Landmark XV-F Source Recovery from the Registered Full Texts for the Frozen Joint Graph, the Reconstruction-Fixed Ambiguity of Descending Extensions, and the F0 Stop

XV-F continues XV-E at its gate E0. Its first task is to resolve the source gap that XV-E documented: either recover, from the supplied record, the core definitions needed to state the preliminary lift of the frozen six-branch joint target and its correction class, or report exactly which items remain unavailable.

Five retrieval passes are recorded. The second reproduces XIV-H's coordinate-response line, the $K_8$, $K_9$ kernels, XIII-C's collision sets and partial-diagonal class, and XII-A's Hadamard matrix; the third returns locators only; the fourth reaches the registered full texts and reproduces XIV-I's reconstruction rule, XIV-I's retarded kernel $R$ and its square, XIV-J's grouped cross-line operator, mixed vertex Hessian, and output convention; the fifth yields exactly one identification—XIV-I's adapted chart $w_{\pm }=(u_{\pm },{\zeta }_{\pm },q_{\pm },r_{\pm },y)$ in terms of $(a,b,z,y)$—three located refusals of the equality $U_{\mathrm{can}}=W\setminus \overline{{\mathcal{N}}_{\mathrm{att}}}$, and four candidate displays from the XII sheet (the fiber tensor $\mathfrak{a}$, the observation weight $f$ as a compactly supported test function, XII-E's quadratic coefficient $a_2$, and the scalar stress tensor $T_0$). Every reproduced display is checked. New checks: in the adapted charts both $K_5$ placements and $K_6$ have exactly the form of the reconstruction rule with explicit coefficients; the fiber map $\mathfrak{a}$ is minus the trace reversal; the derivative ${\dot{a}}_2$ is recomputed; the polarization of $T_0$ matches the displayed Hessian for one normalization and is off by a factor two for the other; the advanced kernel ${E^{\mathrm{A}}}_m$ is constructed and shown unique for all $m\ge 0$, conditionally on a declaration the XIV texts do not contain. Three corrections to the circulated draft are recorded: the argument of the reconstruction coefficients is the chart point; with chart-function coefficients the generating family is not free, and XIV-J's null relation is an instance of this dependence; and no manuscript contains the symbol $\hat{H}$ returned by the fourth pass. The tensor $\mathsf{A}$, the symbols ${\sigma }_{\epsilon }$, ${\dot{a}}_2$, $\mathsf{D}$, the ingredients of $K_8$, $K_9$, the normalization of the polarized stress, the displays of the target contraction, joint-map class and canonical domain, the residual bundle, the presentation-level correction class, the lift of the collision sets, and all candidate identifications remain unsupplied. The graph-level outcome is Unresolved at F0.

The mathematics on the finite normal-jet module with punctured reconstruction is unchanged: the disjoint classification of null relations; jet-detection as “some extension carries a defect”; the affine space of descending extensions over the reconstruction-fixed freedom, infinite-dimensional in XV-D's example; and the theorem that a normal-order bound alone never makes the puncture-supported ambiguity finite-dimensional. Two phrasing errata to XV-E are recorded. Nothing is transferred to the inherited graph.

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

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

XV-E XVE performed a source recovery pass and found the data defining the full joint characteristic operation incomplete at its gate E0. XV-F asks:

Does the recoverable source material define the frozen six-branch operation well enough to construct its preliminary lift and test descent under the actual reconstruction map?

Exactly one of three endings is permitted for the graph-level problem: Lift defined and descended; Obstructed in the frozen class, with the class proved exhaustive; or Unresolved at a named gate. Source incompleteness is not an analytic obstruction. Descent would not be quantum admission.

F0 stop rule. If a core item remains unavailable, this is said plainly; F1–F4 are not entered for the actual graph; no graph formula with placeholders is written and called a lift. The split into core and downstream items is descriptive and does not redefine XV-D's D0 or XV-E's E0.

Direct inputs: the full texts of XV-A through XV-E XVA,XVB,XVC,XVD,XVE and of XII-B XIIB; the XIV-H, XIV-I, XIV-J excerpts of the first retrieval Hkernel,Hverdict,Ipole,Iamplitude,Ijoint,Irelease,Jdescent, Jnull,Jreference,Jconditional,Jverdict,Jrelease,Jadmission; the XIV-C and XIV-G release statements Crelease,Grelease; the XII-A and XII-D passages transcribed in XV-E XIIA,XIID; the passages of the second retrieval HeqL,Ipackage,Iresponse, XIIAhad,XIIICcoll,XIIICpart,XIIDsource; the one new passage of the third retrieval XIIDrelease; the displays and prose descriptions of the fourth retrieval Iamodule,IeqR,Jcross, Jhessian,Jmapclass,Jtarget,Jrconv,HK9; and the displays, refusals, and prose descriptions of the fifth retrieval Ichart,Hrefuse,Irefuse,Jrefuse,Hfiber,Hq,XIIDfiber, XIIDweight,XIIEa2,Jstress,XIIICvertex,XIICsource. Locators cited without content XIVArel,Ione and the X-B summary XBsum are non-inputs. The coverage account record is documentary context. Its reading rule—similar words in distinct manuscripts do not identify their models, response maps, or sign conventions—is applied throughout, including to the four symbols $\mathcal{P}$ (XII-A), $\mathcal{P}$ (XII-B), $\mathfrak{P}$ (XII-D), $\mathsf{P}$ (XIV-I) of sec:threeP, to the four symbols $R$, $R_1$, ${R_1}^{(1)}$, $\mathcal{R}$ of sec:fourR, and to the letters $a_1$, $X$, $f$, $g$, $E^{\mathrm{A}}$, $T$ shared between the XII and XIV sheets. Nothing from XIV-A, XIV-B, XIII-D, XIII-E, X-B, or manuscripts I–V is imported as a premise.

