Landmark XI-G The free-shell class of the cubic Einstein kernel: the onset covector is inherited by every representative

Landmark XI-F proved that the frozen kernel $u$ of the cubic Einstein insertion ${\mathcal{C}}_3(f,g)={\mathcal{A}}_1[u_3;\varphi ]$ carries the timelike conormal $\pm \mathrm{d}t$ at the initial surface, and left open whether the free-shell class of ${\mathcal{C}}_3$ contains some other representative in XI-E's polynomial microcausal class ${\mathcal{F}}_{\mathrm{mc},\mathrm{pol}}$. We decide that question at fixed regulator width, with ${G_{\mathrm{gr}}}^{\mathrm{R}}=-\mathcal{P}{{\mathsf{E}}_0}^{\mathrm{R}}$ and the frozen prescription unchanged.

Separation. For compact kernels on the ambient Minkowski space, vanishing on all triples of free solutions is equivalent to membership in the bulk equation ideal ${\square }_1{\mathcal{E}}^{\prime }+{\square }_2{\mathcal{E}}^{\prime }+P_{m,3}{\mathcal{E}}^{\prime }$, and the free-shell class is encoded faithfully by the trisolution ${\mathsf{W}}_u=({\mathsf{E}}_0\otimes {\mathsf{E}}_0\otimes {\mathsf{E}}_m)u$. The class meets ${\mathcal{F}}_{\mathrm{mc},\mathrm{pol}}$ if and only if $\mathrm{WF}({\mathsf{W}}_u)$ avoids ${\tilde{V}}_{+}^3\cup {\tilde{V}}_{-}^3$. Pairings with exact null plane waves are class invariants.

Obstruction. For XI-F's admissible observation data, pairing with the pure-trace counter-propagating plane waves $\eta e^{i\lambda k_a\cdot y}$, $k_1+k_2=2\mathrm{d}t$, gives $-i\lambda {\Lambda }_f+O(1)$ with ${\Lambda }_f\ne 0$, whereas every compact kernel in ${\mathcal{F}}_{\mathrm{mc},\mathrm{pol}}$ decays rapidly in that direction, because $(k_1,k_2)$ lies in ${\tilde{V}}_{+}^2$. Hence $[u{]}_{\mathrm{sol}}\cap {\mathcal{F}}_{\mathrm{mc},\mathrm{pol}}=\varnothing$: every distribution reproducing ${\mathcal{C}}_3(f,g)$ on ${\mathcal{C}}_{\mathrm{sol}}$ has a wavefront point with two co-oriented null covectors, one in each metric slot and zero in the scalar slot. These are the two halves of XI-F's $\pm \mathrm{d}t$. The leading coefficient is the Cauchy pairing $B_{\Sigma }$, computed explicitly on the pure-trace sector. Transverse (Einstein-shell) pairs are blind to the $00$-component by the Bianchi identity; the witness lies on the gauge-fixed shell and off the Einstein shell.

The cubic source is not ramped, the Cauchy surface is not moved, no interacting differential, retarded quantum product, loop, state, or sharp-width limit is constructed, and the empty Einstein jet is not repaired. The classical ledger $-I+I+0=0$ of XI-F is untouched.

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

Figure fig:scope-XI-G records that separation. The drawing adds no hypothesis.

: four gates, worst ending governs{ The question decided here is

G1 (Cauchy pairing): Constructed.

$B_{\Sigma }$ is written as a bidifferential form in Cauchy data, printed on the pure-trace sector, and identified as the leading onset coefficient of the class invariant.

G2 (separation): Constructed.

Agreement on the free shell is exactly the bulk equation ideal; the class is the trisolution ${\mathsf{W}}_u$; the class meets ${\mathcal{F}}_{\mathrm{mc},\mathrm{pol}}$ iff $\mathrm{WF}({\mathsf{W}}_u)$ avoids the two causal cones.

G3 (alternative): Proved obstructed.

For every frozen $g$, every $m$, and XI-F's observation data, every compact distribution reproducing ${\mathcal{C}}_3(f,g)$ on ${\mathcal{C}}_{\mathrm{sol}}$ has covectors $\pm (k_1,k_2,0)$, $k_1,k_2$ null, co-oriented, $k_1+k_2=2\mathrm{d}t$. Quantifier: the copied class ${\mathcal{F}}_{\mathrm{mc},\mathrm{pol}}$, not every bimodule.

