Landmark X-A Classical covariance of one fixed-regulator switched relational scalar through order κ 2

We freeze one classical prescription. A minimally coupled scalar lives on the formal metric jet $q=\eta +\kappa h+{\kappa }^{2}k$; the curved-minus-flat scalar density is multiplied by one compact switch $g$ before variation. Four coordinate functions are defined by the divergence-form equation of an edge-regulated auxiliary density ${\mathfrak{A}}_{\epsilon }[q]={\eta }^{-1}+{\sigma }_{\epsilon }({\mathfrak{A}}_q-{\eta }^{-1})$, where ${\sigma }_{\epsilon }$ is an explicit smooth time ramp of width $\epsilon$ and ${\mathfrak{A}}_q$ is the metric density of $q$, with fixed Cauchy data on the initial surface. The gauge algebra consists of vector fields supported in a plateau $U$ whose closure lies inside $\{g=1\}\cap \{t>\epsilon \}$.

The regulator preflight is passed exactly: the regulated first coordinate coefficient returns $\xi$ on every pure gauge $K\xi$ of the frozen algebra. An explicit witness shows that the same identity fails for gauge parameters entering the edge zone $0<t<\epsilon$, so the plateau condition is part of the freeze and not removable.

The relational scalar $\Phi =\varphi \circ {{\mathsf{X}}_{\mathsf{q}}}^{-1}$, smeared by one fixed label-space test, is proved gauge invariant through ${\kappa }^2$ at fixed $\epsilon >0$, off shell in the metric and in the scalar, by two independent routes: covariance of the regulated coordinate operator and of the switched Euler operator under plateau-supported transformations together with coefficientwise Cauchy uniqueness; and an exact varied-action identity in which every switch term is displayed and cancels only through $g=1$ and $\mathrm{d}g=0$ on the gauge support. The switched scalar solution with fixed free input intertwines the same action, so the on-shell evaluation is invariant as well. The pure-gauge two-jet and a sourced diagnostic are checked. The linear coordinate kernel is smooth: the regulator replaces the hypersurface delta of the sharp prescription by a ramp term.

No quantum product, quantum Ward identity, sharp-regulator limit, boundary algebra, admission topology, or identification with the VII–IX or cone constructions is claimed.

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

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

claim: X-A only, ending (1) Constructed{ For the frozen switched action, regulator, Cauchy prescription, configuration domain, and plateau-supported gauge algebra of Section sec:sheet, the declared relational scalar is gauge invariant through order ${\kappa }^2$ at fixed $\epsilon >0$ (Theorem thm:main). No quantum product, quantum Ward identity, sharp-regulator limit, boundary algebra, or identification with the VII–IX or cone constructions is claimed.}

A relational scalar built from Cauchy-prescribed harmonic coordinates is a classical object before it is a quantum insertion. The IX track admitted the sharp zero-Cauchy insertion to a boundary-sensitive carrier; VII-B excluded the same insertion from an off-shell microcausal polynomial domain. Neither manuscript constructed a regulated classical coordinate problem, and neither proved covariance of such a problem. That identity has to be proved, or obstructed, here.

X-A therefore opens a new classical line. One switched classical prescription, one explicit regulator, one Cauchy prescription, and one plateau-supported gauge algebra are frozen in Section sec:sheet. The relational scalar is expanded through ${\kappa }^2$, and its gauge covariance, including regulator and Cauchy data, is decided on the frozen algebra.

1. Constructed. Equations, regulator, data, and observable identities are proved through ${\kappa }^2$ at fixed $\epsilon >0$.

2. Proved obstructed. An explicit nonzero switch or regulator defect holds on the frozen group.

3. Unresolved. The regulator is unspecified, uniqueness is missing, or the second-order cancellation is unfinished.

The ending reached is (1). The preflight of Section sec:preflight decides it: the frozen regulator is exact on every pure gauge of the frozen algebra, and the full second-order mechanism is then available. A proved defect appears only for gauge parameters outside the frozen algebra, in the edge zone of the regulator; it is recorded as the boundary of the freeze, not as a theorem on the frozen group.

metric does not mean linear gauge action. The metric jet $\eta +\kappa h+{\kappa }^{2}k$ is retained with its complete second-order pullback law. The second metric coefficient $k$ transforms by ${\mathcal{L}}_{\xi }h+K\zeta +\frac{1}{2}{{\mathcal{L}}_{\xi }}^2\eta$, and this term enters the second-order coordinate coefficient. No second-order relational expansion is combined with only the first term of the metric transformation.

