XIX-B A Witness Mismatch Between Two Coordinate Cauchy Prescriptions Unswitched Harmonic Coordinates and Prepared Coordinate Completion

We compare the finite-time harmonic coordinate prescription of Landmark VI with the prepared coordinate candidate recorded in XVI-D. We first prove the scalar-wave/density identity needed to relate VI's equation to XIX-A's displayed comparison equation, including the conversion between their second-order coefficient conventions. This authenticates the coordinate equation on their common domain, not the entire predecessor observable.

For the explicit counter-propagating witness source, let $V_{\lambda }$ be its zero-data time integral and retain the response sign $R_{\lambda }=-{\mathcal{P}}_{+}{V}_{\lambda }$. After a declared overall metric-sign transport, VI's first coordinate coefficient has normalized forcing $-{\partial }_t{V}_{\lambda }^{0\mu }$, whereas the prepared candidate has forcing $-{\partial }_t(\sigma {V_{\lambda }}^{0\mu })$. Their difference is the nonzero smooth residual ${\partial }_t((\sigma -1){V_{\lambda }}^{0\mu })$. An explicit compact-test construction detects that residual. The coordinate fields consequently differ on the common slab. The same response, placed as VI's first metric coefficient, also fails VI's vacuum first-metric equation; this domain failure is calculated separately.

The result is a first-order mismatch for the specified direct comparison. It is not VI's scalar-switch defect, a universal obstruction to another connecting construction, or a quantum anomaly. Smooth no-decay uniqueness, operator-domain extension, observable matching, and full-kernel determination retain their separate roles. No second-order or positive-formal-quantum-order term repairs the fixed first-order mismatch.

Keywords: harmonic coordinates; Cauchy prescription; prepared response; trace reversal; relational observables; no-decay uniqueness.

XIX-A separated time-related constructions and left the VI–XVI-D coordinate identification undecided XIXA. Its displayed density equation was a comparison prescription. It was not, merely by its label, the registered VI coordinate construction.

The present paper makes one specified comparison and reaches the mismatch alternative. The comparison is:

1. retain XVI-D's response tensor $R_{\lambda }=-{\mathcal{P}}_{+}{V}_{\lambda }$;

2. regard that smooth response as a metric argument of the first harmonic-coordinate map;

3. transport the metric signature explicitly;

4. compare the resulting first harmonic coefficient with XVI-D's $X_{\mathrm{cand}}$, on the same slab and with the same ordinary coordinate-time zero data.

All mathematical definitions used in this comparison are printed below. Neither a separation edge nor an unavailable companion proof is a premise.

decisive distinction. The supplied VI prescription is unswitched:

The supplied XVI-D candidate uses a prepared density forcing:

Equality must be tested after linearization and sign normalization, not inferred from the words “harmonic”, “retarded”, or “Cauchy”.

result is coefficientwise. The comparison parameter below probes the coordinate prescription. It does not regrade the native gravitational coupling order of the response $R_{\lambda }$, which is bilinear in its two tensor inputs. If a different comparison places that response in VI's second metric coefficient, the additional second-order terms must be included. That different comparison is not decided here.

context. Relational-coordinate constructions and perturbative invariant observables are established subjects BFHPR,Frob. Their existence does not identify two particular boundary or preparation prescriptions. Likewise, Cheung's operational characterization of measurement concerns measurement strategies, not the Cauchy forcing studied here CheungEnter.

false identification of the obstruction. VI's compact-switch defect concerns a switched scalar assignment. It is not an obstruction to existence or uniqueness of VI's coordinate jet. Section sec:switch proves the corresponding classical identity separately; no interaction switch $g$ enters the coordinate mismatch theorem.

Fix $T>0$ and $0<\epsilon <T$. Set

All coefficient fields are smooth on a neighborhood of the slab. No spatial decay is imposed unless it is explicitly stated.

Keep both background signatures visible:

For a covariant symmetric tensor $a$, define

and

For a contravariant tensor $B$, trace reversal means

In four dimensions, ${{\mathcal{P}}_{\pm }}^2=\mathrm{id}$.