G4: Recorded.

No product, no differential, no loop.

Overall ending: Proved obstructed (Gate 3). The frozen-kernel theorem of XI-F stands; its covector is inherited, in split form, by the whole class.}

XI-F released two gates for the cubic Einstein response and stopped at a third XIF. Gate F1 proved the frozen representative of ${\mathcal{C}}_3$ obstructed: zero Cauchy data for the second coefficient together with an unswitched, unramped cubic source produce a sharp onset $\theta (y^0)$ in a metric slot, and $\pm \mathrm{d}t\in \mathrm{WF}$ at $\Sigma$. Gates F2–F3 constructed the classical plateau sum with ledger $-I+I+0=0$. The on-shell class was left open. This paper decides it, and nothing else.

1. Restrict both metric slots to the gauge-fixed free shell and write the boundary pairing produced by $\theta (y^0)$ explicitly in Cauchy data.

2. Prove that agreement with $u$ on the free shell is a bulk equation term, identify the class invariant, and show that no bulk term removes what the shell sees.

3. Either construct $\tilde{u}\in {\mathcal{F}}_{\mathrm{mc},\mathrm{pol}}$ agreeing with $u$ on the shell, or prove that every reproducing distribution meets a forbidden covector, or record the missing estimate.

4. Say what the result is not.

Three endings apply to each gate: Constructed, Proved obstructed, Unresolved. The worst governs the release. A missing representative is not a no-go; obstruction of the frozen kernel is not obstruction of the class. The theorems below are stated with the quantifier written out.

1. The sheet proposed the Cauchy pairing $B_{\Sigma }$ as the remainder to be tested. The class invariant is the trisolution ${\mathsf{W}}_u=({\mathsf{E}}_0\otimes {\mathsf{E}}_0\otimes {\mathsf{E}}_m)u$; bulk equation terms are exactly the kernel of the map $u\mapsto {\mathsf{W}}_u$ on compact kernels (Lemma lem:kernel). $B_{\Sigma }$ is the leading onset coefficient of ${\mathsf{W}}_u$ in the counter-propagating null directions (Proposition prop:planewave), not the invariant itself.

2. The sheet expected the forbidden covector of the class to be $\pm \mathrm{d}t$ at $\Sigma$. Pairings with solutions see only null covectors (Lemma lem:transport). The conormal $\pm \mathrm{d}t$ is the sum $k_1+k_2$ of two co-oriented null covectors, one per metric slot; that pair is what every representative inherits (Corollary cor:covector).

3. The witness is a pair of exact null plane waves on the gauge-fixed shell, not a Cauchy-data expansion. The tools are XI-F's (Definition 8.1.2, Proposition 8.1.3, Theorem 8.2.12 of Hormander) together with Theorems 8.2.4, 8.2.9, 8.2.13 and 8.3.1. No restriction theorem to $\Sigma$ is used.

4. Transverse plane-wave pairs, which lie on the Einstein shell $A\gamma =0$, do not detect XI-F's data: the Bianchi identity makes the leading symbol conserved, and the total covector $2\mathrm{d}t$ annihilates its $0\nu$-components (Remark rem:einstein). The detecting polarization is pure trace, $ϵ=\eta$, which is on XI-E's gauge-fixed shell and off the Einstein shell.

5. XI-F's one-slot rewriting of a boundary trace as a bulk functional is the $n=1$ case of the kernel lemma $\ker {\mathsf{E}}_{\mu }\cap {\mathcal{E}}^{\prime }=(\square +{\mu }^2){\mathcal{E}}^{\prime }$ (Corollary cor:oneslot). It is a corollary of G2, not the two-slot proof.

6. The constructed branch of G3 is characterized by an iff criterion on $\mathrm{WF}({\mathsf{W}}_u)$ (Theorem thm:criterion) but is not instantiated: the criterion fails for XI-F's data (Theorem thm:obstruction), and for all data whose boundary trace of $\beta$ pairs nontrivially with the counter-propagating symbol (Proposition prop:general).