regulator passes its own covariance test. Smoothing the dressing is not assumed to preserve the gauge identity. The frozen regulator multiplies the deviation of the metric density from flat by a smooth time ramp ${\sigma }_{\epsilon }$ inside the divergence-form coordinate operator. For a pure gauge supported where ${\sigma }_{\epsilon }=1$ and $\mathrm{d}{\sigma }_{\epsilon }=0$, the regulated first coefficient equals the exact first coefficient and returns $\xi$. For a pure gauge supported in the ramp zone it does not. Both statements are proved in Section sec:preflight before any second-order formula is written.

Fix the data of Section sec:sheet: the slab $M_T$, the formal metric jet $q=\eta +\kappa h+{\kappa }^{2}k$, the switch $g$, the regulator ${\sigma }_{\epsilon }$, the plateau $U$ with $\overline{U}\subset P_g\cap \{\epsilon <t<T\}$, the gauge group ${\mathcal{G}}_U$, the switched scalar action ${S_g}^{\mathrm{sc}}$, the regulated coordinate prescription of Definition def:coordinates, and one label-space test $f\in {C_c}^{\infty }({M_T}^{\circ })$. Then:

1. the regulated coordinate problem has a unique formal solution ${\mathsf{X}}_q=x+\kappa {\mathsf{X}}_1+{\kappa }^2{\mathsf{X}}_2(\mathrm{mod}{\kappa }^3)$, with explicit zero-data Cauchy coefficients;

2. ${\mathsf{X}}_1[K\xi ]=\xi$ for every $\xi \in {\mathfrak{g}}_U$;

3. for every $F\in {\mathcal{G}}_U$, ${\mathsf{X}}_{F^{\ast }q}=F^{\ast }{\mathsf{X}}_q(\mathrm{mod}{\kappa }^3)$, with coefficient laws (eq:X1law)–(eq:X2law);

4. for every configuration $(h,k,\varphi )$ and every $F\in {\mathcal{G}}_U$,

   $${\mathcal{O}}_{f,g,\epsilon }[F^{\ast }q,F^{\ast }\varphi ]={\mathcal{O}}_{f,g,\epsilon }[q,\varphi ](\mathrm{mod}{\kappa }^3);$$

5. the switched scalar solution with fixed free input satisfies ${\rho }_g[F^{\ast }q;{\varphi }_0]=F^{\ast }{\rho }_g[q;{\varphi }_0](\mathrm{mod}{\kappa }^3)$, and ${\mathcal{O}}_{f,g,\epsilon }[q,{\rho }_g[q;{\varphi }_0]]$ is invariant under ${\mathcal{G}}_U$;

6. the varied switched action is an explicit sum of terms carrying a factor $1-g$ or $\partial g$, and vanishes identically on ${\mathcal{G}}_U$;

7. the coefficients depend on the configuration only in ${\mathcal{D}}_f=J^{-}(\mathrm{supp}f)\cap M_T$, are continuous in the smooth topologies at fixed $\epsilon$, and their first functional derivatives at fixed configuration are smooth compactly supported kernels.

No quantum object is part of the statement.

The proof occupies Sections sec:preflight–sec:tests; the handoff statements of (vii) are in Section sec:decision.

These are the statements actually delivered by the earlier manuscripts, copied in their own conventions. None is a theorem about the regulated operator introduced below. Shared vocabulary is not a transfer.

-A, copied. Signature $(-,+,+,+)$. Minimally coupled scalar on the formal metric $\mathbf{g}=\eta +\kappa h+{\kappa }^{2}k$, compactly switched scalar action, unswitched harmonic coordinates with fixed Cauchy data. Admitted transformations are supported strictly inside a region where the interaction switch equals one and are the identity near the initial surface. On that restricted algebra: exact first-order cancellation and a second-order classical intertwiner, with all switch derivatives retained. The proof uses covariance of the switched density-valued Euler operator and uniqueness of the coefficientwise Cauchy problems. The first-order plateau identity is

for $\xi$ supported in the plateau. Outside that support,

and a higher-order coefficient cannot repair this first-order defect. The metric equations in the named Einstein–scalar jet are unswitched; the scalar equation is switched. A causal support bound and a classical comparison map between switches are proved. Equality for switches sharing only an observation plateau is not asserted. No renormalized product, quantum Ward identity, or counterterm separation.