The superscript $\mu$ on a coordinate function labels one of four scalar functions. It does not turn that function into a vector component under a change of metric-sign convention.

The registered VI coefficient convention is

without factorials at order two. Its coordinate conditions are

Thus $w,v,{\partial }_{t}w,{\partial }_{t}v$ vanish at $t=0$. The first equation is

These are ordinary coordinate-time data, not metric-dependent unit-normal data. VI's coordinate proposition is stated for every smooth $h,j$, whereas its descended solution-jet observable has the additional equation-space restriction CheungVI (:coordinates, :DK, :descent).

The necessary first metric condition in VI's named ${\mathcal{J}}_T$ is

where its fixed linearized Einstein tensor is

Here lowering in the latter formula uses ${\eta }_{-}$, and parentheses denote symmetrization with factor $1/2$. This normalization is sufficient for our domain test.

The higher metric coefficient, scalar coefficients, and their equations impose further conditions in ${\mathcal{J}}_T$. We do not assert that an arbitrary smooth metric argument extends to that solution-jet space CheungVI (:jet-space).

consequence. Failure of (eq:VI-vacuum) prevents applying VI's solution-jet descent theorem to that input. It does not prevent solving the independent smooth coordinate Cauchy problem (eq:VI-data).

For a formal metric $q$ with invertible zeroth-order coefficient, write

For a scalar $u$, the scalar-wave definition gives

On $M_T$, use the same metric jet, the same initial surface, and the same ordinary coordinate-time data.

Then the density-coordinate equation of XIX-A and the scalar-wave coordinate equation of VI are equivalent coefficientwise through order two. For the conventions used in XIX-A,

The equivalence applies on the shared domain, including XIX-A's spatially compact metric subclass. It does not identify the complete scalar observables, enlarge XIX-A's original domain, or turn its label ${\mathsf{C}}_{\mathrm{VI}}$ into an authentication of every VI construction.

Proof. The zeroth coefficient of ${\rho }_q$ is $1$, so ${\rho }_q$ has an invertible formal power series modulo ${\kappa }^3$. Multiplication by it preserves vanishing modulo ${\kappa }^3$. Equation (eq:density-identity) therefore proves equivalence of the two coordinate equations.

For completeness, in the convention $q={\eta }_{-}+\kappa h+{\kappa }^{2}j$, put

Matrix inversion and determinant expansion give

All mixed indices in (eq:a2) use ${\eta }_{-}$. The density equation for $X=x+\kappa w+{\kappa }^{2}v$ has coefficients

The initial coefficient data are unchanged.

XIX-A wrote the second metric and coordinate coefficients with a factor $1/2$. Substitution of $k=2j$ and $W_2=2v$ therefore yields the displayed conversion. The argument uses equations and data only; it does not compare scalar dynamics or observable domains.

The density method is also the method recorded for VII-A, with no interaction switch in its coordinate equation CheungVIIA,CheungXA. VII-A is a method comparison here, not an existence theorem imported into VI.

The witness below is spatially homogeneous and noncompact. It does not belong to the spatially compact metric subclass originally printed in XIX-A. We consequently evaluate VI's smooth coordinate prescription directly and independently construct its no-decay solution. Lemma lem:A alone is not used to extend the old XIX-A domain.

Let $\sigma \in C^{\infty }(\mathbb{R};\mathbb{R})$ satisfy

No limit in $\epsilon$ is taken. Continuity guarantees a nonempty open interval on which $0<\sigma <1$.

Fix $\lambda >0$ and a Euclidean unit vector $\mathbf{n}\in {\mathbb{R}}^3$. The native plus-signature tensor inputs are

Their product phase is $e^{2i\lambda t}$.

We take the following explicit smooth source as frozen data:

Thus

These formulas are the supplied XVI-B/XVI-D witness source CheungXVIB (:pair) CheungXVID (:Slambda). Neither $Q$ nor $d_1$ contains an additional frequency factor.

The source definition is sufficient for the present proof. We do not derive the full gravitational source polynomial $J_D$, its Euler origin, or its complete tensor-input domain.