Displays, paraphrases, locators, refusals. A reproduced display with file, label, and line numbers is a definition for the purposes of the manifest (status T). A prose description of a located definition, with locators but without its display, is a paraphrase (status P); it fixes what the definition is about and does not pass F0. A locator, a search term, or a sentence stating where a definition should be found is not a definition (status R). A located sentence stating that an equality is not asserted is a refusal; it is recorded as such and closes the corresponding search.

Standard distribution theory (pullback by submersions, tensor products, convolution under support conditions, point-supported distributions, Paley–Wiener–Schwartz) is used with reference to Horm.

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

is a later project gate and is not asserted here. One companion rendering of the record writes the output space with a different macro; the source line Jadmission is (eq:admission) with ${\mathcal{E}}_H$, and no second output space is introduced.

(E1) Disjoint classification. XV-E's Definition def:kinds calls $a\in H$ universal if $\beta (a)=0$, a flat-annihilator relation if each ${\beta }_{\gamma }(a)$ is flat on $Z$, and jet-detected otherwise. As worded, every universal relation is also flat-annihilator. The disjoint convention is Definition def:kinds; every theorem of XV-E holds under either reading, since each statement about the first two classes is a vanishing statement.

(E2) Jet-detection is necessary, not sufficient. The sentence in XV-E preceding its Lemma lem:cutoff, “jet-detected relations produce a defect,” overstates. What XV-E's Theorem thm:jetdep(ii) proves is that a nonzero defect is produced only by jet-detected relations. Theorem thm:jetchar settles the exact relationship.

(E3) Multiplier statement retained. XV-E's Proposition prop:erratum is retained unchanged: for $v\in C^{\infty }(S_0)$, nonextendability of $vc$ implies nonextendability of $c$; a failure sequence for $vc$ transfers to a test family for $c$ that excludes extension but need not remain normalized in every increasing seminorm.

(F1) The argument of the reconstruction coefficients. The circulated draft glossed the symbol $w$ in XIV-I's reconstruction rule (eq:Recrule) as “the remaining variables.” The fifth pass Ichart shows that $w_{\pm }$ is XIV-I's name for the full adapted chart point $(u_{\pm },{\zeta }_{\pm },q_{\pm },r_{\pm },y)$. The coefficients $c_{j\beta }(w)$ are therefore smooth functions on the chart, including of $q_{\pm }$ and $r_{\pm }$, unless XIV-I's unreproduced coefficient class restricts them.

(F2) The free-module claim. The circulated draft's Lemma lem:recrule(iii) asserted that ${\mathcal{K}}_{\mathrm{int}}$ is free on the family $\{{\delta }^{(j)}(q){{\partial }_r}^{\beta }R(r)\}$ and that $\ker \mathsf{Rec}$ is the coefficient kernel. This is correct only for coefficients independent of $(q,r)$. Over chart functions the family is dependent—$q{\delta }^{(j)}(q)=-j{\delta }^{(j-1)}(q)$ and $r^{2}R(r)=0$—and XIV-J's null relation $(qi\theta +1)M_{\mathrm{R}}$ is the instance $j=1$ of the first identity. Lemma lem:recrule states both cases; manifest items D0.3 and D0.4 are revised.

{(F3) The symbol $\hat{H}$.} The fourth pass returned XIV-J's grouped cross-line operator with hatted replacement blocks ${\hat{H}}_{\gamma \gamma }$, ${\hat{H}}_{\varphi \varphi }$. The fifth pass reports that no manuscript in the corpus contains a symbol $\hat{H}$ (the only hatted symbols in XIV-I are Fourier transforms and ${\hat{s}}_0$, which XIV-I excludes from the present construction). The content of (eq:cross)—pullback by $\pi$ of a difference of two tensor products of blocks—is retained; the replaced blocks are written ${H^{\prime }}_{\gamma \gamma }$, ${H^{\prime }}_{\varphi \varphi }$ as this document's placeholder, and D0.11 carries the annotated status T${}^{\ast }$\ (display reproduced in content; one symbol disputed between passes).

The core items of XVD, Definition D0.1–D0.20, are those required to state a preliminary lift and its correction class: D0.1 (${\mathcal{P}}_{\mathrm{int}}$), D0.2 ($\mathsf{Rec}$ on ${\mathcal{P}}_{\mathrm{int}}$), D0.3 (${\mathcal{K}}_{\mathrm{int}}$, topology), D0.4 ($\ker \mathsf{Rec}$), D0.5 ($W$, $E_{\mathrm{res}}$), D0.8 ($O=U_{\mathrm{can}}$), D0.10 ($m^{\circ }$), D0.11 (${\mathcal{C}}_{\mathrm{cross}}$), D0.12 (vertex Hessian, weights), D0.13 (tensor maps, $L$, $R$, $f$, ${\sigma }_{\epsilon }$), D0.14 ($K_8$, $K_9$), D0.16 (candidate class), D0.17 (correction class). The downstream items are D0.6, D0.7, D0.9 (loci and strata), D0.15 (normalization), D0.18 ($v_{\alpha }$, $c_{D,\beta }$), D0.19 (target cone, estimates), D0.20 ($Q_H$, ${\mathcal{E}}_H$, output operations).

Status codes: S supplied; T transcribed from a reproduced display or excerpt; T${}^{\ast }$\ transcribed in content with a symbol disputed between passes; P paraphrased (located definition described in prose, display not reproduced); C candidate (formula for a differently named object, identification not supplied); R reported only (name, locator, or sentence); U unavailable.