, copied. Signature $(-,+,+,+)$. Second-order classical descent for a specified relational Einstein–scalar jet, and a separate exact first-order compact-switch defect for the identification $(h,{\varphi }_0)\sim (h+K\xi ,{\varphi }_0)$:

with $[P,g]u=({\square }_{\eta }g)u+2({\partial }^{\mu }g){\partial }_{\mu }u$. No correction beginning at second order removes it while the identification and the switch remain as stated. No renormalized response and no quantum Ward identity.

, copied; a different equation space. Signature $(-,+,+,+)$. An unchanged free-product quotient does not receive the Poisson-dressed symbol. Moving the scalar does not preserve the massive free shell for general compact gauges. On a specified coupled first-order jet with forced scalar correction the functional descends, with a unique retarded harmonic section. A compact coupled Ward pair is nonzero. No quantum representation.

-B and IX-A, non-transfer. VII-B (signature $(+,-,-,-)$) excludes the sharp zero-Cauchy insertion from its stated off-shell microcausal polynomial domain: the adjoint $L^{\ast }a=\theta C^{\ast }b+\delta (t)\mathsf{B}(b{|}_{{\Sigma }_0})$ has timelike conormal wavefront covectors. IX-A (signature $(-,+,+,+)$) admits that same insertion in a different carrier with boundary-trace topology ${\tau }_{\Sigma }$, and proves that unchanged regular approximants do not converge to it in that topology. Neither defines a regulated classical coordinate problem, and neither constructs a quantum product. The regulated kernel of Lemma lem:kernel below is not identified with either sharp object.

prerequisites. V's external $R_{1,1}$, oscillator coefficient, and Newton–wave cubic; II's finite contact matrix; IV's three Hilbert models; VII-E/F and VIII-A–C conical certificates and token promises; IX-Free-2's partial free product; IX-D-v3's conditional subtraction; IX-E-2's $\mathrm{Im}K$ formula; IX-E-3's conditional support bridge. No arrow from those manuscripts into X-A is a theorem.

The relational-coordinate literature BFR,Frob is comparison background only. It does not supply covariance of this switched, regulated prescription.

The first deliverable is a sheet. Every later identity refers to it.

Indices are raised and lowered with $\eta$; parentheses on index pairs include the factor $\frac{1}{2}$. Put

Equality modulo ${\kappa }^3$ means coefficient equality; no numerical remainder estimate is implied. For a covector field ${\xi }_{\mu }$ and vector field ${\xi }^{\mu }={\eta }^{\mu \nu }{\xi }_{\nu }$,

Here $D_{\xi }$ is the Lie derivative of a scalar density represented relative to ${\mathrm{d}}^{4}x$.

The configuration space is

All fields are smooth on a neighbourhood of $M_T$ and are used on $M_T$. The case of a $\kappa$-independent scalar is ${\varphi }_1={\varphi }_2=0$.

Write $h={\eta }^{\mu \nu }{h}_{\mu \nu }$ for the trace where no confusion can arise, and set

The symmetric bilinear polarizations $a_2[h,h^{\prime }]$ and $b_2[h,h^{\prime }]$ are defined by $a_2[h,h]=a_2[h]$, $b_2[h,h]=b_2[h]$.

With $\sqrt{-q}=\sqrt{-\det q}$ and ${{\mathfrak{A}}_{\mathfrak{q}}}^{\mu \nu }=\sqrt{-q}{q}^{\mu \nu }$,

Proof. The inverse metric is $q^{\mu \nu }={\eta }^{\mu \nu }-\kappa h^{\mu \nu }+{\kappa }^2(h^{\mu }{}_{\alpha }{h}^{\alpha \nu }-k^{\mu \nu })$. Expanding $\exp (\frac{1}{2}\mathrm{tr}\log (\mathrm{id}+\kappa {\eta }^{-1}h+{\kappa }^2{\eta }^{-1}k))$ gives (eq:volexp). Multiplication gives (eq:Aexp).

Two identities used repeatedly follow from (eq:a):

For the second, $\mathrm{tr}K\xi =2{\partial }_{\rho }{\xi }^{\rho }$ and ${\partial }_{\nu }(K\xi {)}^{\nu \mu }=\square {\xi }^{\mu }+{\partial }^{\mu }{\partial }_{\rho }{\xi }^{\rho }$.