The cubic source $J_3$ is not multiplied by ${\sigma }_{\epsilon }$ or by $g$. The metric Cauchy surface stays at $\Sigma =\{t=0\}$. X-E's normalization $-4\mathcal{P}{{\mathsf{E}}_0}^{\mathrm{R}}$ and X-F's three-tree sum are not used. VII-C's hypotheses (E1)–(E4) are not the admission interface; XI-E's Definition of ${\mathcal{F}}_{\mathrm{mc},\mathrm{pol}}$ is. No reflected residue or finite part is subtracted. The empty fixed-input Einstein jet ${\mathcal{J}}_g({\varphi }_m)=\varnothing$ remains XI-A's result XIA. No mixed loop is opened: the cubic source is not contracted against a Hadamard line. No ${\hat{s}}_{\mathrm{int}}$, antibracket, or master identity is built. No $\epsilon \to 0$ limit is taken.

Indices are moved with $\eta$; parentheses include $\frac{1}{2}$; $k\cdot y={\eta }_{\mu \nu }{k}^{\mu }{y}^{\nu }$. A covector $p$ is null if ${\eta }^{\mu \nu }{p}_{\mu }{p}_{\nu }=0$; ${\tilde{V}}_{+}$ (${\tilde{V}}_{-}$) is the closed cone of future (past) directed causal covectors with zero included (XI-E); for a null covector $p\ne 0$, $p\in {\tilde{V}}_{+}$ iff $p_0>0$. The ramp ${\sigma }_{\epsilon }$, switch $g$, plateau $U$, ${\mathfrak{g}}_U$, and the observation data $f\in {C_c}^{\infty }({M_T}^{\circ })$, ${\mathcal{D}}_f=J^{-}(\mathrm{supp}f)\cap M_T$, are XI-F's. Trace reversal $\mathcal{P}r=r-\frac{1}{2}\eta {\mathrm{tr}}_{\eta }r$, ${\mathcal{P}}^2=\mathrm{id}$; $D^{\mu }r={\partial }_{\nu }(\mathcal{P}r{)}^{\nu \mu }$; $(Kv{)}_{\mu \nu }={\partial }_{\mu }{v}_{\nu }+{\partial }_{\nu }{v}_{\mu }$; $A=2G_{\mathrm{lin}}$;

For $\mu \in \{0,m\}$ let ${{\mathsf{E}}_{\mu }}^{\mathrm{R}},{{\mathsf{E}}_{\mu }}^{\mathrm{A}}$ be the Minkowski retarded and advanced inverses of $\square +{\mu }^2$ on $M$; on compact interior sources they restrict to XI-B's slab inverses ${{\mathsf{E}}_0}^{\mathrm{R}},{{\mathsf{E}}_0}^{\mathrm{A}},{{\mathsf{E}}_{\mathsf{m}}}^{\mathrm{R}},{{\mathsf{E}}_{\mathsf{m}}}^{\mathrm{A}}$ XIB. Put

which is XI-B's causal kernel ${\delta }_{\mu }$. The operators ${{\mathsf{E}}_{\mu }}^{\mathrm{R}/A}$ act on distributions with past/future compact support, satisfy $(\square +{\mu }^2){{\mathsf{E}}_{\mu }}^{\mathrm{R}}w=w$ and ${{\mathsf{E}}_{\mu }}^{\mathrm{R}}(\square +{\mu }^2)r=r$ for $r$ compactly supported, $\mathrm{supp}{{\mathsf{E}}_{\mu }}^{\mathrm{R}}w\subset J^{+}(\mathrm{supp}w)$, and the retarded solution with past-compact support of $(\square +{\mu }^2)\Phi =w$ is unique BGP (Ch. 3). For a distribution $w$ on $M^3$ whose support is compact in the slot-$i$ direction, $i\in \{1,2,3\}$, the slot operators ${{\mathsf{E}}^{\mathrm{R}}}_{\mu ,i}$, ${{\mathsf{E}}^{\mathrm{A}}}_{\mu ,i}$, ${\mathsf{E}}_{\mu ,i}$ are defined by transposition in slot $i$, for instance $⟨{\mathsf{E}}_{\mu ,i}w,\phi ⟩=-⟨w,{\mathsf{E}}_{\mu ,i}\phi ⟩$ for $\phi \in {C_c}^{\infty }(M^3)$; they commute with multiplication by smooth functions of, and differential operators in, the other slots. We write ${\mathsf{E}}_0$ for the metric slots and ${\mathsf{E}}_m$ for the scalar slot, and ${\mathcal{E}}^{\prime }$ for compactly supported distributions on $M^3$ with the slot type of $u$ below.