{}{@{}p{0.055}p{0.07} p{0.09}Xp{0.18}@{}} Item & Group & Status & Record content & Change in XV-F
D0.1 & core & T/P & Finite retarded-pole module ${\mathcal{A}}^m(V)$ Iamodule; reconstruction rule (eq:Recrule) reproduced; adapted chart (eq:chart) reproduced Ichart; generators and coefficient class paraphrased. & Pass 4: rule; pass 5: chart.
D0.2 & core & T/P & Rule (eq:Recrule); agrees with the two XIV-J values (Lemma lem:recrule(i)); linearity of $\mathsf{Rec}$ over chart functions not reproduced. & Pass 4: rule.
D0.3 & core & P & ${\mathcal{K}}_{\mathrm{int}}=\mathrm{im}\mathsf{Rec}$ is the span of the family $\{{\delta }^{(j)}(q_{\pm }){{\partial }_r}^{\beta }R(r_{\pm })\}$ over the unreproduced coefficient class; free iff that class is $(q,r)$-independent (Lemma lem:recrule); topology not named. & Pass 5: revised (F2).
D0.4 & core & T (1)/P & Relation $(qi\theta +1)M_{\mathrm{R}}$ Jnull, an instance of $q{\delta }^{\prime }(q)=-\delta (q)$; further chart identities (Lemma lem:recrule(iv)) are null if $\mathsf{Rec}$ is $C^{\infty }$-linear; the general kernel depends on the coefficient class. & Pass 5: revised (F2).
D0.5 & core & R/P & $W$ carries coordinates $(a,b,z,y)\in {\mathbb{R}}^{16}$ Grelease,Ichart; $E_{\mathrm{res}}$ not defined. & Pass 5: coordinates confirmed.
D0.6 & downstr. & R & Name; $\dim N=8$; codimension-four stratum named Crelease. & None.
D0.7 & downstr. & R/C & Name; codimension five; $q_1,q_2$ in (eq:referenceWF). Candidate (eq:Srespcand). & Pass 2: candidate; pass 4: support refined.
D0.8 & core & P & “The inherited domain on which the specified products are already proved”; the equality $U_{\mathrm{can}}=W\setminus \overline{{\mathcal{N}}_{\mathrm{att}}}$ is refused at three located places Hrefuse,Irefuse,Jrefuse; XIII-C's ${\mathcal{U}}_{\Gamma }$ (Hörmander sum condition) is not identified with the complement of the collision diagonals. & Pass 5: refusals located.
D0.9 & downstr. & T/R & Collision sets ${\Delta }_I$ in XIII-C's graph variables XIIICcoll; connected subsets of the cycle-vertex table XIIICvertex (e.g.\ $\{y,u,v\}$, $\{y,v\}$ on $B_1$); lift to $W$ not supplied. & Pass 5: table located.
D0.10 & core & P & eq:target: “the formal six-branch contraction against the grouped Hadamard difference” Jtarget. & Pass 4.
D0.11 & core & T${}^{\ast }$/C & ${\mathcal{C}}_{\mathrm{cross}}$ as the submersion pullback (eq:cross) Jcross; symbol of the replaced blocks disputed (F3); identification with XII-A's admissible replacement XIIAhad is a candidate. & Pass 4: display; pass 5: (F3).
D0.12 & core & T/P/C & Mixed Hessian (eq:hess) Jhessian; $T_0$ displayed (eq:T0) Jstress; normalization of the polarized $T^{\mu \nu }(\varphi ,\psi )$ two-valued (Lemma lem:polar); remaining weights not reproduced. & Pass 4: Hessian; pass 5: $T_0$.
D0.13

& core & T/C & eq:L (eq:L) HeqL; eq:R (eq:R)–(eq:RR) IeqR, verified; $\mathsf{A}$ defined as the matrix of XIV-H's $a_1$ (eq:a1H) Hfiber, $a_1$ not reproduced; candidates $\mathfrak{a}$ (eq:a1), $f\in {C^{\infty }}_c$, ${\dot{a}}_2$ (eq:a2dot), ${\sigma }_{\epsilon }$ (sec:candidates). & Pass 2: $L$; pass 4: $R$; pass 5: role of $\mathsf{A}$, three candidate displays.
D0.14 & core & T/P/C & $K_8$, $K_9$ (eq:K8)–(eq:K9) Iresponse; $\mathcal{B}$ explicit; ${q^{L,I,J}}_{\alpha \beta }$ paraphrased with $|\alpha |+|\beta |\le 2$ Hq; ${E^{\mathrm{A}}}_m$ candidate (eq:EA); $\mathsf{P}$, $\mathfrak{b}$, $g$ not reproduced; existence proved (Theorem thm:exist). & Pass 5: order bound; advanced candidate.
D0.15 & downstr. & T & eq:Rconvention (eq:Rconv) Jrconv. & Pass 4.
D0.16 & core & P & def:mapclass: four clauses paraphrased Jmapclass; def:class not reproduced. & Pass 4.
D0.17 & core & T/R & ${\mathcal{C}}_{\mathrm{part}}$ defined XIIICpart (Definition def:Cpart); $D_{\Gamma }$, $w_{\Gamma }$, $S_{\Gamma /I}$ are roles, not formulas (confirmed by pass 5); relation to $\mathrm{Hom}({\mathcal{P}}_{\mathrm{int}},Y)$ not supplied. & Pass 2.
D0.18 & downstr. & R & Symbols in (eq:reference). & None.
D0.19 & downstr. & T (partial) & Bound (eq:referenceWF); $\mathcal{L}$ named. & None.
D0.20 & downstr. & R/T & $Q_H$, ${\mathcal{E}}_H$ named Jadmission; output convention (eq:Rconv). & Pass 4.

The third retrieval pass returned the excerpts already audited, one new passage, and a locator guide. The new passage is XII-D's release boundary XIIDrelease: it lists what is constructed on the XII sheet and states that no carrier enlargement, quantum retarded product, state, loop, interacting differential, or $\epsilon \to 0$ limit is opened. It names none of $L$, $R$, $\mathsf{A}$, ${\sigma }_{\epsilon }$, ${\mathcal{C}}_{\mathrm{cross}}$, $U_{\mathrm{can}}$, $m^{\circ }$ and is a non-input. The locator guide changed no status; it was used as the search plan for the fourth and fifth passes.