Fix

Fix the ramp

which is smooth, nondecreasing, and satisfies ${\chi }^{\prime }(s)>0$ for $0<s<1$: with $a=e^{-1/s}$, $b=e^{-1/(1-s)}$, ${\chi }^{\prime }=(a^{\prime }b-a{b}^{\prime })/(a+b{)}^2$, $a^{\prime }=a/s^2>0$, $b^{\prime }=-b/(1-s{)}^2<0$. Fix $0<\epsilon <T$ and

Then ${\sigma }_{\epsilon }=0$ on $\{t\le 0\}$, ${\sigma }_{\epsilon }=1$ on $\{t\ge \epsilon \}$, ${\partial }_{\mu }{\sigma }_{\epsilon }={{\delta }^0}_{\mu }{\sigma }_{\epsilon }^{\prime }$ with ${\sigma }_{\epsilon }^{\prime }(t)={\epsilon }^{-1}{\chi }^{\prime }(t/\epsilon )$, and ${\sigma }_{\epsilon }^{\prime }>0$ exactly on $\{0<t<\epsilon \}$.

Fix a nonempty open, relatively compact $U$ with

and set

It is a Lie algebra under the bracket of vector fields. For $\xi ,\zeta \in {\mathfrak{g}}_U$ let $V_{\kappa }=\kappa \xi +{\kappa }^2\zeta$ and let $F_{\kappa }={\mathrm{Fl}}_{V_{\kappa }}^1$ be its time-one flow, a diffeomorphism of $M$ for every real $\kappa$.

1. $F_{\kappa }$ is the identity outside a compact subset of $U$; in particular on a neighbourhood of $\{t\le \epsilon \}$ and of $\{t\ge T\}$. It maps $U$ onto $U$ and $M_T$ onto $M_T$.

2. $F_{\kappa }^{\ast }g=g$ and $F_{\kappa }^{\ast }{\sigma }_{\epsilon }={\sigma }_{\epsilon }$.

3. For every smooth tensor field or tensor density $\mathsf{T}$,

   $$F_{\kappa }^{\ast }\mathsf{T}=\mathsf{T}+\kappa {\mathcal{L}}_{\xi }\mathsf{T}+{\kappa }^2({\mathcal{L}}_{\zeta }+\frac{1}{2}{{\mathcal{L}}_{\xi }}^2)\mathsf{T}+O({\kappa }^3),$$

   with the Lie derivative of the appropriate type and the remainder $O({\kappa }^3)$ in the smooth topology on compact sets.

4. The formal operators $\exp (\kappa {\mathcal{L}}_{\xi }+{\kappa }^2{\mathcal{L}}_{\zeta })(\mathrm{mod}{\kappa }^3)$, $\xi ,\zeta \in {\mathfrak{g}}_U$, form a group ${\mathcal{G}}_U$ under composition.

Proof. (a) The flow of a vector field supported in a compact $K_0\subset U$ is the identity outside $K_0$ and preserves every set containing $K_0$ whose complement it fixes. (b) On $U$ both $g$ and ${\sigma }_{\epsilon }$ equal one by (eq:U); outside $U$, $F_{\kappa }=\mathrm{id}$. (c) Let $\Psi (s)=({\mathrm{Fl}}_{V_{\kappa }}^s{)}^{\ast }\mathsf{T}$. Then ${\partial }_s\Psi =({\mathrm{Fl}}_{V_{\kappa }}^s{)}^{\ast }{\mathcal{L}}_{V_{\kappa }}\mathsf{T}$, so

because ${{\mathcal{L}}_{{\mathcal{V}}_{\kappa }}}^3=O({\kappa }^3)$. Insert ${\mathcal{L}}_{V_{\kappa }}=\kappa {\mathcal{L}}_{\xi }+{\kappa }^2{\mathcal{L}}_{\zeta }$. (d) By the Baker–Campbell–Hausdorff formula modulo ${\kappa }^3$,

and $[\xi ,{\xi }^{\prime }]\in {\mathfrak{g}}_U$. Inverses are obtained with $(\xi ,\zeta )\mapsto (-\xi ,-\zeta )$.

