Landmark XVIII-B A Prepared Massive-Scalar Stress-Response Benchmark: Spectral Evaluation, Transverse Contact Freedom, and a Fixed-Preparation Residual

We evaluate one external-source benchmark on four-dimensional Minkowski space. The source is classical, and the quantum field is a free real scalar of fixed mass $m>0$, with its minimally coupled canonical stress tensor Wick ordered in the Minkowski vacuum. A two-particle calculation gives the exact spin-two and spin-zero spectral densities. A once-subtracted dispersion prescription for the reduced form factors defines a real, causal, tempered response distribution of scaling degree at most eight. We classify the complete parity-even, Lorentz-covariant, pair-symmetric, doubly transverse contact freedom in that extension class: one quadratic and two quartic polynomial tensors. A zero-momentum condition fixes all three coefficients uniquely. Both linear Ward identities hold distributionally.

The four closed-time-path kernels are constructed consistently at quadratic source order. Smooth preparation is incorporated by multiplying the external source by a fixed time ramp. The resulting response is finite on arbitrary compact joint tests. We calculate its transformation residual, prove plateau-supported intertwining, and construct a late, contact-independent detecting test for an unrestricted preparation-layer transformation.

Spectral detection is not a measurement instrument, a classical record, or a quantum-return decoder. Documentary comparisons retain their individual models and versions and are not premises of the calculation. No relational scalar two-point coefficient, inherited characteristic graph, graviton–ghost inventory, interacting BRST/BV identity, bulk restoration, or ultraviolet completion is established.

target and proof boundary{ The completed object is the connected, renormalized vacuum two-stress response of the printed linear external-source model, at quadratic source order. The tensor source and Minkowski metric are classical. The only quantum field is one free scalar with $m>0$.

No theorem identifies this response with the second metric variation of a nonlinear gravitational action or with the companion's prepared relational scalar. Gravitons, ghosts, nonlinear metric seagulls, and relational coordinate insertions are outside the target. No sharp-preparation, massless, adiabatic, or ultraviolet completion limit is taken.}

The supplied material contains the revision brief and companion excerpts, but not the original XVIII-B source. Accordingly, this document prints a complete independent specification rather than asserting identity with an unavailable version. Its choice of the canonical minimally coupled stress tensor, Fourier convention, source sign, and extension class is explicit below. Authorship and registration are not inferred.

The coefficient ${\mathcal{R}}_{\epsilon }(j,h)$ is a response, not a classical record; a record claim would require a separately declared measurement instrument, and Manuscript I's reported destructive-readout characterization in a stated class—where record compatibility is not residual compatibility—is only a documentary comparison, not an imported uniqueness theorem CheungI.

A decoder is a quantum-return map from a declared accessible factor; the preservation/recovery distinctions of Hawk04-A/B do not identify either such a map or a measurement channel with the quadratic closed-time-path bookkeeping of Proposition prop:ctp CheungHawk04A,CheungHawk04B.

The positive vacuum norm in Proposition prop:detector detects spectral weight, not an Enter instrument or an observation boundary; VII-E's separately specified spectator, record, residual-instrument, and continuation requirements are a comparison explaining why a pairing alone is not observation CheungVIIE.

This benchmark does not reconstruct spacetime, derive a metric, identify an entanglement wedge, or construct a Page curve or an island, and it imports no reconstruction map from the separately identified XVII-A finite-matrix AdS${}_2$ benchmark CheungXVIIA.

stops. The inherited XV-B quantum-admission exit is not opened. XVI-N and Route B are not discharged. O.1 remains NOT_CLOSED. No quantum coefficient of the inherited companion response is admitted. No successful compilation is asserted. These are scope statements, not nonexistence theorems CheungXIVJ,CheungXVB,CheungXVIN,CheungXVIO,CheungXVIP.

Indices are raised and lowered with $\eta$. Write $p^2={\eta }_{ab}{p}^a{p}^b$ and use

Thus ${\partial }_a$ has multiplier $-\mathrm{i}{p}_a$.