XIV-A's relative master equation XIVArel and manuscript I's context-transport passage Ione are not imported. The XII-D local-source passage XIIDsource identifies none of the required symbols and is not used. The X-B retrieval summary XBsum concerns the scalar-only one-contraction normalization of a different track; its retarded-product line has the same $\frac{i}{\hslash }(\cdot -\cdot \star \cdot )$ form as (eq:Rconv), and by the reading rule this is not an identification; it is a non-input.

module and reconstruction (D0.1–D0.3). XIV-I, def:amodule, lines 873–901 Iamodule: the presentation module is the finite retarded-pole module ${\mathcal{A}}^m(V)$ on the adapted chart $w=w_{\pm }$ of (eq:chart) below, reconstructed by

a finite sum, $q=q_{\pm }$, $r=r_{\pm }$, with coefficients $c_{j\beta }$ smooth on the chart and determined by $A$. The generators of ${\mathcal{A}}^m(V)$ and the coefficient class (in particular whether $c_{j\beta }$ may depend on $q$, on $r$, or on both) are described in prose and not reproduced (status P).

kernel (D0.13). XIV-I, eq:R, lines 329–335 IeqR:

cross-line operator (D0.11). XIV-J, lines 1112–1123 Jcross: with $\pi :W\to {\mathbb{R}}^4\times {\mathbb{R}}^4$, $\pi (a,b,z,y)=(y-a,y-b)$,

the pullback by the submersion $\pi$ of the difference of two tensor products of blocks written as distributions in the difference variables; $H^{\prime }$ is this document's placeholder for XIV-J's replaced blocks (F3). Whether these are XII-A's frozen blocks (eq:Hblocks) and an admissible replacement in XII-A's sense is a candidate.

vertex Hessian (D0.12). XIV-J, lines 1128–1145 Jhessian:

bilinear in the metric direction $h$ and the scalar direction $\psi$ at the scalar background $\varphi$, with $g$ the switch. XIV-J calls $T^{\mu \nu }(\varphi ,\psi )$ the inherited polarized tensor and does not expand it.

convention (D0.15). XIV-J, eq:Rconvention, lines 631–636 Jrconv:

This is a map on functionals; it is not the kernel $R$ of (eq:R) (sec:fourR).

{Paraphrased items (status P).} XIV-J, eq:target, lines 720–730 Jtarget: $m^{\circ }$ is “the formal six-branch contraction against the grouped Hadamard difference”; its display is not reproduced. XIV-J, def:mapclass, lines 732–762 Jmapclass: the joint-map class consists of linear maps on reconstructed kernels that (a) contract with the cross-lines, (b) retain both $K_5$ placements, (c) agree with the canonical expression on the proved domain, and (d) meet $N$; the display and the meaning of (d) are not reproduced. XIV-H's own eq:K9 at line 1055 HK9 is located and not reproduced.

adapted chart (D0.1, D0.5). XIV-I, lines 828–855 Ichart: the two placements are

diffeomorphisms onto their images; near the null cone $q_{\pm }={d_{\pm }}^2$ and ${\zeta }_{\pm }\in {\mathbb{R}}^3$ are ordinary coordinates; with $u_{+}=a$, $u_{-}=z$, the adapted chart is $w_{\pm }=(u_{\pm },{\zeta }_{\pm },q_{\pm },r_{\pm },y)$, of dimension sixteen, and ${\mathcal{A}}^m(V)$ is defined on it. The particular choice of ${\zeta }_{\pm }$ is not reproduced. This relates the chart domain $V$ to $W$; it does not identify $V$ with $W$ and does not lift ${\Delta }_I$ to $W$.

refusals (D0.8). XIV-H, lines 1661–1667 Hrefuse; XIV-I, Proposition prop:canonical, lines 1492–1496 Irefuse; XIV-J, lines 1305–1309 Jrefuse: each states that no equality $U_{\mathrm{can}}=W\setminus \overline{{\mathcal{N}}_{\mathrm{att}}}$ is asserted. $U_{\mathrm{can}}$ remains the inherited domain on which the specified products have already been proved. XIII-C's ${\mathcal{U}}_{\Gamma }$ is the open set on which the Hörmander sum condition holds, and XIII-C states that this does not identify ${\mathcal{U}}_{\Gamma }$ with the complement of the collision diagonals.

map (D0.13). XIV-H writes Hfiber

so that $\mathsf{A}$ is the coefficient matrix of XIV-H's linear fiber map $a_1$; $a_1$ itself is not reproduced. XII-D, lines 519–525 XIIDfiber, and XII-A, lines 974–976 XIIA, display

The candidate is ${\mathsf{A}}^{\alpha \mu ,I}={\mathfrak{a}}^{\alpha \mu ,ab}$ with $I$ the symmetric pair $(ab)$; no sentence in XIV-H states that its $a_1$ is XII-D's.

weight (D0.13). XII-D, lines 545–552 XIIDweight, takes an arbitrary $f\in {C^{\infty }}_c({\mathring{M}}_T)$ on the open slab and sets

On the XII sheet $f$ is a compactly supported test weight, not a missing formula. Whether XIV-H's $f$ in (eq:K5a)–(eq:K9) is this $f$ is a candidate (sec:candidates).

