Landmark X-B-ii Scope closure for the scalar-only retarded map: the one-contraction boundary, the normalization sheet, and the Branch B input list

This document closes four scope items left as statements in Landmark X-B, where the free quantum field is the scalar $\varphi$ and the metric coefficients $h,k$ are classical off-shell backgrounds. First, a scope proposition records that every object constructed in X-B is a field-degree-one element of the free algebra with polynomial classical background dependence and no $\hslash$; the non-goals are listed. Second, the one-contraction check is placed on its exact boundary: the auxiliary kernel ${\Delta }_{\mathrm{F}}={\Delta }_{+}-i{{\mathsf{E}}_{\mathsf{m}}}^{\mathrm{A}}$ is proved symmetric, the check is re-derived, the pair passes the restricted causal-factorization and symmetry requirements, and the pointwise square of ${\Delta }_{\mathrm{F}}$ is proved to fail the wavefront product criterion at every diagonal point, so the tools of the sheet define no second pair and no time-ordered family. Third, the normalization sheet is recorded on the whole off-shell insertion domain and proved to determine the single map $R_1(V_{1,g}[h];\cdot )$: a degree-one class in the free algebra vanishes if and only if it is annihilated by every spatially compact smooth solution, hence any contact modification compatible with the classical normalization is zero in the free algebra, and no contact can repair or break the first-order identity. Fourth, the data that an independent Branch B would require are listed, with a non-relabeling rule; no existence, failure, or correction is asserted for that branch. Nothing is constructed beyond X-B.

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

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

: four closure items, nothing new constructed{ C1 (scope): closed by Proposition prop:scope. C2 (one-contraction boundary): closed by Theorem thm:nonextension; the check is an identity for one pair and does not extend by the sheet's tools. C3 (normalization sheet): closed by Theorem thm:unique and Corollary cor:contact; the classical normalization on ${\mathcal{L}}_{\mathrm{off}}$ determines $R_1$, and sheet-compatible contact modifications vanish in ${\mathfrak{A}}_0$. C4 (Branch B): recorded as an input list (Section sec:branchB); not opened.

No time-ordered family, graviton action, graviton propagator, contraction domain for a quantum metric, second-order quantum map, loop, state, or sharp limit is defined. X-B's release theorem is neither enlarged nor withdrawn.}

X-B XB constructs one normalized retarded map $R_1(V_{1,g}[h];\cdot )$ for the scalar-only field content and records four scope statements without separate proof documents. They are closed here as follows.

1. Scope. Restate the boxed field content as a proposition about the constructed objects and list the non-goals.

2. One-contraction boundary. Decide whether the one-contraction check (X-B, eq:contraction-check) is promoted to a family of time-ordered maps. The ending chosen is the explicit non-extension note: the check is proved for its one pair, and it is proved that the tools of the sheet define no second pair.

3. Normalization sheet. Record the one map, state what its transposed notation means, and settle the status of contact terms relative to the classical normalization.

4. Branch B. List, without opening, the data an independent quantum-metric branch would require, and the non-relabeling rule.

For C1–C3 the ending Closed means that the deliverable is a proved statement about objects already defined in X-B. For C4 the only permitted ending is Recorded: a list is not a construction, an obstruction, or a correction. No item is allowed to end by defining a new product, a new vertex, or a new field.

The input is the full text of X-B-v3.tex XB. The classical premises that X-B copies from X-A.tex XA are used only as X-B states them. The retrieval summary that motivated the four items is documentary context, not evidence. XII-B.tex, which concerns the ramped cubic Einstein source on a different track, is not an input and no statement of it is used. Standard references are used only for standard facts: Green operators and spatially compact solutions BGP, wavefront calculus Hormander, the Hadamard wavefront set Radzikowski, and the terminology of time-ordered and retarded products FR,DF,BF.

$M={\mathbb{R}}^4$, $\eta =\mathrm{diag}(1,-1,-1,-1)$, $\square ={{\partial }_t}^2-\Delta$, $P_m=\square +m^2$, $m\ge 0$. $\mathcal{D}=\mathcal{D}(M)={C_c}^{\infty }(M;\mathbb{C})$ with its LF topology. The retarded and advanced Green operators ${{\mathsf{E}}_{\mathsf{m}}}^{\mathrm{R}},{{\mathsf{E}}_{\mathsf{m}}}^{\mathrm{A}}:\mathcal{D}\to C^{\infty }(M)$ satisfy

and $({{\mathsf{E}}_{\mathsf{m}}}^{\mathrm{R}}{)}^{\mathsf{T}}={{\mathsf{E}}_{\mathsf{m}}}^{\mathrm{A}}$, i.e.\ $⟨{{\mathsf{E}}_{\mathsf{m}}}^{\mathrm{A}}c,d⟩=⟨c,{{\mathsf{E}}_{\mathsf{m}}}^{\mathrm{R}}d⟩$ for $c,d\in \mathcal{D}$. Put ${\Delta }_S={{\mathsf{E}}_{\mathsf{m}}}^{\mathrm{A}}-{{\mathsf{E}}_{\mathsf{m}}}^{\mathrm{R}}$; then ${{\Delta }_S}^{\mathsf{T}}=-{\Delta }_S$, and ${\Delta }_S\mathcal{D}$ is exactly the space of smooth solutions of $P_{m}u=0$ with spatially compact support BGP (Thm. 3.4.7). The sign conventions of X-B are

A Hadamard bisolution ${\Delta }_{+}\in {\mathcal{D}}^{\prime }(M\times M)$ is fixed once, with

where in addition the first covector lies in one closed half of the null cone Radzikowski (Thm. 5.1); in particular no element of $\mathrm{WF}({\Delta }_{+})$ has a zero covector in either slot.