The stress tensor is

Wick ordering is with respect to $\Omega$, so $⟨T_{ab}{⟩}_{\Omega }=0$. The free field equation gives ${\partial }^a{T}_{ab}=0$ as an operator-valued distribution on the usual finite-particle domain.

Let ${{\Pi }^{\mathrm{R}}}_{abcd}$ be a renormalized retarded splitting of the vacuum stress commutator, with convention

For compact smooth symmetric $j,h$, set

This is the linear response of the insertion $-g\mathsf{T}(j)$ to the source $V_{{\sigma }_{\epsilon }h}$, hence it has quadratic source order. The corresponding first-order stress response is $-g{\Pi }^{\mathrm{R}}\ast ({\sigma }_{\epsilon }h)$.

The double integral in (eq:response) is distributional. The function $F_{j,H}$ is compact and smooth; no pointwise multiplication of singular kernels is being used.

. The selected connected source-degree-two coefficient consists of the two Wick contractions between the two normal-ordered stress insertions, together with its completely classified local extension freedom.

The distinction between a two-stress correlator and the full second variation of a metric-dependent action, including local variation terms, is important in the external linear-response literature AMP,Mottola. Those works motivate a comparison; they do not insert seagulls into the present linear-source definition.

For $s=p^2\ne 0$, define

These are the scalar and traceless symmetric projectors on the three-dimensional transverse space for timelike $p$.

To avoid any division by $p^2$ in the final kernel, put

and define the polynomial tensors

The quadratic polynomial tensor is

For $s\ne 0$,

Equation (eq:E), not this rational representation, defines $\mathsf{E}$ also on $s=0$.

Every tensor $\mathsf{E},{\mathsf{Q}}_2,{\mathsf{Q}}_0$ is symmetric in each index pair and under exchange of the pairs. Each satisfies

as a polynomial identity on all of ${\mathbb{R}}^4$.

Proof. For ${\mathsf{Q}}_2,{\mathsf{Q}}_0$, use $p^a{L}_{ab}=0$. For $\mathsf{E}$, contraction of (eq:E) cancels the terms proportional to $p_b{\eta }_{cd}$, $p_c{\eta }_{bd}+p_d{\eta }_{bc}$, and $p_b{p}_c{p}_d$. The symmetries follow from the displayed formulas.

Let ${\omega }_{\mathbf{k}}=\sqrt{|\mathbf{k}{|}^2+m^2}$ and

Use

The unordered two-particle completeness relation has factor $1/2!$. This factor is retained below.

Define

For on-shell positive-energy $k,q$, define

The vacuum matrix element is $\hslash {\tau }_{ab}$, and

Moreover $(k+q{)}^a{\tau }_{ab}=0$.

Proof. The two annihilation operators in (eq:stress) give the two derivative contractions $-(k_a{q}_b+k_b{q}_a)$. The trace term gives ${\eta }_{ab}(k\cdot q+m^2)$. The factor $\hslash$ comes from the two field factors. Inserting the two-particle identity gives (eq:phaseformula) and its factor $1/2$.

Using $k^2=q^2=m^2$, contraction with $k+q$ cancels the two terms.

For timelike future $p$, $s=p^2>4m^2$, write

In the rest frame of $p$,

and the angular distribution of $\mathbf{k}=K\mathbf{n}$ is uniform. Its normalized moments are

Proof. The spatial delta sets $\mathbf{q}=-\mathbf{k}$. The energy delta $\delta (\sqrt{s}-2{\omega }_{\mathbf{k}})$ has radial Jacobian ${\omega }_{\mathbf{k}}/(2K)$. Combining the measure factors gives $v/(8\pi )$, with normalized angular measure $\mathrm{d}\Omega /(4\pi )$.

Rotation invariance makes the second moment a multiple of ${\delta }_{ij}$, determined by $|\mathbf{n}{|}^2=1$. The fourth moment is a multiple of the three displayed Kronecker products; double contraction determines $1/15$.

Set $a=4m^2$ and, for $u\ge a$, define

Extend them by zero below $a$. Then

Both densities are strictly positive above threshold. The corresponding projector densities are

The formula defines a positive, tempered vacuum two-point distribution.