{Quadratic coefficient (D0.13, $K_6$).} XII-E, lines 401–403 XIIEa2:

XIV uses ${\dot{a}}_2$ in the definition of $\mathsf{D}$ in (eq:K6) and does not record its expansion; Lemma lem:a2 computes the derivative of (eq:a2).

coefficients (D0.14). XIV-H, lines 653–665 Hq: the ${q^{L,I,J}}_{\alpha \beta }$ are the coefficients of the actual differentiated source, including ramp derivatives, with $|\alpha |+|\beta |\le 2$. The undifferentiated source is XII-C's density-weighted cubic XIICsource, not a displayed polynomial. Expanding it is a calculation, not a located display; it is not performed here.

factor (D0.14). The record reports a normalization line $\square {E_0}^{\mathrm{R}}=\mathrm{id}$ in XIV-I (locator not returned) alongside (eq:R). XII-D sets $v_f=E^{\mathrm{A}}f$ and does not display $E^{\mathrm{A}}$. No XIV text declares ${E^{\mathrm{A}}}_m$ to be the advanced inverse of $\square +m^2$. Lemma lem:advanced constructs that inverse and proves its uniqueness; the identification remains a candidate.

stress (D0.12). XIII-C and XIV-J display Jstress

and the record reports the vertex as $-\frac{1}{2}\int g{\gamma }_{\mu \nu }{T_0}^{\mu \nu }$ (locator not returned). XIV-J does not display the normalization of $T^{\mu \nu }(\varphi ,\psi )$ in (eq:hess); Lemma lem:polar shows the factor is two-valued.

replacement (D0.11). XII-A, Definition def:admissible, lines 518–528 XIIAhad, defines an admissible $H^{\prime }$ by four clauses: the same shell equations in both slots, the same graded antisymmetric part $i\Delta$, a smooth difference $H^{\prime }-H$, and vanishing unless $|A|+|B|$ is even and the ghost numbers sum to zero. No manuscript equates a XIV object with this $H^{\prime }$ (F3).

sets (D0.9, D0.17). XIII-C, lines 603–620 XIIICvertex: the ${\Delta }_I$ are the connected collision subsets of the cycle-vertex table, for instance $\{y,u,v\}$ and $\{y,v\}$ on $B_1$. $D_{\Gamma }$, $w_{\Gamma }$, $S_{\Gamma /I}$ remain roles.

Four maps are written with the letter P: XII-A's $\mathcal{P}$ in $H_{\gamma \gamma }=-\mathcal{P}{h}_0$ XIIAhad, with ${\mathcal{P}}^2=\mathrm{id}$ stated there; XII-B's $\mathcal{P}$, defined explicitly as trace reversal $\mathcal{P}r=r-\frac{1}{2}\eta {\mathrm{tr}}_{\eta }r$ XIIB; XII-D's $\mathfrak{P}$ in ${G^{\mathrm{R}}}_{\mathrm{gr}}=-\mathfrak{P}{E_0}^{\mathrm{R}}$ XIID; and XIV-I's constant map ${\mathsf{P}}^L{}_M$ in (eq:T) Iresponse. XII-B's trace reversal satisfies ${\mathcal{P}}^2=\mathrm{id}$ in four dimensions: ${\mathrm{tr}}_{\eta }(\mathcal{P}r)={\mathrm{tr}}_{\eta }r-2{\mathrm{tr}}_{\eta }r=-{\mathrm{tr}}_{\eta }r$, so $\mathcal{PP}r=\mathcal{P}r+\frac{1}{2}\eta {\mathrm{tr}}_{\eta }r=r$. This is consistent with XII-A's statement and identifies nothing. Lemma lem:fiber adds that XII-D's $a_1$ equals $-\mathcal{P}$ on symmetric tensors, a relation among XII objects only.

Four symbols are written with the letter R: the kernel $R$ of (eq:R); the output map $R_1$ of (eq:Rconv); the admission coefficient ${R_1}^{(1)}$ of (eq:admission), whose relation to $R_1$ is not reproduced; and XII-B's $\mathcal{R}=\frac{1}{2}K$ XIIB. None is identified with another. Likewise XII-D's $a_1$, $X_1$, $f$, $E^{\mathrm{A}}$ and XIV-H's $a_1$, $X$, $f$, ${E^{\mathrm{A}}}_m$ are distinct symbols until a sentence identifies them.

Throughout, $\eta =\mathrm{diag}(+,-,-,-)$, $\square ={\eta }^{\lambda \kappa }{\partial }_{\lambda }{\partial }_{\kappa }={{\partial }_t}^2-\Delta$ on ${\mathbb{R}}^4$, $J^{+}=\{(t,\mathbf{x}):t\ge |\mathbf{x}|\}$, $J^{-}=-J^{+}$, $\partial J^{+}=\{t=|\mathbf{x}|\}$, and ${\mathcal{F}}_{\mathbf{x}}$ the partial Fourier transform in $\mathbf{x}$, ${\mathcal{F}}_{\mathbf{x}}\varphi (t,\mathbf{p})=\int e^{-i\mathbf{p}\cdot \mathbf{x}}\varphi (t,\mathbf{x}){\mathrm{d}}^3\mathbf{x}$, extended to ${\mathcal{S}}^{\prime }({\mathbb{R}}^4)$ as an isomorphism; $\rho =|\mathbf{p}|$.

1. (eq:R) defines the positive measure $⟨R,\varphi ⟩=\frac{1}{4\pi }{{\int }_0}^{\infty }t{\int }_{S^2}\varphi (t,t\omega )\mathrm{d}\omega \mathrm{d}t$, equal to $\Theta (t)\delta (t-|\mathbf{x}|)/(4\pi |\mathbf{x}|)$; it is tempered, with $\mathrm{supp}R=\partial J^{+}$.

2. ${\mathcal{F}}_{\mathbf{x}}R(t,\mathbf{p})=\Theta (t)\sin (\rho t)/\rho$.

3. $\square R={\delta }_0$. If $R^{\prime }\in {\mathcal{D}}^{\prime }({\mathbb{R}}^4)$ satisfies $\square R^{\prime }={\delta }_0$ and $\mathrm{supp}{R}^{\prime }\subseteq \{t\ge c\}$ for some $c\in \mathbb{R}$, then $R^{\prime }=R$. In particular $R$ is the unique fundamental solution supported in $J^{+}$.

