative). Composition with the continuous $u$ gives $u[G]\in {\mathcal{D}}^{\prime }$. Continuity in $G$ is the same estimate read in the other factor.

In each of the six displayed entries $K^{I,J}(a,b,z)$ of sec:entries, the index $I$ is carried by the variable $a$ and the index $J$ by the variable $z$; the cross-line pullback $\pi (a,b,z,y)=(y-a,y-b)$ of XIV-J Jcross attaches the two cross-lines to $a$ and $b$. After contraction with the cross-lines, $J$ at $z$ is the only free tensor index, and $y$ is the only variable not consumed by an internal line.

Proof. Read off: in (eq:K5a), ${\mathsf{A}}^{\alpha \beta ,I}$ stands with ${\sigma }_{\epsilon }(a)$ and ${\partial }_a$, while $L^{\mu ,J}(a,z)={\sigma }_{\epsilon }(z){\mathsf{A}}^{\lambda \mu ,J}{\partial }_{z^{\lambda }}R(a-z)$ carries $J$ with ${\sigma }_{\epsilon }(z)$; in (eq:K5z) the roles of $(a,I)$ and $(z,J)$ are exchanged consistently; in (eq:K47), $L^{\mu ,J}(b,z)$ and $L^{\nu ,I}(b,a)$; in (eq:K6), XIV-I's defining display (eq:Dfrom) assigns $I$ to $h$ and $J$ to $G$, and ${\sigma }_{\epsilon }(a)$ accompanies $I$; in (eq:Q)–(eq:K9), the ordering $(\cdot ;a,z)$ of $Q^{L,I,J}(v;a,z)$ pairs $I$ with $a$ and $J$ with $z$. The statement about $\pi$ is the display (eq:cross) of the patch.

Lemma lem:index is the internal evidence for C1(i): the free metric index after cross-line contraction sits at $z$, which is the variable of XIV-G's residual test $G$, and the output is a symmetric contravariant tensor at $z$ (upper index pair $J=(ab)$), pairing with the covariant $G_{ab}$. The scalar slot at $y$ is the trivial line and does not change the rank. This is a reading of the displays, not a source identification; C1 remains a convention.

The constant metric $\eta$ induces an isomorphism ${\mathrm{Sym}}^2{T}^{\ast }M\cong {\mathrm{Sym}}^{2}TM$ and a fiber metric on ${\mathrm{Sym}}^2{T}^{\ast }M$, whose fiber dimension is also ten. Neither is part of C1. The pairing of C1(iv) is the duality pairing between the value bundle $E_{\mathrm{res}}$ and the test bundle ${E_{\mathrm{res}}}^{\ast }\otimes |\Lambda {|}_W$; the coincidence of ranks does not identify the two roles.

Let $u\in {\mathcal{D}}^{\prime }(W;E_{\mathrm{res}})$ and $G\in {C^{\infty }}_c(Z;{\mathrm{Sym}}^2{T}^{\ast }M)$. Then

where $G_{ab}{u}^{ab}={\sum }_{a,b}{G}_{ab}(z)u^{ab}\in {\mathcal{D}}^{\prime }(W)$. In particular, any wavefront of $u$ whose covector has nonzero $z$-component—for instance a conormal to the attachment diagonal $\{a=z\}$ in the $z$-direction—is invisible to every smooth residual test.