${\mathcal{C}}_3(f,g)[\gamma ,\varphi ]={\int }_{[0,T]\times {\mathbb{R}}^3}{\beta }_{\mu \nu }(y){J_3}^{\mu \nu }[\gamma ](y){\mathrm{d}}^{4}y$, with $\beta ={{\mathsf{E}}_0}^{\mathrm{A}}(\mathcal{P}\alpha )$ and $\mathrm{supp}\beta \subset {\mathcal{D}}_f$. The degree-$(2,1)$ kernel has the form

The factor $\theta (y^0)$ is forced: ${{\mathsf{E}}_0}^{\mathrm{R}}$ truncates every source at $t=0$, and the Einstein–Hilbert density carries neither $g$ nor ${\sigma }_{\epsilon }$.

Here $\alpha =\alpha [f,g,\varphi ]$ is XI-F's smooth kernel of ${\mathcal{A}}_1$, linear in $\varphi$, and $j_3$ is the constant-coefficient bilinear local kernel of $J_3=2{\mathcal{H}}_2$, so that for all smooth ${\gamma }_1,{\gamma }_2,\varphi$ on $M$,

where $J_3(\cdot ,\cdot )$ is the symmetric bilinear form with $J_3[\gamma ]=J_3(\gamma ,\gamma )$. The kernel $u$ is compactly supported on $M^3$, in $\{y_1=y_2\in {\mathcal{D}}_f\cap \{y^0\ge 0\}\}\times {\mathcal{D}}_f$; no spatial decay of ${\gamma }_a$ is needed in (eq:upair), exactly as in XI-F's witness.

Theorem F1 (XI-F). For every frozen switch and every mass there are admissible observation data for which $\pm \mathrm{d}t$ lies in the wavefront set of this kernel at $\Sigma$. The kernel leaves ${\mathcal{F}}_{\mathrm{mc},\mathrm{pol}}$. The failure is not finite rank and not a loop.

The data of that theorem, called the F1 data below, are: $t_g:=\min \{t:(t,\mathbf{x})\in \mathrm{supp}g\}>0$; $f\in {C_c}^{\infty }({M_T}^{\circ })$, $f\ge 0$, $f\not\equiv 0$, $\mathrm{supp}f\subset (0,t_g)\times {\mathbb{R}}^3$, $\cos (mt)>0$ on $\mathrm{supp}f$; $\varphi ={\varphi }_m$ with ${\varphi }_m=\sin (mt)/m$ for $m>0$ and ${\varphi }_m=t$ for $m=0$. XI-F's Step 1 computes, for these data,

the last because ${\beta }^{0i}={{\mathsf{E}}_0}^{\mathrm{A}}(-\frac{1}{2}{\sigma }_{\epsilon }{\partial }_i{b}^0)$ with $b^0$ of compact spatial support per slice. $B$ is smooth on $[0,T]$ and vanishes near $T$.

The open remark (XI-F). On the free shell, a single-slot boundary trace equals a bulk functional with a source flat at $\Sigma$, hence has a smooth representative. The boundary term carried by $u$ is a bilinear form in the Cauchy data of both metric slots, weighted by $\beta {|}_{\Sigma }$. Whether it has a representative satisfying the microcausal exclusion after both slots are restricted to free data is not proved.

${\mathcal{F}}_{\mathrm{mc},\mathrm{pol}}$ consists of finite graded polynomials $F(\Phi )={\sum }_n⟨u_n,{\Phi }^{\otimes n}⟩$ on ambient configurations on $M$, with compact, graded symmetric distributional kernels $u_n$ on $M^n$ satisfying

where a total zero covector is never a wavefront point and finite field indices do not change the criterion. The free shell ${\mathcal{C}}_{\mathrm{sol}}$ is

on $M$; smooth slab solutions extend to $M$ by their Cauchy data. Admission is $Q_H(F)=[F{|}_{{\mathcal{C}}_{\mathrm{sol}}}]\in {\mathcal{E}}_H$. XI-E admitted ${\mathcal{B}}_f$ and $\mathcal{T}$; it did not admit ${\mathcal{C}}_3$.