For $\kappa$-dependent objects $\mathsf{T}={\sum }_j{\kappa }^j{\mathsf{T}}_j$ the formal pullback is applied coefficientwise and truncated; this is the Taylor expansion of the genuine pullback of the genuine $\kappa$-dependent field by $F_{\kappa }$. Explicitly,

all modulo ${\kappa }^3$. Equation (eq:metricpull) is the complete second-order metric law. Setting $\zeta =0$ and reading the $\lambda$-linear part of the one-parameter family $\exp (\lambda \kappa {\mathcal{L}}_{\xi })$ gives the infinitesimal law ${\delta }_{\xi }h=K\xi$, ${\delta }_{\xi }k={\mathcal{L}}_{\xi }h$, equivalently, in the notation $q=\eta +\kappa h$ with $\kappa$-dependent $h$, ${\delta }_{\xi }h=K\xi +\kappa {\mathcal{L}}_{\xi }h$.

Let $\mu \ge 0$, $s\in C^{\infty }(M_T)$, $u_0,u_1\in C^{\infty }({\mathbb{R}}^3)$. There is a unique $u\in C^{\infty }(M_T)$ with

For $y\in M_T$, $u(y)$ depends only on the restrictions of $s,u_0,u_1$ to $J^{-}(y)\cap M_T$. If $s,u_0,u_1$ vanish outside a compact set $K_0$ and its causal future, then $\mathrm{supp}u\subset J^{+}(K_0)\cap M_T$. For every compact $K\subset M_T$ and every $N$ there are $N^{\prime }$ and $C$ with

Proof. For compactly supported smooth data and source, existence, uniqueness, the support bound, and continuous dependence are standard BGP (Chapter 3). Finite propagation for the constant-coefficient operator $\square +{\mu }^2$ is the energy argument of Evans (2.4.3), with ${\mu }^2{u}^2$ included in the energy density: a smooth solution of the homogeneous equation whose data vanish on $J^{-}(y)\cap \Sigma$ and whose source vanishes on $J^{-}(y)\cap M_T$ vanishes at $y$.

For general data, fix a compact $K\subset M_T$. The set ${\mathcal{D}}_K=J^{-}(K)\cap M_T$ is compact. Choose $\varrho \in {C_c}^{\infty }(M)$ equal to one on a neighbourhood of ${\mathcal{D}}_K$, and let $u_K$ be the compact-data solution for $(\varrho s,\varrho u_0,\varrho u_1)$. If ${\varrho }^{\prime }$ is another such cutoff, the difference $u_K-u_K^{\prime }$ has source and data vanishing on ${\mathcal{D}}_K$, hence vanishes on $K$ by finite propagation. Define $u=u_K$ on $K$. Compatibility over different $K$ follows in the same way, so $u$ is a well-defined smooth solution. Uniqueness and the dependence statement are finite propagation again. The estimate is the compact-case estimate applied to $(\varrho s,\varrho u_0,\varrho u_1)$.

Write ${\mathsf{E}}_{\mu }s$ for the solution with zero data; ${\mathsf{E}}_0$ is the wave case and ${\mathsf{E}}_m$ the Klein–Gordon case. In spatial Fourier variables, with ${\Omega }_{\mu }=\sqrt{-\Delta +{\mu }^2}$,

interpreted locally by finite propagation; the sine quotient has its continuous value $t-s^{\prime }$ at zero frequency when $\mu =0$. For a compactly supported smooth $a$ on $M$, ${\mathsf{E}}_{\mathrm{A}}a$ denotes the advanced solution of $(\square +{\mu }^2)b=a$, smooth and supported in $J^{-}(\mathrm{supp}a)$ BGP (Chapter 3).

Let $a\in {C_c}^{\infty }({M_T}^{\circ })$ and $s\in C^{\infty }(M_T)$, scalar or componentwise. Then

Proof. Put $u={\mathsf{E}}_{\mu }s$ and $b={\mathsf{E}}_{\mathrm{A}}a$. The set $J^{-}(\mathrm{supp}a)\cap M_T$ is compact and $b$ vanishes on a neighbourhood of $\{t\ge T\}$ in $M_T$, since $\mathrm{supp}a\subset \{t\le T-\delta \}$ for some $\delta >0$. Green's identity on $M_T$ gives

At $t=T$ the integrand vanishes because $b$ does; at $t=0$ it vanishes because $u$ has zero data. There is no spatial boundary term because $b$ is compactly supported in each slice.