Proof. $u[G]={\mathrm{pr}}_{\ast }(G_{ab}{u}^{ab})$ for the projection $\mathrm{pr}(a,b,z,y)=(a,b,y)$, which is proper on $\mathrm{supp}(G_{ab}{u}^{ab})\subseteq {\mathbb{R}}^{12}\times \mathrm{supp}G$. Write ${\mathrm{pr}}_{\ast }$ as the operator with Schwartz kernel $K(x^{\prime };x)=\delta (x^{\prime }-\mathrm{pr}(x))$, whose wavefront is $\{(x^{\prime },x;\theta ,-{}^{t}d\mathrm{pr}\theta ):x^{\prime }=\mathrm{pr}(x)\}$ by Horm (Thm. 8.2.4), with ${}^{t}d\mathrm{pr}({\xi }_a,{\xi }_b,{\xi }_y)=({\xi }_a,{\xi }_b,0,{\xi }_y)$. The set ${\mathrm{WF}}_X(K)$ of Horm (Thm. 8.2.12) is empty since ${}^{t}d\mathrm{pr}$ is injective, and that theorem gives $\mathrm{WF}({\mathrm{pr}}_{\ast }v)\subseteq \{(\mathrm{pr}(x),\theta ):(x,{}^{t}d\mathrm{pr}\theta )\in \mathrm{WF}(v)\}$, which is the display. The last sentence is the case ${\xi }_z\ne 0$.

The order-two statement of Definition def:gtest concerns the order of a distribution and is not a joint principal symbol GG34. By Lemma lem:erase, smooth residual testing can hide attachment-diagonal wavefront, so no wavefront statement for a joint object follows from residual testing alone. Gates G3–G5 of XIV-G are not entered.

No direct sheet defines $E_{\mathrm{res}}$. Convention def:Eres specifies one fiber (rank ten), one density, and one pairing, compatible with Definition def:gtest through Lemma lem:partial and with the displayed entries through Lemma lem:index. Q1 is discharged with status convention; it passes F0 only if C1 is accepted as the frozen choice.

Proof. The first sentence is the record (Gresidual; the patch's D0.5). The rest cites the results above.

XIV-J Jdescent works with a linear reconstruction $\mathsf{Rec}:{\mathcal{P}}_{\mathrm{int}}\to {\mathcal{D}}^{\prime }(W)$ with image ${\mathcal{K}}_{\mathrm{int}}=\mathsf{Rec}({\mathcal{P}}_{\mathrm{int}})$ and proves: a linear candidate $\tilde{\mathsf{M}}:{\mathcal{P}}_{\mathrm{int}}\to {\mathcal{D}}^{\prime }(W)$ defines a map on reconstructed kernels if and only if

in which case the induced map $\mathsf{M}(\mathsf{Rec}A)=\tilde{\mathsf{M}}(A)$ is unique; and agreement of $\tilde{\mathsf{M}}$ with canonical contraction on $U_{\mathrm{can}}$ proves (eq:descent) there only. XIV-J presents ${\mathcal{P}}_{\mathrm{int}}$ as the finite direct sum of the internal coefficient presentation spaces of the six operational entries; the summands are named and not displayed in the retrieved record (status open for the summands, identified for (eq:descent)).

Write $E=\{47,5a,5z,6,8,9\}$ for the six entries, ${\mathcal{P}}_{\mathrm{int}}={⨁}_{e\in E}{\mathcal{P}}_e$, $\mathsf{Rec}={\sum }_e{\mathsf{Rec}}_e$ with ${\mathsf{Rec}}_e:{\mathcal{P}}_e\to {\mathcal{D}}^{\prime }(W)$, ${\mathcal{K}}_e={\mathsf{Rec}}_e({\mathcal{P}}_e)$.

$\ker \mathsf{Rec}\supseteq {⨁}_e\ker {\mathsf{Rec}}_e$, with equality if and only if the sum ${\sum }_e{\mathcal{K}}_e\subseteq {\mathcal{D}}^{\prime }(W)$ is direct. If ${\mathcal{K}}_e\cap {\mathcal{K}}_{e^{\prime }}\ne 0$ for some $e\ne e^{\prime }$, then $\ker \mathsf{Rec}$ contains elements $(p_e,-p_{e^{\prime }})$ with ${\mathsf{Rec}}_e{p}_e={\mathsf{Rec}}_{e^{\prime }}{p}_{e^{\prime }}\ne 0$, and by (eq:descent) every joint map must take equal values on the $e$- and $e^{\prime }$-presentations of one and the same kernel.