Define

Then

Because the source is ambient smooth and flat at $t=0$, extension of $V_{\lambda }$ by zero for $t\le 0$ is smooth.

The sign $R_{\lambda }=-{\mathcal{P}}_{+}{V}_{\lambda }$ is retained. It is the polarized response $U_D$ on the witness, also explicitly recorded in XVI-B's account of XVI-D CheungXVIB (:later-response) CheungXVID. The separate action-equation coefficient $r_D=-u_D$ is not silently substituted for the response argument.

For a contravariant response argument $R$, its native first density coefficient is

Consequently,

The prepared coordinate forcing and candidate are

These are the supplied completion interface and candidate CheungXVIB (:coordinate-leg) CheungXVID (:Xcand).

Direct differentiation, with spatial homogeneity, proves

This calculation does not identify an inherited Green symbol outside its declared domain.

Lower the response with ${\eta }_{+}$:

The comparison metric coefficient on VI's background is

This is the transport induced by

For an actual real metric, ${\square }_{-q}=-{\square }_q$. The same identity holds for the formal jets because the determinants agree in four dimensions and the inverses differ by a minus sign. The harmonic equation and ordinary coordinate-time data are therefore transported without changing the coordinate scalar functions.

Complex coefficients are used only through complex linear extension. Real witnesses will be obtained by taking $\mathrm{Re}(\zeta \cdot )$, with $\left|\zeta \right|=1$.

For (eq:htransport),

Hence

Proof. Raising both indices of $-{R_{\lambda }}^{♭}$ with ${\eta }_{-}=-{\eta }_{+}$ gives $-{R_{\lambda }}^{\mu \nu }$. Its minus-signature trace is ${\eta }_{+,\alpha \beta }{R_{\lambda }}^{\alpha \beta }$. Substitution in (eq:trace-reverse-cov) yields $-{\mathcal{P}}_{+}{R}_{\lambda }$. Equation (eq:V) and ${\mathcal{P}}_{+}^2=\mathrm{id}$ give $V_{\lambda }$. Spatial homogeneity then gives the divergence formula.

On the same slab $M_T$, with the transport (eq:htransport) and equal ordinary zero Cauchy data, VI's first coordinate equation becomes

The prepared candidate instead satisfies (eq:candidate-PDE). Their normalized forcing difference is

For every $\lambda >0$, its time component is nonzero as a smooth function. It is supported in the temporal preparation layer $[0,\epsilon ]$, but not in a compact spatial set.

There are a phase $\zeta \in \mathbb{C}$, $\left|\zeta \right|=1$, and a real nonnegative $f\in {C_c}^{\infty }({M_T}^{\circ })$ such that

Thus a real smooth witness detects the forcing mismatch.

Proof. Equation (eq:VI-first) and Lemma lem:transport give

Multiplying by $-1$ gives (eq:VI-normalized). Subtracting its forcing from the candidate forcing gives

which proves (eq:forcing-residual). The sign normalization is essential: the native candidate has ${\square }_{-}{X}_{\mathrm{cand}}=+F_{\mathrm{co}}$, not $-F_{\mathrm{co}}$.

Suppose ${{\mathfrak{r}}_{\lambda }}^0$ were identically zero. Then $(\sigma -1){V_{\lambda }}^{00}$ would be constant. Its value at $0$ is zero, so

Choose a nonempty open interval $J\subset (0,\epsilon )$ with $0<\sigma <1$. On $J$, the last equality implies ${V_{\lambda }}^{00}=0$, hence ${S_{\lambda }}^{00}=({V_{\lambda }}^{00}{)}^{\prime \prime }=0$. But the coefficient

in (eq:S00) has strictly nonzero real part on $J$. The exponential never vanishes. This is a contradiction.

The residual vanishes for $t\le 0$, because $V_{\lambda }=0$, and for $t\ge \epsilon$, because $\sigma =1$. At a time where it is nonzero it is nonzero at every spatial point; its support is therefore not spatially compact.