${\mathcal{A}}_{\mathrm{off}}$ is the algebra of polynomial functionals of $\phi \in C^{\infty }(M)$ with compact derivative kernels, with the Wick product $F\star G={\sum }_{n\ge 0}\frac{{\hslash }^n}{n!}⟨F^{(n)},{\Delta }_{+}^{\otimes n}{G}^{(n)}⟩$. For $c\in \mathcal{D}$, $F_c(\phi )=\int c\phi {\mathrm{d}}^{4}x$ and ${\mathcal{L}}_{\mathrm{off}}=\{F_c\}\cong \mathcal{D}$. ${\mathfrak{J}}_0$ is the pointwise ideal generated by $\{F_{P_{m}u}:u\in \mathcal{D}\}$; it is a star ideal; ${\mathfrak{A}}_0={\mathcal{A}}_{\mathrm{off}}/{\mathfrak{J}}_0$ and $\hat{\varphi }(c)=[F_c]$. Elements of ${\mathfrak{J}}_0$ vanish on smooth free solutions. Field degree is the polynomial degree in $\phi$; ${{\mathcal{A}}_{\mathrm{off}}}^{(n)}$ denotes the homogeneous part of degree $n$, and ${{\mathfrak{A}}_0}^{(1)}$ the image of ${{\mathcal{A}}_{\mathrm{off}}}^{(1)}$ in ${\mathfrak{A}}_0$, i.e.\ ${{\mathfrak{A}}_0}^{(1)}=\{\hat{\varphi }(d):d\in \mathcal{D}\}$.

The ramp ${\sigma }_{\epsilon }$, switch $g\in {C_c}^{\infty }((0,T)\times {\mathbb{R}}^3;[0,1])$, label test $f$, plateau $P_g$, region $U$, and algebra ${\mathfrak{g}}_U={C_c}^{\infty }(U;TM)$ are those of X-B. Backgrounds $(h,k)$ range over $\mathcal{B}=C^{\infty }(M;S^2{T}^{\ast }M{)}^2$. With $a_1[h]$, $b_1[h]$ as in X-B,

a formally self-adjoint operator with smooth coefficients supported in $\mathrm{supp}g$. The vertex is $V_{1,g}[h,\phi ]=-\frac{1}{2}⟨\phi ,{\mathsf{L}}_{1,g}[h]\phi ⟩$, so that $V_{1,g}^{\prime }=-{\mathsf{L}}_{1,g}[h]\phi$ and $V_{1,g}^{\prime \prime }=-{\mathsf{L}}_{1,g}[h]$. For smooth $u$, ${\rho }_{1,g}[h;u]=-{{\mathsf{E}}_{\mathsf{m}}}^{\mathrm{R}}{\mathsf{L}}_{1,g}[h]u$. The single normalized map is

The frozen insertion tests $c_0=f$, $c_1[h]$, $c_2[h,k]$ are smooth, supported in $K=\mathrm{supp}f$, and polynomial and continuous in $(h,k)$ at fixed $\epsilon$ (X-B, Proposition prop:linear). The completed coefficient is ${\hat{A}}_1[h]=\hat{\varphi }(c_1[h]-{\mathsf{L}}_{1,g}[h]{{\mathsf{E}}_{\mathsf{m}}}^{\mathrm{A}}f)$ and the retained mixed term is $\mathcal{M}[h]=R_1(V_{1,g}[h];F_{c_1[h]})$. The general insertion pair of X-B is

1. $V=-\frac{1}{2}⟨\phi ,\mathsf{L}\phi ⟩$ with $\mathsf{L}$ formally self-adjoint with smooth compactly supported coefficients;

2. $F_c$ with $c\in \mathcal{D}$.

The Branch A sheet is the tuple

together with the maps $(h,k)\mapsto c_j[h,k]$, $j=0,1,2$, and $h\mapsto {\mathsf{L}}_{1,g}[h]$. Every object of X-B is built from ${\mathfrak{S}}_A$ by the operations: smooth multiplication, differentiation, application of ${{\mathsf{E}}_{\mathsf{m}}}^{\mathrm{R}}$, ${{\mathsf{E}}_{\mathsf{m}}}^{\mathrm{A}}$, ${{\mathsf{E}}_0}^{\mathrm{R}}$ to compactly supported smooth sources, evaluation of background slots on classical configurations, formation of $F_c$, and passage to ${\mathfrak{A}}_0$.

Each of the objects

of X-B has the following properties.

1. It is a finite sum of elements $\hat{\varphi }(d)$ (or $F_d$ before passage to the quotient) with $d\in \mathcal{D}$; its field degree is one.

2. The test $d$ depends polynomially on $(h,k)$, of degree at most two in $h$ and one in $k$, and continuously in the smooth background topology at fixed $\epsilon$.

3. No $\hslash$ occurs in $d$. The constant $\hslash$ and the kernel ${\Delta }_{+}$ enter X-B only through the commutator $[\hat{\varphi }(c),\hat{\varphi }(d)]=i\hslash ⟨c,{\Delta }_{S}d⟩$ and through the one-contraction check of Section sec:boundary, in which ${\Delta }_{+}$ cancels.