4. If $E:\mathcal{D}({\mathbb{R}}^4)\to {\mathcal{D}}^{\prime }({\mathbb{R}}^4)$ is linear with $\square E\varphi =\varphi$ and $\mathrm{supp}E\varphi \subseteq \{t\ge c_{\varphi }\}$ for every $\varphi$, then $E\varphi =R\ast \varphi$.

In particular (eq:R) agrees with XIV-I's pole chart Ipole, which characterizes $R$ by the transform in (ii).

Proof. (i) In polar coordinates $\mathbf{x}=r\omega$, for $t>0$ the composite $\delta (t^2-r^2)$ as a distribution in $r>0$ equals $\delta (r-t)/(2t)$, so $\frac{1}{2\pi }\int \Theta (t)\delta (t^2-r^2)\varphi r^2\mathrm{d}r\mathrm{d}\omega \mathrm{d}t=\frac{1}{4\pi }{{\int }_0}^{\infty }t{\int }_{S^2}\varphi (t,t\omega )\mathrm{d}\omega \mathrm{d}t$; the same computation with $\delta (t-r)/(4\pi r)$ gives the same value. The mass of $R$ in the ball of radius $\varrho$ is at most ${\varrho }^2/2$, so $R$ is tempered. The support is the cone $t=r$.

(ii) For $\psi \in \mathcal{S}({{\mathbb{R}}^4}_{(t,\mathbf{p})})$, by (i) and Fubini, $⟨R,{\mathcal{F}}_{\mathbf{x}}\psi ⟩={{\int }_0}^{\infty }\int \psi (t,\mathbf{p})\left[\frac{t}{4\pi }{\int }_{S^2}{e}^{-it\mathbf{p}\cdot \omega }\mathrm{d}\omega \right]{\mathrm{d}}^3\mathbf{p}\mathrm{d}t$, and ${\int }_{S^2}{e}^{-it\mathbf{p}\cdot \omega }\mathrm{d}\omega =4\pi \sin (\rho t)/(\rho t)$.

(iii) Let $u(t)=\sin (\rho t)/\rho$; $u(0)=0$, $u^{\prime }(0)=1$, $u^{\prime \prime }+{\rho }^{2}u=0$. Then $({{\partial }_t}^2+{\rho }^2)(\Theta u)=\Theta (u^{\prime \prime }+{\rho }^{2}u)+u(0){\delta }^{\prime }+u^{\prime }(0)\delta =\delta (t)$ for every $\mathbf{p}$, including $\rho =0$ where $u=t$. Since ${\mathcal{F}}_{\mathbf{x}}(\square R)=({{\partial }_t}^2+{\rho }^2){\mathcal{F}}_{\mathbf{x}}R$, $\square R={\delta }_0$. Uniqueness: $v=R-R^{\prime }$ satisfies $\square v=0$ and $\mathrm{supp}v\subseteq \{t\ge c\}$. The addition map is proper on $J^{+}\times \{t\ge c\}$: if $x\in J^{+}$, $y^0\ge c$ and $x+y$ lies in a compact set, then $x^0\le C-c$, $|\mathbf{x}|\le x^0$, so $x$ and hence $y$ are bounded. Thus $R\ast v$ is defined and $v={\delta }_0\ast v=(\square R)\ast v=\square (R\ast v)=R\ast (\square v)=0$ Horm (4.2).

(iv) $E\varphi -R\ast \varphi$ solves $\square (\cdot )=0$ with support in a half-space $\{t\ge c\}$; apply the argument of (iii).

$R\ast R$ is well defined and equals $\mathcal{B}=\frac{1}{8\pi }\Theta (t)\Theta (t^2-|\mathbf{x}{|}^2)=\frac{1}{8\pi }\Theta (t-|\mathbf{x}|)$ almost everywhere. Hence (eq:RR) holds, $\mathcal{B}\in L^{\infty }({\mathbb{R}}^4)$, $\mathrm{supp}\mathcal{B}=J^{+}$, $\square \mathcal{B}=R$, and $\mathcal{B}$ is constant on the interior of $J^{+}$. On $\{t>0\}$, ${\partial }_{\alpha }\mathcal{B}=\frac{1}{4\pi }{x}_{\alpha }\delta (x^2)$, where $x_{\alpha }={\eta }_{\alpha \nu }{x}^{\nu }$ and $x^2=t^2-|\mathbf{x}{|}^2$.

Proof. $R\ast R$ exists since both factors are supported in $J^{+}$, and $\mathrm{supp}(R\ast R)\subseteq J^{+}+J^{+}=J^{+}$, $\square (R\ast R)=(\square R)\ast R=R$. For $\mathcal{B}$: ${\mathcal{F}}_{\mathbf{x}}\mathcal{B}(t,\mathbf{p})=\frac{\Theta (t)}{8\pi }{\int }_{|\mathbf{x}|<t}{e}^{-i\mathbf{p}\cdot \mathbf{x}}{\mathrm{d}}^3\mathbf{x}=\frac{\Theta (t)}{8\pi }\cdot \frac{4\pi }{\rho }{{\int }_0}^{t}r\sin (\rho r)\mathrm{d}r=\Theta (t)\frac{\sin (\rho t)-\rho t\cos (\rho t)}{2{\rho }^3}=:\Theta (t)u_2(t)$, with $u_2(0)=0$, $u_2^{\prime }(t)=t\sin (\rho t)/(2\rho )$, $u_2^{\prime }(0)=0$, and $u_2^{\prime \prime }+{\rho }^2{u}_2=\sin (\rho t)/\rho$. Hence $({{\partial }_t}^2+{\rho }^2){\mathcal{F}}_{\mathbf{x}}\mathcal{B}={\mathcal{F}}_{\mathbf{x}}R$ (no boundary terms; at $\rho =0$, $u_2=t^3/6$), i.e.\ $\square \mathcal{B}=R$. Then $v=\mathcal{B}-R\ast R$ satisfies $\square v=0$, $\mathrm{supp}v\subseteq J^{+}$, so $v=0$ by Lemma lem:R(iii). The last formula is the chain rule for $\Theta (x^2)$ on $\{t>0\}$, where $x\mapsto x^2$ has nonvanishing differential away from $x=0$.