Choose $t_c\in (0,\epsilon )$ with ${{\mathfrak{r}}_{\lambda }}^0(t_c)\ne 0$, and choose

By continuity, $\mathrm{Re}(\zeta {{\mathfrak{r}}_{\lambda }}^0)>0$ on a sufficiently small interval about $t_c$. Take a nonnegative nonzero time bump $b$ supported there and a nonnegative nonzero spatial bump $c$. Then $f(t,\mathbf{x})=b(t)c(\mathbf{x})$ proves (eq:forcing-test). All operations are real linear after taking $\mathrm{Re}(\zeta \cdot )$.

the calculation excludes. The equality required for this direct identification is, in VI's native sign,

Equivalently, in the common plus-sign normalization it is

Lemma lem:B disproves that equality on the witness. It does not use the compact interaction switch $g$.

For $t\in (\epsilon ,T)$,

Its other components vanish on that interval. In particular, $h_{\lambda }$ fails (eq:VI-vacuum). For every nonzero phase $\zeta$, the real coefficient $\mathrm{Re}(\zeta h_{\lambda })$ also fails that equation on the slab.

Consequently, placing this response as VI's first metric coefficient does not produce an element of ${\mathcal{J}}_T$, whatever higher metric and scalar coefficients are chosen.

Proof. For a spatially homogeneous tensor $h$ with $h_{0i}=0$, write $H_s={\sum }_k{h}_{kk}$. Substitution in (eq:Einstein) gives

For example, ${\overline{h}}_{-}^{00}=(h_{00}+H_s)/2$; the $h_{00}^{\prime \prime }$ terms cancel from the spatial equation. This verifies the formula without imposing any gauge condition.

From (eq:htransport) and (eq:V), $h_{\lambda }=({\mathcal{P}}_{+}{V}_{\lambda }{)}^{♭}$. On the plateau,

so

It follows that

This proves (eq:Einstein-witness).

The matrix $n_i{n}_j-{\delta }_{ij}$ is nonzero. For every $\zeta \ne 0$, $\mathrm{Re}(\zeta e^{2i\lambda t})$ is not identically zero on an open time interval. Thus the corresponding real Einstein coefficient is nonzero. The first metric equation in ${\mathcal{J}}_T$ is independent of higher coefficients, which proves the final assertion.

Proposition prop:jet-domain is an equation-space failure for the descended on-shell observable. Lemma lem:B is a forcing mismatch between well-defined off-shell coordinate prescriptions. Neither is a failure of smooth coordinate existence.

Both $\mathrm{Re}{F_{\mathrm{co}}}^0$ and $\mathrm{Im}{F_{\mathrm{co}}}^0$ are spatially noncompact on the slab. In particular, $F_{\mathrm{co}}\notin {C_c}^{\infty }({M_T}^{\circ };{\mathbb{C}}^4)$.

Proof. On the plateau,

Neither the real nor the imaginary part of the last expression vanishes identically on $(\epsilon ,T)$. Hence neither corresponding part of ${F_{\mathrm{co}}}^0$ can vanish identically there. Each is spatially constant, and is nonzero at every spatial point at some plateau time. It cannot have compact spatial support.

This agrees with the supplied XVI-D support boundary CheungXVID (:coord). It is not the cause of the forcing mismatch: Section sec:lemmaC constructs the smooth no-decay solution operator. Extending a source domain cannot replace one forcing by another.

Let $F_1,F_2$ be smooth scalar, vector, or tensor forcings on $M_T$. Suppose $u_1,u_2$ are smooth solutions of

with equal ordinary coordinate-time Cauchy data. Then

The statement holds componentwise over $\mathbb{C}$, without spatial decay or finite total energy.

For different operators, uniqueness of each separate problem does not by itself identify their solutions. A proposed common solution must satisfy both equations.

Proof. Equality of solutions implies equality of forcings by application of ${\square }_{+}$.