4. Every two-point kernel that occurs is a scalar Green kernel (${{\mathsf{E}}_{\mathsf{m}}}^{\mathrm{R}}$, ${{\mathsf{E}}_{\mathsf{m}}}^{\mathrm{A}}$, ${{\mathsf{E}}_0}^{\mathrm{R}}$) applied to a compactly supported smooth source, or ${\Delta }_{+}$ as in (iii). Every background slot is evaluated on a classical configuration; no background slot is contracted against another background slot through a kernel.

5. No functional of $h$ or $k$ alone (in particular no gravitational action) is formed.

Proof. (i)–(ii) for ${\mathcal{O}}_{\kappa }$: the $c_j$ are the frozen tests of X-B, Proposition prop:linear, each in ${\mathcal{D}}_K(M)$, with $c_1$ linear in $h$ and $c_2$ quadratic in $h$ and linear in $k$ through ${\mathsf{X}}_1[h]$ and ${\mathsf{X}}_2[h,k]$; continuity is the handoff statement. For $R_1$: $d_c[h]=-{\mathsf{L}}_{1,g}[h]{{\mathsf{E}}_{\mathsf{m}}}^{\mathrm{A}}c$ is linear in $h$ through the coefficients of (eq:Lone), and ${{\mathsf{E}}_{\mathsf{m}}}^{\mathrm{A}}c$ is smooth, so $d_c[h]\in \mathcal{D}$ with support in $\mathrm{supp}g$; continuity is X-B, Theorem thm:retarded. For ${\hat{A}}_1$ and $\mathcal{M}$: sums and compositions of the previous two with $c$ replaced by $f$ or $c_1[h]$; the degree in $h$ of $\mathcal{M}$ is two. For ${\delta }_{\xi }{\hat{A}}_1$: X-B, (eq:defect-class-ref) below is the difference of two such elements. Field degree one is the statement that each object is a class of a functional $F_d$, which has first derivative $d$ and vanishing higher derivatives.

(iii) The tests $d$ are defined by (eq:Rone) and the formulas for $c_j$, none of which contains $\hslash$ or ${\Delta }_{+}$. X-B uses ${\Delta }_{+}$ in the Wick product and in the one-contraction check; Proposition prop:check below shows that the check is independent of ${\Delta }_{+}$.

(iv) By inspection of Definition def:sheet: the kernels ${G_0}^{\mathrm{R}}$ inside ${\mathsf{X}}_1$, ${\mathsf{X}}_2$, and ${\Lambda }_{\epsilon }$ are applied to the classical sources ${\partial }_{\mu }({\sigma }_{\epsilon }{a_1}^{\mu \rho }[h])$ and their second-order analogues; ${{\mathsf{E}}_{\mathsf{m}}}^{\mathrm{A}}$ is applied to $c$. The background slots of $c_2$ are evaluated on $(h,h)$ and $k$: two slots of the same kernel are both evaluated classically; neither is integrated against a two-point distribution in the metric variables.

(v) Every object has field degree one in $\phi$; a functional of $h,k$ alone would have degree zero.

For reference in the proof, X-B's defect formula is

both terms of which are elements $\hat{\varphi }(d)$.

-goals (recorded, not inferred).

1. No quantum metric propagator: no bidistribution in two metric slots is present in ${\mathfrak{S}}_A$.

2. No gravitational action and no graviton vertex: no functional of degree zero in $\phi$ is formed.

3. No loop: no object contains a kernel contracted against itself; the only two-point kernels are applied to smooth compact sources.

4. No hidden contraction: by Proposition prop:scope(iii) the constructed classes do not depend on ${\Delta }_{+}$.

5. No second-order quantum map: ${\rho }_{2,g}$ is displayed in X-B as classical input and is not an object of Proposition prop:scope.

6. Branch B (Section sec:branchB) is a different construction. Treating a graviton line as latent in $R_1$ would contradict (N1)–(N4).

Let ${\Delta }_{\mathrm{F}}={\Delta }_{+}+i{{\Delta }_S}^{\mathrm{A}}={\Delta }_{+}-i{{\mathsf{E}}_{\mathsf{m}}}^{\mathrm{A}}$, regarded as the Schwartz kernel in ${\mathcal{D}}^{\prime }(M\times M)$.

1. ${{\Delta }_{\mathrm{F}}}^{\mathsf{T}}={\Delta }_{\mathrm{F}}$, i.e.\ ${\Delta }_{\mathrm{F}}(x,y)={\Delta }_{\mathrm{F}}(y,x)$.

2. $(P_m\otimes 1){\Delta }_{\mathrm{F}}=-i{\delta }_{\Delta }$, where ${\delta }_{\Delta }$ is the diagonal delta, $⟨{\delta }_{\Delta },\psi ⟩=\int \psi (x,x){\mathrm{d}}^{4}x$. Equivalently $P_m({\Delta }_{\mathrm{F}}c)=-ic$ for $c\in \mathcal{D}$.

3. For $c\in \mathcal{D}$, both ${\Delta }_{+}c$ and ${\Delta }_{\mathrm{F}}c$ are smooth functions.

4. $\mathrm{WF}({\Delta }_{\mathrm{F}})\supset \{(x,x;\xi ,-\xi ):x\in M,\xi \ne 0\}$.