Since $\mathcal{P}$ is an invertible constant matrix, $-\square \mathcal{P}\gamma =0$ is $\square \gamma =0$ componentwise: every $\gamma =ϵe^{ik\cdot y}$ with constant symmetric $ϵ$ and null $k$ is a free metric configuration, for every polarization $ϵ$.

$S_{\mathrm{EH}}=-\frac{2}{{\kappa }^2}\int \sqrt{-q}R[q]$, $q=\eta +\kappa \gamma$, $\sqrt{-q}{G}^{\mu \nu }(q)=\kappa {\mathcal{H}}_1+{\kappa }^2{\mathcal{H}}_2+O({\kappa }^3)$, $A\gamma =2{\mathcal{H}}_1$, $J_3=2{\mathcal{H}}_2$, with XI-E's finite prescription for ${\mathcal{H}}_2$ XIE. Two structural facts are used: $J_3$ is a bilinear differential polynomial with constant coefficients, every monomial of which carries exactly two derivatives; and the contracted Bianchi identity ${\nabla }_{\mu }{G}^{\mu \nu }=0$, written as ${\partial }_{\mu }(\sqrt{-q}{G}^{\mu \nu })+\sqrt{-q}{{\Gamma }^{\nu }}_{\mu \rho }{G}^{\mu \rho }=0$, gives at order ${\kappa }^2$

consistent with XI-E's gravitational Noether coefficient.

XI-F's classical ledger is retained and not re-proved:

and, when $A\gamma =0$ on $U$, ${{\mathcal{A}}_2}^{(3)}[\gamma +K\xi ]-{{\mathcal{A}}_2}^{(3)}[\gamma ]={\mathcal{A}}_1[K{\zeta }_{\mathrm{gf}};{\varphi }_0]$, zero under the causal plateau condition, with witness value $-I+I+0=0$. Nothing below changes a line of it.

Let $u$ be the frozen kernel of ${\mathcal{C}}_3(f,g)$. A compactly supported distribution $v$ on $M^3$, with two symmetric-tensor slots and one scalar slot and symmetric in the tensor slots, belongs to $[u{]}_{\mathrm{sol}}$ if

Real and complex solutions are equivalent here by bilinearity. We say that $[u{]}_{\mathrm{sol}}$ {meets ${\mathcal{F}}_{\mathrm{mc},\mathrm{pol}}$} if some $v\in [u{]}_{\mathrm{sol}}$ satisfies (eq:mc) with $n=3$.

If $F\in {\mathcal{F}}_{\mathrm{mc},\mathrm{pol}}$ satisfies $F{|}_{{\mathcal{C}}_{\mathrm{sol}}}={\mathcal{C}}_3(f,g){|}_{{\mathcal{C}}_{\mathrm{sol}}}$, then $[u{]}_{\mathrm{sol}}$ meets ${\mathcal{F}}_{\mathrm{mc},\mathrm{pol}}$. Conversely, if $v\in [u{]}_{\mathrm{sol}}$ satisfies (eq:mc), then $F=⟨v,\gamma \otimes \gamma \otimes \varphi ⟩\in {\mathcal{F}}_{\mathrm{mc},\mathrm{pol}}$ and $F{|}_{{\mathcal{C}}_{\mathrm{sol}}}={\mathcal{C}}_3(f,g){|}_{{\mathcal{C}}_{\mathrm{sol}}}$.

Proof. Restrict $F$ to configurations in ${\mathcal{C}}_{\mathrm{sol}}$ with $c=\bar{c}=0$, which is allowed since zero is a free ghost; ${\mathcal{C}}_3$ has no ghost dependence, so the ghost-free part of $F$ already reproduces it. On ${\mathcal{C}}_{\mathrm{sol}}$, substitute $b=D\gamma$ in every $b$-slot: each kernel becomes $(\cdots \otimes D^{\mathsf{T}}\otimes \cdots )u_n$, compact, with wavefront set not enlarged by a differential operator Hormander ((8.1.11)), hence still satisfying (eq:mc). The shell is a linear space closed under the independent scalings $\gamma \mapsto s\gamma$, $\varphi \mapsto r\varphi$; comparing the coefficient of $s^{2}r$ isolates a single symmetric kernel $v$ of slot type $(\gamma ,\gamma ,\varphi )$ with $⟨v,\gamma \otimes \gamma \otimes \varphi ⟩={\mathcal{C}}_3(f,g)[\gamma ,\varphi ]$ on the shell. Polarization gives $v\in [u{]}_{\mathrm{sol}}$. The converse is Definition def:domain.