Fix $f\in {C_c}^{\infty }({M_T}^{\circ };\mathbb{R})$. It is a function of relational labels. It is never transformed, and no spacetime volume density is inserted when it is paired with the relational scalar.

1. The density representative is frozen before multiplication by $g$. The switched scalar density is ${\mathfrak{L}}_g=(1-g){\mathfrak{L}}_{\eta }+g{\mathfrak{L}}_q$ with ${\mathfrak{L}}_q$ the natural curved density (eq:Lq). An integration by parts performed before switching need not equal one performed afterwards; the multiplied-equation prescription $g\cdot (\text{curved operator})$ is not the Euler operator of the switched action (Appendix app:audit).

2. The convention is not $q=\eta +\kappa gh$. That substitution produces different powers of $g$ and different switch-derivative terms. It is not used anywhere below.

The coordinate prescription uses an auxiliary density different from the density of the switched action: the edge-regulated density ${\mathfrak{A}}_{\epsilon }[q]$ of (eq:Aeps), regulated by ${\sigma }_{\epsilon }$ and not by $g$. This split is declared here. It is the analogue of VII-A's split between switched scalar action and unswitched harmonic equation, with the unswitched equation replaced by the edge-regulated one.

Let ${\eta }^{-1}$ denote the constant tensor density ${{\mathfrak{A}}_{\eta }}^{\alpha \beta }={\eta }^{\alpha \beta }$ (note $\sqrt{-\eta }=1$).

For the formal metric $q=\eta +\kappa h+{\kappa }^{2}k$, set

The output of ${\mathsf{D}}_{\epsilon }[q]$ is a scalar density relative to ${\mathrm{d}}^{4}x$.

The coordinate functions ${{\mathsf{X}}_{\mathsf{q}}}^{\mu }=x^{\mu }+\kappa {{\mathsf{X}}_1}^{\mu }+{\kappa }^2{{\mathsf{X}}_2}^{\mu }$, $\mu =0,1,2,3$, are the formal solutions of

In the notation of the production plan, the operator is ${\mathcal{H}}_{g,\epsilon }[h]={\mathsf{D}}_{\epsilon }[q]$, the boundary operator is Cauchy restriction on $\Sigma$, and the data ${d^{\mu }}_{g,\epsilon }=(x^{\mu }{|}_{\Sigma },{{\delta }^{\mu }}_0)$ are independent of the configuration and of $g$.

The fixed background objects are: $\eta$, $\Sigma$, ${\sigma }_{\epsilon }$, and the data. The switch $g$ does not enter. The index $\mu$ labels four scalar functions; it does not indicate a vector field.

${\mathsf{D}}_{\epsilon }[q]=\square +\kappa {\mathcal{V}}_{\epsilon }[h]+{\kappa }^2{\mathcal{W}}_{\epsilon }[h,k](\mathrm{mod}{\kappa }^3)$, with

The regulator enters with at most one derivative; there are no ${\sigma }_{\epsilon }^{\prime \prime }$ terms and no $g$ terms.

Proof. Insert (eq:Aexp) into (eq:Aeps) and expand the divergence; ${\partial }_{\alpha }{\sigma }_{\epsilon }={{\delta }^0}_{\alpha }{\sigma }_{\epsilon }^{\prime }$.

Each ${{\mathsf{X}}_{\mathsf{q}}}^{\mu }$ is a critical point of $\frac{1}{2}\int {\mathfrak{A}}_{\epsilon }[q{]}^{\alpha \beta }{\partial }_{\alpha }{\mathsf{X}}^{\mu }{\partial }_{\beta }{\mathsf{X}}^{\mu }$ under compactly supported variations. The regulated prescription is therefore the action-level switching of VII-A with $g$ replaced by ${\sigma }_{\epsilon }$, not the multiplied equation $\square {\mathsf{X}}_1={\sigma }_{\epsilon }C[h]$. The two differ by the ${\sigma }_{\epsilon }^{\prime }$ term in (eq:V); see Lemma lem:kernel. The density ${\mathfrak{A}}_{\epsilon }[q]$ is the density of a formal auxiliary metric, since in four dimensions $\det \mathfrak{A}=\det q$ and $q^{\mu \nu }={\mathfrak{A}}^{\mu \nu }/\sqrt{-\det \mathfrak{A}}$; this fact is not used.