5. ${\Delta }_{\mathrm{F}}-{\Delta }_{+}=-i{{\mathsf{E}}_{\mathsf{m}}}^{\mathrm{A}}$ and ${\Delta }_{\mathrm{F}}-{\Delta }_{+}^{\mathsf{T}}=-i{{\mathsf{E}}_{\mathsf{m}}}^{\mathrm{R}}$.

Proof. (i) Using $({{\mathsf{E}}_{\mathsf{m}}}^{\mathrm{A}}{)}^{\mathsf{T}}={{\mathsf{E}}_{\mathsf{m}}}^{\mathrm{R}}$ and ${\Delta }_{+}^{\mathsf{T}}={\Delta }_{+}-i{\Delta }_S$,

(ii) ${\Delta }_{+}$ is a bisolution, so $(P_m\otimes 1){\Delta }_{+}=0$; $P_m{{\mathsf{E}}_{\mathsf{m}}}^{\mathrm{A}}c=c$ for all $c$ says $(P_m\otimes 1){{\mathsf{E}}_{\mathsf{m}}}^{\mathrm{A}}={\delta }_{\Delta }$ for the kernel of ${{\mathsf{E}}_{\mathsf{m}}}^{\mathrm{A}}$. Hence $(P_m\otimes 1){\Delta }_{\mathrm{F}}=-i{\delta }_{\Delta }$.

(iii) ${{\mathsf{E}}_{\mathsf{m}}}^{\mathrm{A}}c$ is smooth by the Green operator property. For ${\Delta }_{+}c$: since $\mathrm{WF}({\Delta }_{+})$ contains no point $(x,\xi ;y,0)$ with $\xi \ne 0$, the kernel maps $\mathcal{D}$ into $C^{\infty }$ Hormander (Thm. 8.2.13).

(iv) ${\delta }_{\Delta }$ is the pullback of ${\delta }_0\in {\mathcal{D}}^{\prime }({\mathbb{R}}^4)$ under the submersion $(x,y)\mapsto x-y$, so $\mathrm{WF}({\delta }_{\Delta })=\{(x,x;\xi ,-\xi ):\xi \ne 0\}$ Hormander (Thm. 8.2.4). A differential operator with smooth coefficients does not enlarge the wavefront set Hormander ((8.1.11)), so $\mathrm{WF}({\delta }_{\Delta })=\mathrm{WF}((P_m\otimes 1){\Delta }_{\mathrm{F}})\subset \mathrm{WF}({\Delta }_{\mathrm{F}})$.

(v) The first is the definition. For the second, ${\Delta }_{\mathrm{F}}-{\Delta }_{+}^{\mathsf{T}}={\Delta }_{+}-i{{\mathsf{E}}_{\mathsf{m}}}^{\mathrm{A}}-{\Delta }_{+}+i{\Delta }_S=-i{{\mathsf{E}}_{\mathsf{m}}}^{\mathrm{A}}+i{{\mathsf{E}}_{\mathsf{m}}}^{\mathrm{A}}-i{{\mathsf{E}}_{\mathsf{m}}}^{\mathrm{R}}=-i{{\mathsf{E}}_{\mathsf{m}}}^{\mathrm{R}}$.

For the pair (A1)–(A2) define, as in X-B,

Both pairings are of a compactly supported smooth function against a smooth function: $V^{\prime }(\phi )=-\mathsf{L}\phi$ is smooth with support in the compact set carrying the coefficients of $\mathsf{L}$, and ${\Delta }_{\mathrm{F}}c$, ${\Delta }_{\mathrm{F}}{V}^{\prime }$ are smooth by Lemma lem:DF(iii) and because $V^{\prime }$ is itself a compact smooth test for each $\phi$.

For the pair (A1)–(A2),

Proof. ${F_c}^{(1)}=c$ and ${F_c}^{(n)}=0$ for $n\ge 2$, so the Wick sum has only the terms $n=0,1$. For the second formula, $⟨c,{\Delta }_{+}{V}^{\prime }⟩=⟨V^{\prime },{\Delta }_{+}^{\mathsf{T}}c⟩$.

For the pair (A1)–(A2),

and for $V^{\prime }=-\mathsf{L}\phi$ the right-hand side equals $-F_{\mathsf{L}{{\mathsf{E}}_{\mathsf{m}}}^{\mathrm{A}}c}(\phi )$. In particular, for $V=V_{1,g}[h]$ it equals $F_{d_c[h]}(\phi )$, whose class is $R_1(V_{1,g}[h];F_c)$. The identity is independent of ${\Delta }_{+}$ and of $\hslash$. Moreover ${\mathcal{C}}_{\mathrm{F}}(V,F_c)={\mathcal{C}}_{\mathrm{F}}(F_c,V)$.