Proof. If ${\sum }_e{\mathsf{Rec}}_e{p}_e=0$ and the sum of images is direct, each ${\mathsf{Rec}}_e{p}_e=0$; conversely a nontrivial relation ${\sum }_e{k}_e=0$, $k_e\in {\mathcal{K}}_e$ not all zero, lifts to an element of $\ker \mathsf{Rec}$ outside $⨁\ker {\mathsf{Rec}}_e$. The last sentence is (eq:descent) applied to $(p_e,-p_{e^{\prime }})$.

Lemma lem:directsum is the reason the plan requires the generators of every entry before $\ker \mathsf{Rec}$ can be computed: the per-entry kernels do not suffice when images overlap, and Theorem thm:normalorder below shows that the images of the $K_6$-, $K_8$-, and $K_9$-presentations all lie in one and the same finite normal-jet space.

The retained kernel and the supported sector are II2, Jreference

The coordinate-response line is XIV-H's HeqL

The displays, copied for enumeration (copying is not instantiation; see sec:patch3), are

from XIV-H Hkernel and XIV-I II2, where XIV-I defines

“the constant bilinear fiber map determined by” this identity; and from XIV-I's input statement copied from XIV-H Iresponse, with the order bound of XIV-H Hq,

(In (eq:K8) the inner tensor index is written $\kappa$ to keep it distinct from the multi-index $\alpha$ of (eq:Q); the source writes $\alpha$ for both.) The cross-line operator of the patch is recalled for Lemma lem:index:

1. {XVFpatch (Thm. thm:exist)} For any constant arrays $\mathsf{A}$, $\mathsf{D}$, $\mathsf{P}$, $\mathfrak{b}$, any $m\in \mathbb{C}$, and any $f,g,F,{\sigma }_{\epsilon },q_{\alpha \beta }\in C^{\infty }({\mathbb{R}}^4)$, each of (eq:K5a)–(eq:K9) is a distribution on ${{\mathbb{R}}^{12}}_{(a,b,z)}$; the $v$-integrals in (eq:T), (eq:K8) are the distributions ${\mathcal{T}}_0$, ${\mathcal{T}}_{\mathcal{B}}$ with $⟨{\mathcal{T}}_0,\varphi ⟩=⟨Q,{{\Phi }^R}_{\varphi }⟩$, ${{\Phi }^R}_{\varphi }(v,a,z)=⟨R,\varphi (a,v+\cdot ,z)⟩$, and likewise for $\mathcal{B}$.

2. {XVFpatch (Lemma lem:Q)} For smooth $\Phi$ with compact $v$-support near the diagonal, $⟨Q,\Phi ⟩={\sum }_{\alpha ,\beta }\int q_{\alpha \beta }(v)({{\partial }_z}^{\alpha }{{\partial }_a}^{\beta }\Phi )(v,v,v){\mathrm{d}}^{4}v$.

3. {XVFpatch (Prop. prop:support)} $\mathrm{supp}{K}_6,\mathrm{supp}{K}_9\subseteq S_{\partial }$, $\mathrm{supp}{K}_8\subseteq S_{J^{+}}$.

4. {XVFpatch (Lemma lem:chart), source Ichart} With $u=a$, $d=b-a$, $q=d^2$, $\zeta =\mathbf{d}\in {\mathbb{R}}^3$, $r=a-z$, the map $w_{+}=(u,\zeta ,q,r,y)$ is a global chart of $V_{+}:=\{(b-a{)}^0>0\}\subset W$ onto ${\mathbb{R}}^4\times {\Omega }_0\times {\mathbb{R}}^4\times {\mathbb{R}}^4$, ${\Omega }_0=\{q+|\zeta {|}^2>0\}$, $d^0=\sqrt{q+|\zeta {|}^2}$; the second placement $w_{-}$ uses $u=z$, $d=b-z$, $r=z-a$ on $V_{-}=\{(b-z{)}^0>0\}$. On $V_{+}$, $R(d)=\frac{1}{2\pi }\delta (q)$, $({\partial }_{\alpha }R)(d)=\frac{1}{\pi }{d}_{\alpha }{\delta }^{\prime }(q)$, and $\mathcal{B}(d)=\frac{1}{8\pi }\Theta (q)$.