Conversely, if the forcings agree, the difference $z=u_1-u_2$ satisfies ${\square }_{+}z=0$ and zero data. It suffices to treat a real scalar component. Fix $0<t_{\ast }<T$, ${\mathbf{x}}_{\ast }\in {\mathbb{R}}^3$, and $\rho >0$. Set

These are compact integration regions, so $E(s)$ is finite. The wave equation gives

The boundary moves inward with speed one. Therefore

Zero initial position implies zero initial spatial gradient, and hence $E(0)=0$. Nonnegativity gives $E(s)=0$. Thus $z_t(s,{\mathbf{x}}_{\ast })=0$ for $0\le s\le t_{\ast }$, and

The point was arbitrary. Smoothness includes the endpoints. Real and imaginary parts, and then all components, give the stated extensions.

For different operators, the difference of two solutions need not satisfy one homogeneous equation. The preceding uniqueness argument cannot be applied until the equations have been matched or the candidate has been tested in both.

This is the no-decay discipline of the supplied VI and XVI-D coordinate statements CheungVI,CheungXVID. We have supplied the proof rather than using the phrase “no spatial decay” as an identification theorem.

For every smooth source $F$, define

Then $G_{+}F$ is the unique smooth zero-data solution of ${\square }_{+}u=F$. It requires no spatial decay and agrees with the local compact-cutoff construction. Moreover,

is the corresponding zero-data inverse of ${\square }_{-}$. For spatially homogeneous $F(t)$,

Proof. Let $M_{r}F(s,\mathbf{x})$ denote the normalized spherical mean. For smooth $F$, the spherical-mean identity is

It follows by the divergence theorem, equivalently from ${{\partial }_r}^2{M}_{r}F+(2/r){\partial }_r{M}_{r}F={\Delta }_{\mathbf{x}}{M}_{r}F$, with its smooth value at $r=0$.

Differentiate

The first endpoint term vanishes. The second derivative has endpoint term $F(t,\mathbf{x})$, because ${\partial }_r(r{M}_{r}F){|}_{r=0}=F$. The spherical-mean identity gives

The position and first time derivative vanish at $0$. Smoothness follows by differentiation on the compact spheres and bounded time intervals. Lemma lem:C proves uniqueness.

For a compact observation set, the integration points in (eq:Kirchhoff) lie in a compact truncated causal past. Multiplication of $F$ by a smooth spatial cutoff equal to one on that region leaves the answer unchanged there. This proves the local-cutoff statement. The sign relation follows from ${\square }_{-}=-{\square }_{+}$. A homogeneous source has spherical mean $F$, which proves (eq:homogeneous-inverse).

This is the finite-time interpretation stated in VI CheungVI (:Gzero); general wave Cauchy theory is treated in BGP.

Set

viewed by smooth ambient zero extension, and let $\mathcal{H}$ consist of ambient smooth homogeneous sources $q(t)$ that vanish for $t\le 0$. Their intersection is zero: a nonzero spatially constant source cannot be compactly supported in space.

If $E_{0,\mathrm{int}}^{+}$ is the compact-interior zero-data inverse, define

The direct-sum decomposition makes this well defined. Proposition prop:local-inverse and Lemma lem:C show that it is the restriction of $G_{+}$ to ${\mathcal{D}}_{\mathrm{int}}\oplus \mathcal{H}$. This reproves the mathematical content of the supplied restricted extension CheungXVID (:restricted-extension).

Agreement of two linear extensions on ${\mathcal{D}}_{\mathrm{int}}$ alone does not determine their values on $\mathcal{H}$. For example, for $b\in {C_c}^{\infty }((0,T))$ and an interior time $t_{\ast }$, the map

agrees on ${\mathcal{D}}_{\mathrm{int}}$ and retains zero initial data. In general it is not a right inverse of ${\square }_{+}$.

By contrast, two extensions that both return smooth solutions of the same wave equation with the same data must agree on their common source domain, by Lemma lem:C. The equation, smoothness, and initial data are the decisive additional requirements.

This is exactly why the conditional coordinate-compatibility clause of XVI-D cannot be replaced by an appeal to agreement on compact sources CheungXVID (:coordinate-compatibility). Here the smooth inverse is constructed, but its construction does not rename every inherited occurrence of a Green symbol.