Proof. By Lemma lem:wick and Lemma lem:DF(v), ${\mathcal{C}}_{\mathrm{F}}(V,F_c)-V\star F_c=\hslash ⟨V^{\prime },({\Delta }_{\mathrm{F}}-{\Delta }_{+})c⟩=-i\hslash ⟨V^{\prime },{{\mathsf{E}}_{\mathsf{m}}}^{\mathrm{A}}c⟩$. Multiplying by $i/\hslash$ gives $⟨V^{\prime },{{\mathsf{E}}_{\mathsf{m}}}^{\mathrm{A}}c⟩$, since $i\cdot (-i)=1$. The second equality is ${{\Delta }_S}^{\mathrm{A}}=-{{\mathsf{E}}_{\mathsf{m}}}^{\mathrm{A}}$. The third is transposition, $⟨V^{\prime },{{\mathsf{E}}_{\mathsf{m}}}^{\mathrm{A}}c⟩=⟨{{\mathsf{E}}_{\mathsf{m}}}^{\mathrm{R}}{V}^{\prime },c⟩=-⟨c,{{\Delta }_S}^{\mathrm{R}}{V}^{\prime }⟩$, legitimate because $V^{\prime }$ is compactly supported and smooth. For $V^{\prime }=-\mathsf{L}\phi$: $⟨-\mathsf{L}\phi ,{{\mathsf{E}}_{\mathsf{m}}}^{\mathrm{A}}c⟩=-⟨\phi ,\mathsf{L}{{\mathsf{E}}_{\mathsf{m}}}^{\mathrm{A}}c⟩$ by formal self-adjointness of $\mathsf{L}$ with compact coefficients and smoothness of ${{\mathsf{E}}_{\mathsf{m}}}^{\mathrm{A}}c$; this is $-F_{\mathsf{L}{{\mathsf{E}}_{\mathsf{m}}}^{\mathrm{A}}c}(\phi )$, and with $\mathsf{L}={\mathsf{L}}_{1,g}[h]$ it is $F_{d_c[h]}(\phi )$. Independence of ${\Delta }_{+}$ and $\hslash$ is visible in the result. Symmetry: by Lemma lem:DF(i), $⟨c,{\Delta }_{\mathrm{F}}{V}^{\prime }⟩=⟨{{\Delta }_{\mathrm{F}}}^{\mathsf{T}}c,V^{\prime }⟩=⟨V^{\prime },{\Delta }_{\mathrm{F}}c⟩$, and $V{F}_c=F_{c}V$ pointwise.

Let $V$ be as in (A1) with coefficient support $S=\mathrm{supp}\mathsf{L}\subset M$ compact, and $c\in \mathcal{D}$.

1. If $S\cap J^{-}(\mathrm{supp}c)=\varnothing$, then ${\mathcal{C}}_{\mathrm{F}}(V,F_c)=V\star F_c$, and $R_1(V;F_c)=0$ for $V=V_{1,g}[h]$.

2. If $S\cap J^{+}(\mathrm{supp}c)=\varnothing$, then ${\mathcal{C}}_{\mathrm{F}}(V,F_c)=F_c\star V$.

Proof. (i) ${\mathcal{C}}_{\mathrm{F}}(V,F_c)-V\star F_c=-i\hslash ⟨V^{\prime },{{\mathsf{E}}_{\mathsf{m}}}^{\mathrm{A}}c⟩$, and $\mathrm{supp}{{\mathsf{E}}_{\mathsf{m}}}^{\mathrm{A}}c\subset J^{-}(\mathrm{supp}c)$ is disjoint from $\mathrm{supp}{V}^{\prime }\subset S$. The vanishing of $R_1$ is then (eq:check), or directly ${\mathsf{L}}_{1,g}[h]{{\mathsf{E}}_{\mathsf{m}}}^{\mathrm{A}}c=0$. (ii) By Lemma lem:wick and Lemma lem:DF(v), ${\mathcal{C}}_{\mathrm{F}}(V,F_c)-F_c\star V=\hslash ⟨V^{\prime },({\Delta }_{\mathrm{F}}-{\Delta }_{+}^{\mathsf{T}})c⟩=-i\hslash ⟨V^{\prime },{{\mathsf{E}}_{\mathsf{m}}}^{\mathrm{R}}c⟩$, and $\mathrm{supp}{{\mathsf{E}}_{\mathsf{m}}}^{\mathrm{R}}c\subset J^{+}(\mathrm{supp}c)$ is disjoint from $S$.

Propositions prop:check and prop:causal say exactly what X-B says: the auxiliary expression supplies the usual one-contraction retarded interpretation for this pair, with the symmetry and the two causal-factorization properties that a time-ordered product would have to satisfy on this pair.

A family of time-ordered maps in the sense of DF,BF,FR consists of multilinear maps $T_n$ on $n$-tuples of admissible functionals, satisfying at least

1. symmetry in the entries;

2. causal factorization: $T_{n+k}(A,B)=T_n(A)\star T_k(B)$ whenever the support of $A$ is disjoint from the causal past of that of $B$;

3. a Wick-type expansion in which the $j$-contraction term of $T_2(A,B)$ is $\frac{{\hslash }^j}{j!}⟨A^{(j)},{{\Delta }_{\mathrm{F}}}^{\otimes j}{B}^{(j)}⟩$ wherever this is defined, with a specified extension where it is not;

4. $T_1=\mathrm{id}$, and normalization conditions for the extensions in (T3).

A retarded product is then $R(A;B)=\frac{i}{\hslash }(T_2(A,B)-A\star B)$.

1. The expression ${\mathcal{C}}_{\mathrm{F}}(V,F_c)$ is defined by the sheet for every pair (A1)–(A2), and Propositions prop:check–prop:causal are the restrictions of (T1)–(T3) to this pair; the resulting retarded expression is (eq:Rone).