The question is therefore

Three statements that are not this question: $u\notin {\mathcal{F}}_{\mathrm{mc},\mathrm{pol}}$, already XI-F's Theorem F1; the one-slot rewriting of a boundary trace, true for one slot and not a theorem for two; and the existence of some larger bimodule containing a singular kernel, which XI-E's correction leaves open and which this paper does not touch.

For smooth ${\gamma }_1,{\gamma }_2,\varphi$ on $M$ and $0\le s<T$ put

and

If $\square {\gamma }_1=\square {\gamma }_2=0$, then ${J_3}^{\mu \nu }({\gamma }_1,{\gamma }_2){|}_{\Sigma }=Q^{\mu \nu }(j_1,j_2)$ is a bidifferential form in the Cauchy data $j_a=({\gamma }_a{|}_{\Sigma },{\partial }_t{\gamma }_a{|}_{\Sigma })$, obtained from the jet expression of $J_3$ by replacing every ${{\partial }_t}^2{\gamma }_a$ by $\Delta {\gamma }_a$. Hence $B_{\Sigma }={\int }_{\Sigma }\beta {|}_{\Sigma }\cdot Q(j_1,j_2)$, with the scalar slot entering only through the smooth weight $\beta {|}_{\Sigma }=\beta [f,g,\varphi ]{|}_{\Sigma }$, linearly.

Proof. $J_3$ contains at most second derivatives, and on the shell ${{\partial }_t}^2{\gamma }_a=\Delta {\gamma }_a$. Mixed derivatives ${\partial }_t{\partial }_i{\gamma }_a$ are spatial derivatives of the second datum. The scalar enters $\beta$ linearly through $\alpha [f,g,\varphi ]$.

The scalar slot is a spectator: it carries no derivative of $\varphi$ that could cancel a metric-slot covector, and in every wavefront statement below its covector is zero.

The witness of Section sec:G3 uses pure-trace configurations $\gamma =\psi \eta$. Their cubic source is closed-form.

For smooth $\psi$ and $\gamma =\psi \eta$,

Consequently, for the polarized source,

Proof. $q={\Omega }^2\eta$ with ${\Omega }^2=1+\kappa \psi$ is conformally flat. For $\tilde{g}={\Omega }^{2}g$ in four dimensions with $g$ flat and $w=\ln \Omega$, the standard conformal transformation of the Ricci tensor gives

hence ${\tilde{G}}_{\mu \nu }=-2{\partial }_{\mu }{\partial }_{\nu }w+2{\eta }_{\mu \nu }\square w+2{\partial }_{\mu }w{\partial }_{\nu }w+{\eta }_{\mu \nu }{\partial }_{\rho }w{\partial }^{\rho }w$. Since $q^{\mu \nu }={\Omega }^{-2}{\eta }^{\mu \nu }$ and $\sqrt{-q}={\Omega }^4$, $\sqrt{-q}{G}^{\mu \nu }={\eta }^{\mu \alpha }{\eta }^{\nu \beta }{\tilde{G}}_{\alpha \beta }$ with no further $\Omega$ factor. Expand $w=\frac{1}{2}\ln (1+\kappa \psi )=\frac{\kappa }{2}\psi -\frac{{\kappa }^2}{4}{\psi }^2+O({\kappa }^3)$, so that ${\partial }_{\mu }w=\frac{\kappa }{2}{\partial }_{\mu }\psi -\frac{{\kappa }^2}{2}\psi {\partial }_{\mu }\psi$ and ${\partial }_{\mu }{\partial }_{\nu }w=\frac{\kappa }{2}{\partial }_{\mu }{\partial }_{\nu }\psi -\frac{{\kappa }^2}{2}({\partial }_{\mu }\psi {\partial }_{\nu }\psi +\psi {\partial }_{\mu }{\partial }_{\nu }\psi )$, up to $O({\kappa }^3)$. The order-$\kappa$ coefficient is (eq:H1trace); the order-${\kappa }^2$ coefficient is $(\partial \psi \partial \psi +\psi \partial \partial \psi )-\eta (|\partial \psi {|}^2+\psi \square \psi )+\frac{1}{2}\partial \psi \partial \psi +\frac{1}{4}\eta |\partial \psi {|}^2$, which is (eq:H2trace). Polarizing $2{\mathcal{H}}_2$ gives (eq:J3trace).