5. {XVFpatch (Lemma lem:recrule), source Iamodule,Jnull} XIV-I's module ${\mathcal{A}}^m(V)$ on the chart $w_{\pm }$ is reconstructed by $\mathsf{Rec}(A)={\sum }_{j,\beta }{c}_{j\beta }(w){\delta }^{(j)}(q){{\partial }_r}^{\beta }R(r)$; XIV-J's values $\mathsf{Rec}(M_{\mathrm{R}})=\delta (q)R(r)$, $\mathsf{Rec}(i\theta M_{\mathrm{R}})={\delta }^{\prime }(q)R(r)$ and null relation $(qi\theta +1)M_{\mathrm{R}}$ are instances; the generating family is free over $(q,r)$-independent coefficients and dependent over chart functions.

6. {XVFpatch (Prop. prop:K5chart)} On $V_{+}$, ${K_5}^{[a]}$ and $K_6$ have the form of the rule in (P5) with $j\in \{1,2\}$, $|\beta |=2$; on $V_{-}$ so does ${K_5}^{[z]}$; $K_{47}$ has this form in neither chart.

For $e\in \{5a,5z\}$ the record supplies XIV-I's module ${\mathcal{A}}^m(V)$ on the chart $w_{\pm }$ with its reconstruction rule (P5) and chart (P4); XIV-I states that one explicit module contains the regular diagonal, separated flow-out, and internal transition presentations of both placements II2. The generators of ${\mathcal{A}}^m(V)$, its coefficient class (dependence on $q$, $r$), and the module action (linearity of $\mathsf{Rec}$ over chart functions) are described in prose and not displayed Iamodule. Status: rule and chart identified; generators, coefficient class, and action open. The module is a local chart model on $V_{\pm }$; its relations $q{\delta }^{\prime }(q)=-\delta (q)$, $r^{2}R(r)=0$, and $(qi\theta +1)M_{\mathrm{R}}$ are examples of null relations, not a classification of $\ker \mathsf{Rec}$ on ${\mathcal{P}}_{\mathrm{int}}$ (Lemma lem:directsum).

jets along the null incidence. On $V_{+}$ let $S^{+}:=\{q=0,r=0\}$, so that $S^{+}=S_{\partial }\cap V_{+}$, with normal variables $n=(q,r)\in {\mathbb{R}}^5$ and base variables $x^{\prime }=(u,\zeta ,y)$, $\zeta \ne 0$; $S^{+}$ has codimension five and $S^{+}\cong {\mathbb{R}}^4\times ({\mathbb{R}}^3\setminus 0)\times {\mathbb{R}}^4$. For $b\in {\mathcal{D}}^{\prime }(S^{+})$ and $\gamma \in {\mathbb{N}}^5$ set

Let ${\mathcal{J}}^k$ be the space of finite sums ${\sum }_{|\gamma |\le k}{\mathsf{E}}_{\gamma }{b}_{\gamma }$, $b_{\gamma }\in {\mathcal{D}}^{\prime }(S^{+})$ ({normal order at most $k$}), and ${{\mathcal{J}}^k}_{\infty }\subseteq {\mathcal{J}}^k$ the subspace with $b_{\gamma }\in C^{\infty }(S^{+})$.

1. For $c\in C^{\infty }(V_{+})$ and $\gamma \in {\mathbb{N}}^5$,

   $$c(x^{\prime },n){{\partial }_n}^{\gamma }\delta (n)=\sum _{{\gamma }^{\prime }\le \gamma }(-1{)}^{|\gamma -{\gamma }^{\prime }|}(\genfrac{}{}{0ex}{}{\gamma }{{\gamma }^{\prime }})({{\partial }_n}^{\gamma -{\gamma }^{\prime }}c)(x^{\prime },0){{\partial }_n}^{{\gamma }^{\prime }}\delta (n).$$

   Hence $c\cdot {{\mathcal{J}}^k}_{\infty }\subseteq {{\mathcal{J}}^k}_{\infty }$.