2. For the pair $(V,V)$ with $V$ as in (A1) and $\mathsf{L}\ne 0$, the two-contraction term that (T3) would prescribe, $\frac{{\hslash }^2}{2}⟨V^{\prime \prime },{{\Delta }_{\mathrm{F}}}^{\otimes 2}{V}^{\prime \prime }⟩$, contains the pointwise product ${\Delta }_{\mathrm{F}}\cdot {\Delta }_{\mathrm{F}}$ of the kernel with itself on $M\times M$. This product is not defined by the wavefront product criterion: at every point of the diagonal, $\mathrm{WF}({\Delta }_{\mathrm{F}})$ contains both $(x,x;\xi ,-\xi )$ and $(x,x;-\xi ,\xi )$, whose sum is zero.

3. By contrast the corresponding Wick term $\frac{{\hslash }^2}{2}⟨V^{\prime \prime },{\Delta }_{+}^{\otimes 2}{V}^{\prime \prime }⟩$ is defined by the same criterion, consistently with the definition of ${\mathcal{A}}_{\mathrm{off}}$.

4. Consequently the sheet of X-B defines no $T_2(V,V)$, no $R(V;V)$, and no $T_n$ for $n\ge 2$ on general polynomial pairs. The one-contraction check is an identity for the pair (A1)–(A2) and is not a family of time-ordered maps. Passing the check for this pair does not pass the requirements (T1)–(T4) for any other pair.

Proof. (i) is Propositions prop:check and prop:causal.

(ii) The kernel of $V^{\prime \prime }=-\mathsf{L}$ is $-{\mathsf{L}}_x{\delta }_{\Delta }(x,y)$. In $⟨V^{\prime \prime },{{\Delta }_{\mathrm{F}}}^{\otimes 2}{V}^{\prime \prime }⟩=\int V^{\prime \prime }(x_1,x_2){\Delta }_{\mathrm{F}}(x_1,y_1){\Delta }_{\mathrm{F}}(x_2,y_2)V^{\prime \prime }(y_1,y_2)$, integrating the first delta kernel against $x_2$ identifies the first slots of the two ${\Delta }_{\mathrm{F}}$ factors (with $\mathsf{L}$ acting on one of them), and integrating the second delta kernel against $y_2$ identifies their second slots. The result is the pairing of the compactly supported coefficient distribution of $\mathsf{L}\otimes \mathsf{L}$ against the restriction to $\{x_1=x_2,y_1=y_2\}$ of ${\Delta }_{\mathrm{F}}(x_1,y_1){\Delta }_{\mathrm{F}}(x_2,y_2)$, that is, against a derivative of the pointwise product ${\Delta }_{\mathrm{F}}\cdot {\Delta }_{\mathrm{F}}$ on $M\times M$. The sufficient condition for the existence of a product $uv$ of distributions is that there be no $(z,\zeta )\in \mathrm{WF}(u)$ with $(z,-\zeta )\in \mathrm{WF}(v)$ Hormander (Thm. 8.2.10). With $u=v={\Delta }_{\mathrm{F}}$ and $z=(x,x)$, $\zeta =(\xi ,-\xi )$, Lemma lem:DF(iv) gives both $(z,\zeta )$ and $(z,-\zeta )$ in $\mathrm{WF}({\Delta }_{\mathrm{F}})$. The criterion fails at every diagonal point.

(iii) Every element of $\mathrm{WF}({\Delta }_{+})$ has a nonzero first covector in one fixed closed half of the null cone (Premise prem:free). If $(z,\zeta )$ and $(z,-\zeta )$ were both in $\mathrm{WF}({\Delta }_{+})$, the first components $\xi$ and $-\xi$ would both lie in that closed half cone, forcing $\xi =0$, which is excluded. Hence the criterion holds and ${\Delta }_{+}\cdot {\Delta }_{+}$ is defined Hormander (Thm. 8.2.10).

(iv) A map $T_2$ satisfying (T3) on $(V,V)$ requires a distribution extending ${\Delta }_{\mathrm{F}}\cdot {\Delta }_{\mathrm{F}}$ across the diagonal; the sheet contains no such extension and no normalization condition (T4) selecting one. The existence of such extensions and their freedom is the renormalization problem of BF, which is not opened here. Without $T_2(V,V)$ there is no $R(V;V)$; without $T_2$ on general pairs there is no family. The last sentence is the content of (i)–(iii): the pair (A1)–(A2) never meets the undefined product because ${F_c}^{(2)}=0$.

The boundary is not at ${\Delta }_{\mathrm{F}}$ itself, which is a well-defined distribution and a fundamental solution up to the factor $-i$ (Lemma lem:DF(ii)); it is at the first pointwise product of ${\Delta }_{\mathrm{F}}$ with itself, i.e.\ at the first pair of entries both of field degree at least two. X-B's wording “it does not constitute a claimed family of time-ordered maps” is therefore exact. No statement of this section asserts that such a family does not exist; it asserts that the sheet does not define one.

${\mathfrak{J}}_0$ is graded by field degree, and ${\mathfrak{J}}_0\cap {{\mathcal{A}}_{\mathrm{off}}}^{(1)}=\{F_{P_{m}v}:v\in \mathcal{D}\}$. Consequently, for $d\in \mathcal{D}$, $\hat{\varphi }(d)=0$ in ${\mathfrak{A}}_0$ if and only if $d\in P_m\mathcal{D}$.