Retain the slab, source, response sign, metric-sign transport, and ordinary coordinate-time data in Sections sec:native–sec:witness. For every $\lambda >0$, the direct comparison of VI's first coordinate coefficient on $h_{\lambda }$ with $X_{\mathrm{cand}}[R_{\lambda }]$ is a mismatch.

More precisely:

1. The two normalized flat forcings differ by the nonzero residual (eq:forcing-residual).

2. Their unique smooth zero-data coordinate coefficients are

   $$\begin{array}{rl}{w_{\lambda }}^{\mu }(t)& =-{{\int }_0}^t{V}_{\lambda }^{0\mu }(s)\mathrm{d}s,\\ {d_{\lambda }}^{\mu }(t):={w_{\lambda }}^{\mu }-{X_{\mathrm{cand}}}^{\mu }& =-{{\int }_0}^t(1-\sigma (s)){V_{\lambda }}^{0\mu }(s)\mathrm{d}s.\end{array}$$

   The time component ${d_{\lambda }}^0$ is nonzero somewhere in $(0,\epsilon )$.

3. A real linear combination of the witness coefficients and a compact real test detect that coordinate difference.

4. The response used as a first metric coefficient is not in VI's solution-jet domain, as shown by (eq:Einstein-witness).

This theorem excludes the stated direct first-order identification. It does not exclude a differently declared metric map, preparation, coupling-order comparison, or observable construction. It neither repairs nor contradicts XVI-D's conditional inherited-coordinate clause.

Proof. Item (i) is Lemma lem:B, with both operators normalized to ${\square }_{+}$.

The field in (eq:w-explicit) has zero position and velocity at $0$ and satisfies

Lemma lem:C identifies it with VI's smooth first coefficient. Equation (eq:Xcand) gives the other field; subtraction proves (eq:coordinate-difference).

If ${d_{\lambda }}^0$ vanished identically on $(0,\epsilon )$, then its derivative would give $(1-\sigma ){V_{\lambda }}^{00}=0$ there. On an interval with $0<\sigma <1$, this would force ${S_{\lambda }}^{00}=0$, contradicting (eq:S00), exactly as in Lemma lem:B. Thus item (ii) holds.

Choose $t_c\in (0,\epsilon )$ with ${d_{\lambda }}^0(t_c)\ne 0$ and a unit phase $\zeta$ such that $\mathrm{Re}(\zeta {d_{\lambda }}^0(t_c))>0$. A nonnegative compact test supported in a small neighborhood where that real part remains positive has a strictly positive pairing. Since the constructions are complex linear on the printed source coefficients, this pairing is the coordinate difference for the real metric coefficient $\mathrm{Re}(\zeta h_{\lambda })$. This proves item (iii).

Item (iv) is Proposition prop:jet-domain. None of these calculations uses an interaction switch $g$, a scalar-switch identity, a quantum coefficient, or a theorem from an unregistered sheet.

A correction starting at second coordinate order cannot make

equal modulo ${\kappa }^3$, while their displayed first coefficients remain fixed.

Proof. Equality modulo ${\kappa }^3$ requires equality of the coefficient of $\kappa$. Theorem thm:mismatch disproves that equality.

For $t\ge \epsilon$, the forcing residual vanishes, but

Thus the field difference has a constant preparation-memory value there. This formula does not assert that this integral is nonzero at every frequency. The theorem detects the difference on the common slab and does not require a universal late-time nonvanishing claim.

Replacing $R_{\lambda }$ by a prepared metric argument would alter the linear density coefficient and its divergence. Such a replacement is not allowed silently in the theorem. Likewise, replacing $R_{\lambda }$ by the opposite action-equation coefficient changes the response argument. The result concerns the printed response sign and transport, not every possible map between the manuscripts.

This section is not a hypothesis of Theorem thm:mismatch. It records why VI's switch defect is a different question, and proves the identity directly.

Use ${\eta }_{-}$, fix $m>0$, and put

Let ${\varphi }_0$ satisfy $P{\varphi }_0=0$. For a symmetric tensor $a$, define the density variation

If

then direct expansion gives

For a compact smooth vector field $\xi$ supported in ${M_T}^{\circ }$, set

Calculation gives

Consequently,

for the unswitched coordinate problem, by zero-data uniqueness.