2. Base derivatives ${\partial }_{{x^{\prime }}_i}$ map ${{\mathcal{J}}^k}_{\infty }\to {{\mathcal{J}}^k}_{\infty }$; normal derivatives ${\partial }_{n_j}$ map ${{\mathcal{J}}^k}_{\infty }\to {{\mathcal{J}}^{k+1}}_{\infty }$.

3. Every first-order differential operator with coefficients in $C^{\infty }(V_{+})$ maps ${{\mathcal{J}}^k}_{\infty }\to {{\mathcal{J}}^{k+1}}_{\infty }$, and every such operator with no normal-derivative component maps ${{\mathcal{J}}^k}_{\infty }\to {{\mathcal{J}}^k}_{\infty }$.

Proof. (i) Pair with $\varphi$: $⟨c{\partial }^{\gamma }\delta ,\varphi ⟩=(-1{)}^{|\gamma |}\int {{\partial }_n}^{\gamma }(c\varphi )(x^{\prime },0)\mathrm{d}{x}^{\prime }$; expand by Leibniz and use $⟨{\partial }^{{\gamma }^{\prime }}\delta ,\psi ⟩=(-1{)}^{|{\gamma }^{\prime }|}{\partial }^{{\gamma }^{\prime }}\psi (0)$. The coefficients $({\partial }^{\gamma -{\gamma }^{\prime }}c)(x^{\prime },0)$ are smooth on $S^{+}$. (ii) ${\partial }_{{x^{\prime }}_i}(b\otimes {\partial }^{\gamma }\delta )=({\partial }_{{x^{\prime }}_i}b)\otimes {\partial }^{\gamma }\delta$ and ${\partial }_{n_j}(b\otimes {\partial }^{\gamma }\delta )=b\otimes {\partial }^{\gamma +e_j}\delta$. (iii) follows from (i), (ii).

On $V_{+}$, as operators on ${\mathcal{D}}^{\prime }(V_{+})$ expressed in $w_{+}$,

with $d_{\mu }={\eta }_{\mu \nu }{d}^{\nu }$, $d^0=\sqrt{q+|\zeta {|}^2}$, smooth on $V_{+}$. Each is a first-order operator with smooth coefficients; its normal component is $\pm 2d_{\mu }{\partial }_q$ for ${\partial }_b$, $-{\partial }_{r^{\mu }}$ for ${\partial }_z$, and $-2d_{\mu }{\partial }_q+{\partial }_{r^{\mu }}$ for ${\partial }_a$.

Proof. Chain rule for $u=a$, $d=b-a$, $q={\eta }_{\nu \kappa }{d}^{\nu }{d}^{\kappa }$, ${\zeta }^i=d^i$, $r=a-z$: $\partial q/\partial d^{\mu }=2d_{\mu }$, $\partial {\zeta }^i/\partial d^{\mu }={{\delta }^i}_{\mu }$, and $b$ enters only through $d$ (sign $+$), $z$ only through $r$ (sign $-$), $a$ through $u$ ($+$), $d$ ($-$), $r$ ($+$).

In ${\mathcal{D}}^{\prime }({\mathbb{R}}^4)$, ${\partial }_{\alpha }\mathcal{B}=\frac{1}{2}{x}_{\alpha }R$. Consequently $\mathrm{supp}{\partial }_{\alpha }\mathcal{B}=\partial J^{+}$, and on $V_{+}$, $({\partial }_{\alpha }\mathcal{B})(d)=\frac{1}{4\pi }{d}_{\alpha }\delta (q)$.