Proof. A general element of the pointwise ideal is a finite sum ${\sum }_i{G}_i{F}_{P_m{u}_i}$ with $G_i\in {\mathcal{A}}_{\mathrm{off}}$. Decomposing each $G_i$ into homogeneous parts and using that the pointwise product of homogeneous functionals of degrees $p,q$ is homogeneous of degree $p+q$, every homogeneous component of the sum is again in ${\mathfrak{J}}_0$; so ${\mathfrak{J}}_0$ is graded. Its degree-one part consists of ${\sum }_i{\lambda }_i{F}_{P_m{u}_i}=F_{P_m({\sum }_i{\lambda }_i{u}_i)}$, ${\lambda }_i\in \mathbb{C}$. Finally $F_d\in {\mathfrak{J}}_0$ iff $F_d\in {\mathfrak{J}}_0\cap {{\mathcal{A}}_{\mathrm{off}}}^{(1)}$ iff $F_d=F_{P_{m}v}$ for some $v$, iff $d=P_{m}v$, since $F_d$ determines $d$ (test on $\phi \in \mathcal{D}$).

For $d\in \mathcal{D}$ the following are equivalent:

1. $d\in P_m\mathcal{D}$, i.e.\ $\hat{\varphi }(d)=0$;

2. ${\Delta }_{S}d=0$;

3. $⟨d,u⟩=0$ for every $u\in {\Delta }_S\mathcal{D}$ (every smooth spatially compact solution);

4. $⟨d,u⟩=0$ for every smooth solution $u$ of $P_{m}u=0$.

Proof. (a)$\Rightarrow$(d): $⟨P_{m}v,u⟩=⟨v,P_{m}u⟩=0$, the integration by parts being justified by compactness of $\mathrm{supp}v$. (d)$\Rightarrow$(c) is trivial. (c)$\Rightarrow$(b): for $e\in \mathcal{D}$, $⟨d,{\Delta }_{S}e⟩=⟨{{\Delta }_S}^{\mathsf{T}}d,e⟩=-⟨{\Delta }_{S}d,e⟩$; vanishing for all $e$ forces the smooth function ${\Delta }_{S}d$ to vanish. (b)$\Rightarrow$(a): if ${{\mathsf{E}}_{\mathsf{m}}}^{\mathrm{A}}d={{\mathsf{E}}_{\mathsf{m}}}^{\mathrm{R}}d=:v$, then $\mathrm{supp}v\subset J^{+}(\mathrm{supp}d)\cap J^{-}(\mathrm{supp}d)$, which is compact in Minkowski space (global hyperbolicity), so $v\in \mathcal{D}$ and $P_{m}v=P_m{{\mathsf{E}}_{\mathsf{m}}}^{\mathrm{R}}d=d$. The parenthetical identification in (c) is BGP (Thm. 3.4.7).

For a smooth solution $u$ of $P_{m}u=0$, define ${\mathrm{ev}}_u:{{\mathfrak{A}}_0}^{(1)}\to \mathbb{C}$ by ${\mathrm{ev}}_u\hat{\varphi }(d)=⟨d,u⟩$. This is well defined by Lemma lem:detect, (a)$\Rightarrow$(d), and the family $\{{\mathrm{ev}}_u:u\in {\Delta }_S\mathcal{D}\}$ separates points of ${{\mathfrak{A}}_0}^{(1)}$ by Lemma lem:detect, (c)$\Rightarrow$(a).

For $c\in \mathcal{D}$, $h\in \mathcal{B}$, and every smooth $u$ (not necessarily a solution),

If $u$ is a solution, the left side is $\int c{\rho }_{1,g}[h;u]{\mathrm{d}}^{4}x$ and the right side is ${\mathrm{ev}}_u{R}_1(V_{1,g}[h];F_c)$. Thus the notation $-⟨c,{{\mathsf{E}}_{\mathsf{m}}}^{\mathrm{R}}{\mathsf{L}}_{1,g}[h]\hat{\varphi }⟩$ denotes the element $\hat{\varphi }(d_c[h])\in {{\mathfrak{A}}_0}^{(1)}$, characterized by its classical evaluations; it does not denote the application of ${{\mathsf{E}}_{\mathsf{m}}}^{\mathrm{R}}$ to an unsmeared quantum field.

Proof. ${\mathsf{L}}_{1,g}[h]u$ is smooth with support in the compact set $\mathrm{supp}g$, so ${{\mathsf{E}}_{\mathsf{m}}}^{\mathrm{R}}{\mathsf{L}}_{1,g}[h]u$ is defined and smooth, and $⟨c,{{\mathsf{E}}_{\mathsf{m}}}^{\mathrm{R}}w⟩=⟨{{\mathsf{E}}_{\mathsf{m}}}^{\mathrm{A}}c,w⟩$ for compact $w$. Then $⟨{{\mathsf{E}}_{\mathsf{m}}}^{\mathrm{A}}c,{\mathsf{L}}_{1,g}[h]u⟩=⟨{\mathsf{L}}_{1,g}[h]{{\mathsf{E}}_{\mathsf{m}}}^{\mathrm{A}}c,u⟩$ by formal self-adjointness of ${\mathsf{L}}_{1,g}[h]$, legitimate because its coefficients are compactly supported and ${{\mathsf{E}}_{\mathsf{m}}}^{\mathrm{A}}c$, $u$ are smooth. Hence $-⟨c,{{\mathsf{E}}_{\mathsf{m}}}^{\mathrm{R}}{\mathsf{L}}_{1,g}[h]u⟩=⟨-{\mathsf{L}}_{1,g}[h]{{\mathsf{E}}_{\mathsf{m}}}^{\mathrm{A}}c,u⟩=⟨d_c[h],u⟩$. The rest is Definition def:ev and ${\rho }_{1,g}[h;u]=-{{\mathsf{E}}_{\mathsf{m}}}^{\mathrm{R}}{\mathsf{L}}_{1,g}[h]u$.