Let $g\in {C_c}^{\infty }({M_T}^{\circ };\mathbb{R})$ be a fixed scalar interaction switch. Let $G_{m,0}$ be the smooth zero-data inverse of $P$, and define

The massive Cauchy inverse is the standard smooth wave Cauchy construction; its local uniqueness uses the preceding energy proof with the additional nonnegative term $m^2{z}^2$ BGP.

For the displayed definitions,

where

This is a classical scalar-response identity. It is neither a loop anomaly nor an obstruction to the harmonic-coordinate Cauchy problem.

Proof. Set $u=X_{\xi }{\varphi }_0$. Equations (eq:Qrelation)–(eq:puregauge) and $P{\varphi }_0=0$ give $Q_{K\xi }{\varphi }_0=-Pu$. The density product rule gives

Therefore the scalar-response change is

Since $gPu=P(gu)-[P,g]u$, and $gu$ has zero data, $G_{m,0}P(gu)=gu$. The coordinate change is $\xi$, so the dressing subtracts $u$. Smearing proves (eq:switch-defect).

The coordinate equation contained no $g$. Its transformation was established before the scalar response was introduced.

Equation (eq:switch-defect) has the form of the supplied VI switch identity CheungVI (:switch-defect, :plateau-obstruction), also copied in X-A's prerequisite sheet CheungXA. If $\xi$ is supported strictly where $g=1$ and $\mathrm{d}g=0$, the displayed defect vanishes. For larger supports the expression need not vanish. A correction beginning at second order cannot change its fixed first coefficient.

The first cancellation, larger-support defect, and plateau second-order intertwiner reported for VII-A belong to its own equations and transformations CheungVIIA,CheungXA. We record the analogy; we do not transfer its intertwiner or its nonvanishing witnesses to the comparison theorem.

VI's formal smearing is

Its first coefficient is

The source records descend this expression on the named solution-jet quotient CheungVI (:descent).

The prepared witness completion instead has the shape

on its own completion domain CheungXVID,CheungXVIB. A common integrand shape does not identify the metric argument, coordinate field, scalar response, smearing convention, or equation space.

For the real witness selected in Theorem thm:mismatch, there are a homogeneous real free Klein–Gordon scalar $\varphi$ and a real compact test $f$ for which replacing $w$ by $X_{\mathrm{cand}}$, while keeping the scalar first coefficient fixed, changes the first-order smearing.

Proof. Write $d^0=\mathrm{Re}(\zeta {d_{\lambda }}^0)$, and choose a detecting time $t_c$ at which $d^0(t_c)>0$. Take

Then $P\varphi =0$ and $\dot{\varphi }(t_c)=m>0$. For any common smooth first scalar coefficient ${\Phi }_1$, the difference of the two displayed expressions is

Choose a nonnegative nonzero compact test where both factors are positive. This integral is strictly negative.

The proposition isolates the coordinate contribution. It does not show that independently defined scalar legs cannot compensate a coordinate difference in a particular complete observable. Nor does it apply VI's solution-jet descent theorem to the inadmissible first metric coefficient of Proposition prop:jet-domain.

Even a successful future coordinate match would leave a separate observable comparison: one must match all inputs and both scalar and coordinate legs.

The fields $R_{\lambda }$, $V_{\lambda }$, and $X_{\mathrm{cand}}$ are smooth fields on the slab. They are not compact distributions on a three-slot configuration product. In particular,

is the response tensor, not the coordinate field.

For completeness, the following elementary construction explains the witness boundary used in XVIII-A CheungXVIIIA (:witness,:smoothwitness).

The family of evaluations on all pure-trace tensor pairs does not determine a compact smooth three-slot kernel with symmetric tensor slots.

Proof. Choose a nonzero constant symmetric contravariant tensor $A$ such that $A^{\mu \nu }{\eta }_{+,\mu \nu }=0$, for example $A^{12}=A^{21}=1$ and all other components zero. Choose nonzero real compact smooth functions $b,c$ and set