For $t\le 0$, ${\mathfrak{A}}_{\epsilon }[q]={\eta }^{-1}$; for $t\ge \epsilon$, ${\mathfrak{A}}_{\epsilon }[q]={\mathfrak{A}}_q$. The sharp onset $\theta (t)$ of the zero-Cauchy coupling in the VII-B and IX-A prescriptions is replaced by the ramp ${\sigma }_{\epsilon }$. The Cauchy data remain on $\Sigma =\{t=0\}$; the initial surface is not moved. What changes is the singular structure of the linear coordinate kernel.

For $a\in {C_c}^{\infty }({M_T}^{\circ };T^{\ast }M)$ define ${\mathsf{J}}_{\epsilon }(a)[h]={\int }_{M_T}{a}_{\mu }{{\mathsf{X}}_1}^{\mu }[h]{\mathrm{d}}^{4}x$. Then

The kernel $q_{a,\epsilon }$ is smooth, vanishes on $\{t\le 0\}$, and is supported in $J^{-}(\mathrm{supp}a)\cap \{t\ge 0\}$.

By contrast, the multiplied-equation prescription ${\tilde{\mathsf{X}}}_1={\mathsf{E}}_0({\sigma }_{\epsilon }C[h])$ has kernel $C^{\ast }({\sigma }_{\epsilon }{\mathsf{E}}_{\mathrm{A}}a)={\sigma }_{\epsilon }{C}^{\ast }({\mathsf{E}}_{\mathrm{A}}a)+{\sigma }_{\epsilon }^{\prime }\mathsf{B}({\mathsf{E}}_{\mathrm{A}}a)$, with ${\mathsf{B}}^{\rho \sigma }(v)=-n^{(\rho }{v}^{\sigma )}+\frac{1}{2}{\eta }^{\rho \sigma }{n}_{\mu }{v}^{\mu }$, $n_{\mu }={\partial }_{\mu }t$. The two prescriptions differ by the switch-derivative term ${\sigma }_{\epsilon }^{\prime }\mathsf{B}({\mathsf{E}}_{\mathrm{A}}a)$. Neither is identified with the sharp kernel $\theta C^{\ast }b+\delta (t)\mathsf{B}(b{|}_{\Sigma })$ of VII-B, and no $\epsilon \to 0$ limit is taken here.

Proof. By Lemma lem:Dexp the first coefficient equation is $\square {{\mathsf{X}}_1}^{\mu }=-{\partial }_{\alpha }({\sigma }_{\epsilon }{a_1}^{\alpha \mu }[h])$ with zero data, so ${{\mathsf{X}}_1}^{\mu }=-{\mathsf{E}}_0{\partial }_{\alpha }({\sigma }_{\epsilon }{a_1}^{\alpha \mu }[h])$. Lemma lem:transpose with $b={\mathsf{E}}_{\mathrm{A}}a$ gives

The boundary term at $t=0$ vanishes because ${\sigma }_{\epsilon }(0)=0$; at $t=T$ because $b$ vanishes there; spatially because $b$ is compactly supported in each slice. Inserting (eq:a),

Smoothness and support follow from those of $b$ and ${\sigma }_{\epsilon }$. For the multiplied-equation prescription, $\int a{\mathsf{E}}_0({\sigma }_{\epsilon }Ch)=\int {\sigma }_{\epsilon }bCh=\int C^{\ast }({\sigma }_{\epsilon }b)h$, and the product rule with ${\partial }_{\mu }{\sigma }_{\epsilon }=n_{\mu }{\sigma }_{\epsilon }^{\prime }$ gives the displayed decomposition, exactly as in the VII-B adjoint computation with $\theta$ replaced by ${\sigma }_{\epsilon }$.

Let $\xi \in {C_c}^{\infty }({M_T}^{\circ };TM)$ with $\mathrm{supp}\xi \subset \{t\ge \epsilon \}$. Then

In particular (eq:preflight) holds for every $\xi \in {\mathfrak{g}}_U$. In the notation of the plan, ${\mathcal{E}}_{g,\epsilon }\square \xi =\xi$ with the prescribed data.

Proof. By (eq:a), $a_1[K\xi ]=({\partial }_{\rho }{\xi }^{\rho })\eta -K{\xi }^{\cdot \cdot }$ is supported in $\mathrm{supp}\xi \subset \{t\ge \epsilon \}$, where ${\sigma }_{\epsilon }=1$ and ${\sigma }_{\epsilon }^{\prime }=0$. Hence, by (eq:V) and (eq:C),