Fix $h\in \mathcal{B}$. A map $\mathsf{R}:{\mathcal{L}}_{\mathrm{off}}\to {{\mathfrak{A}}_0}^{(1)}$ satisfies the classical normalization sheet for $V_{1,g}[h]$ if

No linearity or continuity of $\mathsf{R}$ is assumed in this definition.

X-B's product sheet writes the normalization row for the single test $f$; the proof of X-B, Theorem thm:retarded, is for arbitrary $c$. Definition def:sheetN records the normalization on the whole domain ${\mathcal{L}}_{\mathrm{off}}$, which is what that proof establishes.

For every $h\in \mathcal{B}$, the map $R_1(V_{1,g}[h];\cdot )$ of (eq:Rone) satisfies (eq:sheet), and it is the only map ${\mathcal{L}}_{\mathrm{off}}\to {{\mathfrak{A}}_0}^{(1)}$ that does.

Proof. Existence is Lemma lem:transpose. If $\mathsf{R}$ satisfies (eq:sheet), write $\mathsf{R}(F_c)=\hat{\varphi }({d^{\prime }}_c)$ with ${d^{\prime }}_c\in \mathcal{D}$. Then $⟨{d^{\prime }}_c-d_c[h],u⟩=0$ for all $u\in {\Delta }_S\mathcal{D}$, so ${d^{\prime }}_c-d_c[h]\in P_m\mathcal{D}$ by Lemma lem:detect, and $\hat{\varphi }({d^{\prime }}_c)=\hat{\varphi }(d_c[h])$ by Lemma lem:graded.

Let $N:\mathcal{D}\to \mathcal{D}$ be any map and ${\mathsf{R}}^N(F_c)=\hat{\varphi }(d_c[h]+Nc)$.

1. ${\mathsf{R}}^N$ satisfies (eq:sheet) if and only if ${\Delta }_{S}Nc=0$ for all $c$, if and only if $\hat{\varphi }(Nc)=0$ for all $c$; in that case ${\mathsf{R}}^N=R_1(V_{1,g}[h];\cdot )$ as maps into ${\mathfrak{A}}_0$.

2. Example of a sheet-compatible modification: $Nc=P_m(\chi c)$ with $\chi \in C^{\infty }(M)$; it is invisible in ${\mathfrak{A}}_0$.

3. Example of a sheet-incompatible modification: $Nc=wc$ with $w\in C^{\infty }(M)$ real and $w\not\equiv 0$; in particular $w=\lambda g{b}_1[h]$ with $\lambda \ne 0$ whenever $g{b}_1[h]\not\equiv 0$.

Proof. (i) ${\mathrm{ev}}_u{\mathsf{R}}^N(F_c)-{\mathrm{ev}}_u{R}_1(F_c)=⟨Nc,u⟩$; apply Lemma lem:detect to $Nc$ for each $c$, and Lemma lem:graded. (ii) $P_m(\chi c)\in P_m\mathcal{D}$. (iii) Choose $x_0$ with $w(x_0)\ne 0$. The Cauchy problem with smooth compactly supported data $u{|}_{t={x_0}^0}=\psi$, ${\partial }_{t}u{|}_{t={x_0}^0}=0$, $\psi ({\mathbf{x}}_0)\ne 0$, has a unique smooth solution, spatially compact by finite propagation BGP (Thm. 3.2.11); hence $u\in {\Delta }_S\mathcal{D}$ BGP (Thm. 3.4.7), and $u(x_0)\ne 0$. On a small ball around $x_0$ the real function $\mathrm{Re}(w\bar{u})$ or $\mathrm{Im}(w\bar{u})$ has a fixed sign; choose $c\ge 0$, $c\not\equiv 0$, supported in that ball with the corresponding phase so that $⟨wc,u⟩\ne 0$. Thus ${\Delta }_S(wc)\ne 0$ and (i) fails.

Let $N$ be sheet-compatible as in Corollary cor:contact(i) and ${\widehat{A_1}}^N[h]={\mathcal{O}}^{(1)}[h]+{\mathsf{R}}^N(F_f)$. Then ${\widehat{A_1}}^N[h]={\hat{A}}_1[h]$ in ${\mathfrak{A}}_0$ for every $h$, so X-B's Theorem thm:ward holds for ${\widehat{A_1}}^N$ exactly when it holds for ${\hat{A}}_1$. No contact term is selected to repair the identity, and none could break it without violating (eq:sheet).

Proof. Corollary cor:contact(i) with $c=f$.

Theorem thm:unique is uniqueness relative to the classical normalization on ${\mathcal{L}}_{\mathrm{off}}$ and to the target ${{\mathfrak{A}}_0}^{(1)}$. It replaces the sentence of X-B “makes no uniqueness claim among arbitrary additional contact normalizations” by a precise one: every modification of $R_1$ with values in ${{\mathfrak{A}}_0}^{(1)}$ either changes a classical evaluation or is zero in ${\mathfrak{A}}_0$. Two remarks delimit this.