Let $c$ be smooth in $(q,\xi )$ near $q=0$ and $j\ge 0$. Then, as distributions in $q$ depending smoothly on $\xi$,

In particular $q{\delta }^{(j)}(q)=-j{\delta }^{(j-1)}(q)$ and $q\delta (q)=0$.

Proof. $⟨c{\delta }^{(j)},\varphi ⟩=(-1{)}^j{{\partial }_q}^j(c\varphi )(0)={\sum }_i(\genfrac{}{}{0ex}{}{j}{i})(-1{)}^{j-i}{{\partial }_q}^{j-i}c(0)(-1{)}^i{{\partial }_q}^i\varphi (0)$ by Leibniz, and $(-1{)}^i{{\partial }_q}^i\varphi (0)=⟨{\delta }^{(i)},\varphi ⟩$.

1. The two XIV-J values $\mathsf{Rec}(M_{\mathrm{R}})=\delta (q)R(r)$, $\mathsf{Rec}(i\theta M_{\mathrm{R}})={\delta }^{\prime }(q)R(r)$ Jnull are instances of (eq:Recrule) with $c_{00}=1$, resp.\ $c_{10}=1$, all other coefficients zero.

2. If $\mathsf{Rec}$ is linear over multiplication by the chart function $q$ (a property not reproduced), then (eq:Recrule) annihilates the null relation $(qi\theta +1)M_{\mathrm{R}}$ of Jnull: $q{\delta }^{\prime }(q)R(r)+\delta (q)R(r)=0$ by Lemma lem:taylor.

3. Suppose the coefficients $c_{j\beta }$ are independent of $(q,r)$. Then the family $\{{\delta }^{(j)}(q){{\partial }_r}^{\beta }R(r)\}$ is linearly independent over that class, ${\mathcal{K}}_{\mathrm{int}}$ is free on it, and $\mathsf{Rec}(A)=0$ iff all $c_{j\beta }(A)=0$. In particular $\{{{\partial }_r}^{\beta }R{\}}_{\beta }$ is linearly independent over $\mathbb{C}$.

4. Suppose the coefficient class contains the chart functions $q$ or $r^2$. Then the family is linearly dependent over that class:

   $$q{\delta }^{(j)}(q){{\partial }_r}^{\beta }R(r)=-j{\delta }^{(j-1)}(q){{\partial }_r}^{\beta }R(r),r^{2}R(r)=0,r^2{\partial }_{\lambda }R(r)=-2r_{\lambda }R(r).$$

   If $\mathsf{Rec}$ is linear over the coefficient class, each of these is a null relation of ${\mathcal{A}}^m(V)$; XIV-J's relation is the first with $j=1$, $\beta =0$. In this case $\ker \mathsf{Rec}$ strictly contains the coefficient kernel, and ${\mathcal{K}}_{\mathrm{int}}$ is not free on the family.

Proof. (i) is read off; (ii) is Lemma lem:taylor. (iii) First, $\{{{\partial }^{\beta }}_{r}R{\}}_{\beta }$ is independent over $\mathbb{C}$: if ${\sum }_{\beta }{a}_{\beta }{\partial }^{\beta }R=0$, apply the Fourier transform on ${\mathbb{R}}^4$ ($R$ is tempered). From $\square R={\delta }_0$, $\hat{R}$ coincides on the open set $\Omega =\{{p_0}^2\ne |\mathbf{p}{|}^2\}$ with a smooth nowhere-vanishing function. The relation gives $P(p)\hat{R}=0$ with $P(p)={\sum }_{\beta }{a}_{\beta }(\pm ip{)}^{\beta }$, so $P=0$ on $\Omega$, hence $P\equiv 0$ and all $a_{\beta }=0$. Now pair a vanishing sum $\sum c_{j\beta }(\xi ){\delta }^{(j)}(q){\partial }^{\beta }R(r)$ with $\psi (\xi )\chi (q)\theta (r)$, where $\xi$ denotes the chart variables other than $(q,r)$: ${\sum }_{j,\beta }⟨c_{j\beta },\psi ⟩(-1{)}^j{\chi }^{(j)}(0)⟨{\partial }^{\beta }R,\theta ⟩=0$. Choose $\chi$ with ${\chi }^{(j)}(0)={\delta }_{j{j}_0}$ for the finitely many $j$ occurring; then ${\sum }_{\beta }⟨c_{j_0\beta },\psi ⟩{\partial }^{\beta }R=0$, so $⟨c_{j_0\beta },\psi ⟩=0$ for all $\beta ,\psi$. (iv) The first identity is Lemma lem:taylor. $r^{2}R(r)=\frac{1}{2\pi }\Theta (r^0)r^2\delta (r^2)=0$ by $q\delta (q)=0$ pulled back along $r\mapsto r^2$ on $r\ne 0$, and $R$ has no mass at $r=0$ by Lemma lem:R(i). Differentiating $r^{2}R=0$ gives $2r_{\lambda }R+r^2{\partial }_{\lambda }R=0$. The remaining assertions are restatements.

1. ${\Phi }_{\pm }$ in (eq:chart) are linear automorphisms of ${\mathbb{R}}^{16}$.

2. On $\{d\in {\mathbb{R}}^4:d^0>0\}$ the map $\kappa (d)=(d^2,\mathbf{d})=(q,\zeta )$ is a diffeomorphism onto ${\Omega }_0=\{(q,\zeta )\in \mathbb{R}\times {\mathbb{R}}^3:q+|\zeta {|}^2>0\}$, with inverse $d^0=\sqrt{q+|\zeta {|}^2}$, $\mathbf{d}=\zeta$. The set $\partial J^{+}\setminus \{0\}$ is contained in $\{d^0>0\}$ and is $\{q=0\}$ there.