Two checks. First, by (eq:algebra), $\mathcal{P}(\psi \eta )=-\psi \eta$, $D^{\mu }(\psi \eta )=-{\partial }^{\mu }\psi$, $D^{\ast }D(\psi \eta )=\partial \partial \psi -\frac{1}{2}\eta \square \psi$, so $A(\psi \eta )=\square \psi \eta -2\partial \partial \psi +\eta \square \psi =2{\mathcal{H}}_1[\psi \eta ]$, as required by $A\gamma =2{\mathcal{H}}_1$. Second, XI-F's witness $\gamma =t\eta$ gives $q=(1+\kappa t)\eta$, a spatially flat Robertson–Walker metric in conformal time with $a^2=1+\kappa t$, for which $G_{00}=3(a^{\prime }/a{)}^2$; its ${\kappa }^2$ coefficient is $\frac{3}{4}$, and (eq:H2trace) gives ${{\mathcal{H}}_2}^{00}[t\eta ]=\frac{3}{2}-\frac{3}{4}=\frac{3}{4}$.

Let ${\gamma }_a={\psi }_a\eta$ with $\square {\psi }_a=0$ and Cauchy data $({\psi }_a,{\dot{\psi }}_a){|}_{\Sigma }$. Then $B_{\Sigma }={\int }_{{\mathbb{R}}^3}[{\beta }_{00}{J}^{00}+2{\beta }_{0i}{J}^{0i}+{\beta }_{ij}{J}^{ij}](0,\mathbf{x}){\mathrm{d}}^{3}x$ with

all evaluated at $t=0$. For the F1 data the ${\beta }_{ij}$ term is absent.

Proof. Insert $\square {\psi }_a=0$ and ${{\partial }_t}^2{\psi }_a=\Delta {\psi }_a$ into (eq:J3trace), with ${\partial }^0={\partial }_t$, ${\partial }^i=-{\partial }_i$, ${\eta }^{ij}=-{\delta }^{ij}$.

For the spatially homogeneous sector ${\gamma }_{0\mu }=0$, ${\gamma }_{ij}={\gamma }_{ij}(t)$, XI-F's Lemma gives ${J_3}^{0i}=0$ and ${J_3}^{00}=q(\dot{\gamma },\dot{\gamma })$, $q(a,b)=\frac{1}{4}(\mathrm{tr}a\mathrm{tr}b-a_{ij}{b}_{ij})$, so there $B_{\Sigma }={\int }_{\Sigma }{\beta }_{00}q({\dot{\gamma }}_1,{\dot{\gamma }}_2)$. The full twenty-component form of $Q$ is XI-E's ${\mathcal{H}}_2$ with ${{\partial }_t}^2\mapsto \Delta$; the witness needs only the pure-trace sector printed above.

For constant symmetric ${ϵ}_1,{ϵ}_2$ and vectors $k_1,k_2$ let ${\mathsf{q}}^{\mu \nu }({ϵ}_1,k_1;{ϵ}_2,k_2)$ be defined by

Since every monomial of $J_3$ has exactly two derivatives, $\mathsf{q}$ is real, symmetric under $1\leftrightarrow 2$, polynomial, and homogeneous of degree two in $(k_1,k_2)$ jointly: $\mathsf{q}({ϵ}_1,\lambda k_1;{ϵ}_2,\lambda k_2)={\lambda }^2\mathsf{q}({ϵ}_1,k_1;{ϵ}_2,k_2)$. The operator $Q$ of Lemma lem:cauchyred is $\mathsf{q}$ with $k_a\mapsto -i\partial$ acting on ${\gamma }_a$, followed by ${{\partial }_t}^2\mapsto \Delta$.