Proof. Use the partial Fourier transform ${\mathcal{F}}_{\mathbf{x}}$ and the formulas of XVFpatch (Lemmas lem:R, lem:RR): ${\mathcal{F}}_{\mathbf{x}}R=\Theta (t)\sin (\rho t)/\rho$, ${\mathcal{F}}_{\mathbf{x}}\mathcal{B}=\Theta (t)u_2(t)$, $u_2=(\sin \rho t-\rho t\cos \rho t)/(2{\rho }^3)$, $u_2(0)=0$, $u_2^{\prime }=t\sin (\rho t)/(2\rho )$. For $\alpha =0$: ${\mathcal{F}}_{\mathbf{x}}({\partial }_t\mathcal{B})=\Theta (t)u_2^{\prime }(t)=\frac{t}{2}{\mathcal{F}}_{\mathbf{x}}R={\mathcal{F}}_{\mathbf{x}}(\frac{1}{2}tR)$, and $x_0=t$. For $\alpha =i\in \{1,2,3\}$: ${\mathcal{F}}_{\mathbf{x}}({\partial }_i\mathcal{B})=i{p}_i\Theta u_2$, while ${\mathcal{F}}_{\mathbf{x}}(x^{i}R)=i{\partial }_{p_i}{\mathcal{F}}_{\mathbf{x}}R=i\Theta (t)\frac{p_i}{\rho }\frac{d}{d\rho }\frac{\sin \rho t}{\rho }=-2i{p}_i\Theta u_2$, so ${\mathcal{F}}_{\mathbf{x}}(\frac{1}{2}{x}_{i}R)={\mathcal{F}}_{\mathbf{x}}(-\frac{1}{2}{x}^{i}R)=i{p}_i\Theta u_2$. Fourier injectivity on ${\mathcal{S}}^{\prime }$ gives the identity. The support statement follows from $\mathrm{supp}R=\partial J^{+}$ and $x_{\alpha }\ne 0$ somewhere on every open piece of the cone. The chart formula is (P4): $\frac{1}{2}{d}_{\alpha }\cdot \frac{1}{2\pi }\delta (q)$.

Under the hypotheses of (P1), $\mathrm{supp}{K}_8\subseteq S_{\partial }$. Hence $\mathrm{supp}{K}_{\mathrm{supp}}\subseteq S_{\partial }=\{a=z,b-a\in \partial J^{+}\}\subseteq \{(b-a{)}^2=0,a=z\}$, and $K_{\mathrm{supp}}$ vanishes on the interior of the solid cone.

Proof. ${\partial }_{b^{\kappa }}{\mathcal{T}}_{\mathcal{B}}$ is the distribution with $⟨{\partial }_{b^{\kappa }}{\mathcal{T}}_{\mathcal{B}},\varphi ⟩=-⟨Q,{{\Phi }^{\mathcal{B}}}_{{\partial }_{b^{\kappa }}\varphi }⟩$, and ${{\Phi }^{\mathcal{B}}}_{{\partial }_{b^{\kappa }}\varphi }(v,a,z)=\int \mathcal{B}(u)({\partial }_{b^{\kappa }}\varphi )(a,v+u,z)\mathrm{d}u=-⟨{\partial }_{\kappa }\mathcal{B},\varphi (a,v+\cdot ,z)⟩$. Thus ${\partial }_{b^{\kappa }}{\mathcal{T}}_{\mathcal{B}}$ is the $\mathcal{T}$-construction of (P1) with $R$ replaced by ${\partial }_{\kappa }\mathcal{B}=\frac{1}{2}{x}_{\kappa }R$, and the proof of (P3) for $K_9$ applies verbatim with $\mathrm{supp}{\partial }_{\kappa }\mathcal{B}=\partial J^{+}$. Multiplication by smooth functions and the outer ${\partial }_{b^{\mu }}$ preserve the support statement.

Under (P1), as distributions on ${\mathbb{R}}^{12}$,

where $R(b-a){\delta }^{(4)}(a-z)$ denotes the pullback of $R\otimes \delta$ by $(a,b,z)\mapsto (b-a,a-z)$. On $V_{+}$ the bracketed distributions are $\frac{1}{2\pi }{q}_{\alpha \beta }(u){\mathsf{E}}_0(1)$ and $\frac{1}{4\pi }{q}_{\alpha \beta }(u)d_{\kappa }{\mathsf{E}}_0(1)$, elements of ${{\mathcal{J}}^0}_{\infty }$.