3. Hence, with ${\zeta }_{\pm }={\mathbf{d}}_{\pm }$, $w_{\pm }=(u_{\pm },{\zeta }_{\pm },q_{\pm },r_{\pm },y)$ is a global chart of $V_{\pm }:=\{{d_{\pm }}^0>0\}\subset {\mathbb{R}}^{16}$ onto ${\mathbb{R}}^4\times {\Omega }_0\times {\mathbb{R}}^4\times {\mathbb{R}}^4$, of dimension $4+3+1+4+4=16$, consistent with Ichart and with $\dim W=16$ Grelease.

4. On $V_{\pm }$,

   $$R(d_{\pm })=\frac{1}{2\pi }\delta (q_{\pm }),({\partial }_{\alpha }R)(d_{\pm })=\frac{1}{\pi }{d}_{\pm \alpha }{\delta }^{\prime }(q_{\pm }),({\partial }_{\mu }{\partial }_{\alpha }R)(d_{\pm })=\frac{1}{\pi }[{\eta }_{\mu \alpha }{\delta }^{\prime }(q_{\pm })+2d_{\pm \mu }{d}_{\pm \alpha }{\delta }^{\prime \prime }(q_{\pm })],$$

   where $d_{\pm \alpha }={\eta }_{\alpha \nu }{d_{\pm }}^{\nu }$ is smooth on the chart, and $\mathcal{B}(d_{\pm })=\frac{1}{8\pi }\Theta (q_{\pm })$.

Proof. (i) is immediate. (ii) $q=(d^0{)}^2-|\mathbf{d}{|}^2$ and $d^0>0$ give $d^0=\sqrt{q+|\zeta {|}^2}$, smooth on ${\Omega }_0$; bijectivity is clear. On $\partial J^{+}\setminus \{0\}$, $d^0=|\mathbf{d}|>0$ and $q=0$. (iii) follows from (i), (ii). (iv) On $V_{\pm }$, $\Theta (d^0)=1$, and $\delta (d^2)$ is the pullback of $\delta (q)$ under the diffeomorphism $\kappa$; the derivatives follow from the chain rule ${\partial }_{d^{\alpha }}=2d_{\alpha }{\partial }_q+{{\delta }_{\alpha }}^i{\partial }_{{\zeta }^i}$, since $\delta (q)$ is independent of $\zeta$. The statement for $\mathcal{B}$ is Lemma lem:RR.

Let $\mathsf{A}$, $\mathsf{D}$ be constant arrays and $f,{\sigma }_{\epsilon }\in C^{\infty }({\mathbb{R}}^4)$ (any completion). On $V_{+}=\{(b-a{)}^0>0\}$, in the chart $w_{+}$ with $b=a+d_{+}$, $z=a-r_{+}$,

and

On $V_{-}=\{(b-z{)}^0>0\}$, ${K_5}^{[z],I,J}$ has the same form in the chart $w_{-}$ with $(a,I)\leftrightarrow (z,J)$, $d_{-}$, $r_{-}$. All coefficients are smooth on the chart; they are of the form $c_{j\beta }(w_{\pm })$ in (eq:Recrule) with $j\in \{1,2\}$ and $|\beta |=2$. The supports of ${K_5}^{[a]}$, $K_6$ (resp.\ ${K_5}^{[z]}$) meet the complement of $V_{+}$ (resp.\ $V_{-}$) only on $\{b=a\}$ (resp.\ $\{b=z\}$).

Proof. From (eq:L), $L^{\mu ,J}(a,z)={\sigma }_{\epsilon }(z){\mathsf{A}}^{\lambda \mu ,J}{\partial }_{z^{\lambda }}R(a-z)=-{\sigma }_{\epsilon }(z){\mathsf{A}}^{\lambda \mu ,J}({\partial }_{\lambda }R)(r_{+})$, so ${\partial }_{a^{\beta }}{L}^{\mu ,J}(a,z)=-{\sigma }_{\epsilon }(z){\mathsf{A}}^{\lambda \mu ,J}({\partial }_{\beta }{\partial }_{\lambda }R)(r_{+})$; and ${\partial }_{a^{\alpha }}R(b-a)=-({\partial }_{\alpha }R)(d_{+})$. The bracket in (eq:K5a) is therefore $f(b){\sigma }_{\epsilon }(a){\sigma }_{\epsilon }(z){\mathsf{A}}^{\alpha \beta ,I}{\mathsf{A}}^{\lambda \mu ,J}({\partial }_{\alpha }R)(d_{+})({\partial }_{\beta }{\partial }_{\lambda }R)(r_{+})$. With $a,z,y$ fixed, ${\partial }_{b^{\mu }}={\partial }_{d_{+}^{\mu }}$ acts on $f(a+d_{+})$ and on $({\partial }_{\alpha }R)(d_{+})$; insert Lemma lem:chart(iv). For $K_6$, write ${\delta }^{(4)}(a-z)=\delta (r_{+})={\square }_{r}R(r_{+})={\eta }^{\beta \lambda }{\partial }_{\beta }{\partial }_{\lambda }R(r_{+})$ (Lemma lem:R(iii)) and proceed identically; the overall sign is that of ${\partial }_{a^{\alpha }}R(b-a)=-({\partial }_{\alpha }R)(d_{+})$. ${K_5}^{[z]}$ follows by the symmetry of (eq:K5z) with (eq:K5a). The support statement: each kernel contains the factor $({\partial }_{\alpha }R)(d_{\pm })$ or its $b$-derivative, supported in $\{d_{\pm }\in \partial J^{+}\}$, and $\partial J^{+}\setminus \{0\}\subset \{d^0>0\}$ by Lemma lem:chart(ii).