For null $k_1,k_2$,

and it satisfies the Bianchi relation $(k_1+k_2{)}_{\mu }{\mathsf{q}}^{\mu \nu }=-(k_1\cdot k_2)(k_1+k_2{)}^{\nu }$. For the counter-propagating pair ${k_1}^{\mu }=(1,\mathbf{n})$, ${k_2}^{\mu }=(1,-\mathbf{n})$, $|\mathbf{n}|=1$, one has $k_1+k_2=2\mathrm{d}t$ as covectors, $k_1\cdot k_2=2$, and

Proof. Insert ${\psi }_a=e^{i{k}_a\cdot y}$ into (eq:J3trace): ${\partial }^{\mu }{\psi }_1{\partial }^{\nu }{\psi }_2=-{k_1}^{\mu }{k_2}^{\nu }{e}^{i(k_1+k_2)\cdot y}$, ${\psi }_1{\partial }^{\mu }{\partial }^{\nu }{\psi }_2=-{k_2}^{\mu }{k_2}^{\nu }{e}^{i(k_1+k_2)\cdot y}$, $\square {\psi }_a=0$. This is (eq:qtrace). Contracting with $(k_1+k_2{)}_{\mu }$ and using ${k_a}^2=0$, so that $(k_1+k_2)\cdot k_a=k_1\cdot k_2=:\sigma$, gives $-\frac{3}{2}\sigma (k_1+k_2{)}^{\nu }-\sigma (k_1+k_2{)}^{\nu }+\frac{3}{2}\sigma (k_1+k_2{)}^{\nu }=-\sigma (k_1+k_2{)}^{\nu }$. Independently, (eq:bianchi) polarized reads ${\partial }_{\mu }{J_3}^{\mu \nu }({\gamma }_1,{\gamma }_2)=-\frac{1}{2}[{\Gamma }_1{}_{\mu \rho }^{\nu }[{\gamma }_1]A^{\mu \rho }{\gamma }_2+(1\leftrightarrow 2)]$; for ${\gamma }_a={\psi }_a\eta$ with $\square {\psi }_a=0$, $A{\gamma }_a=-2\partial \partial {\psi }_a$ and ${\Gamma }_1{}_{\mu \rho }^{\nu }[{\psi }_1]A^{\mu \rho }{\gamma }_2=-2{\partial }_{\mu }{\psi }_1{\partial }^{\mu }{\partial }^{\nu }{\psi }_2$, which on plane waves is $2i\sigma {k_2}^{\nu }{e}^{i(k_1+k_2)\cdot y}$; the right side becomes $-i\sigma (k_1+k_2{)}^{\nu }{e}^{i(k_1+k_2)\cdot y}$ and the left side $i(k_1+k_2{)}_{\mu }{\mathsf{q}}^{\mu \nu }{e}^{i(k_1+k_2)\cdot y}$. The two computations agree. For the counter-propagating pair, ${k_1}^0{k_2}^i+{k_2}^0{k_1}^i=0$, ${k_1}^0{k_1}^i+{k_2}^0{k_2}^i=0$, ${k_1}^i{k}_2^j+{k_2}^i{k}_1^j=-2n^i{n}^j$, ${k_1}^i{k}_1^j+{k_2}^i{k}_2^j=2n^i{n}^j$, and (eq:qcounter) follows.

If both polarizations are de Donder transverse, $k_a\cdot \mathcal{P}{ϵ}_a=0$, then $A({ϵ}_a{e}^{i{k}_a\cdot y})=0$ by (eq:algebra), and (eq:bianchi) gives $(k_1+k_2{)}_{\mu }{\mathsf{q}}^{\mu \nu }=0$. For $k_1+k_2=2\mathrm{d}t$ this is ${\mathsf{q}}^{0\nu }=0$: linearized Einstein pairs see only ${\beta }_{ij}$, which vanishes for the F1 data by (eq:F1data). The pure-trace polarization $ϵ=\eta$ has $k\cdot \mathcal{P}\eta =-k\ne 0$; it lies on XI-E's gauge-fixed shell $\square \gamma =0$ and off the Einstein shell. This is XI-A's distinction between the two shells, met at the level of the detector.