Proof. In the rest frame $p=(\sqrt{s},0)$, ${\tau }_{00}={\tau }_{0i}=0$ and

Its spatial trace is $-(s+2m^2)$. The traceless part is $2K^2(n_i{n}_j-{\delta }_{ij}/3)$. Lemma lem:phase therefore gives

Multiplying by ${\hslash }^{2}v/(16\pi )$ from (eq:phaseformula) gives

This is (eq:Wspec). Lorentz covariance and transversality extend the rest-frame calculation to the future two-particle cone.

Positivity also follows directly from the squared matrix element in (eq:phaseformula). The polynomial tensor growth is at most quartic, while $r_2,r_0$ are bounded. Hence the spectral distribution is tempered.

The spectral normalization is derived here from the printed field convention and the identical-particle factor. For comparison, the same spin decomposition and massive-scalar spectral functions appear in the external linear-response calculation of Anderson, Molina-París, and Mottola AMP (Appendix B). No semiclassical metric equation or stability theorem from that work is used here.

For every nonzero $f\in {C_c}^{\infty }(M)$, take $j^{ab}={\eta }^{ab}f$. Then

This detects the spectral coefficient and is independent of the finite retarded contact prescription.

It is not a finite measurement instrument, an Enter destructive readout, a classical record, or a decoder.

Proof. The norm follows by vacuum positivity and the Fourier representation of $W^{>}$. Since ${\eta }^{ab}{\eta }^{cd}{{\mathsf{Q}}_2}_{abcd}=0$ and ${\eta }^{ab}{\eta }^{cd}{{\mathsf{Q}}_0}_{abcd}=3(p^2{)}^2$, the second line follows.

The Fourier transform of a nonzero compact smooth function is real analytic on real momentum space and cannot vanish on the entire open future threshold region. The positive integrand is therefore nonzero on an open set. Rapid Fourier decay makes the integral finite.

A finite change in the retarded splitting does not change the already fixed vacuum Wightman distribution. No apparatus or selective map is constructed by this norm.

The interpretation is deliberately spectral. The supplied readout and observation-boundary manuscripts I and VII-E concern different interfaces CheungI,CheungVIIE. They are comparisons, not premises of (eq:vacnorm).

For $i\in \{2,0\}$ and $z\in \mathbb{C}\setminus [a,\infty )$, define

The subtraction is in the reduced form factor. It is not a once-subtracted integral of the growing projector density ${\rho }_i=u^2{r}_i$.

The functions $F_i$ are analytic off the cut, real for real $z<a$, and satisfy $F_i(0)=0$. For $s>a$,

Their real boundary values are continuous at threshold and grow at most logarithmically at infinity.

Proof. The densities are bounded. On compact subsets off the cut, the subtracted integrand is $O(u^{-2})$, uniformly with its required $z$ derivatives. This proves convergence and analyticity. Reality and $F_i(0)=0$ follow directly.

The distributional identity

gives (eq:disc). Near threshold, the densities vanish at least as $(u-a{)}^{1/2}$, so their Cauchy transforms have finite continuous boundary values there. The bounded limiting densities at large $u$ give logarithmic growth. Explicit formulas below also establish these boundary statements directly.

Define the retarded boundary value by analytic continuation from $\mathrm{Im}{p}^0>0$:

On real momenta this is $F_i(p^2+\mathrm{i}0\mathrm{sgn}{p}^0)$ at the spectral cut. The advanced value reverses the sign.

The reference response is

The tensor structures in this formula are polynomials on all momenta; there is no singular projector evaluation at $p^2=0$.

Put $w=z/a$ and

These functions are analytic on the same slit plane. The change $u=a/(1-v^2)$ in (eq:subtracted) gives

For $n\ge 1$,

The regular value at $w=0$ is $J_n(0)=1/(2n+1)$.

For real $w<1$, $w\ne 0$,

For $w>1$, set $v_{\ast }=\sqrt{1-1/w}$. Then

The apparent singularities of the intermediate $J_0$ representation cancel in the combinations (eq:F2closed)–(eq:F0closed) at $w=0,1$.

The printed form factors obey

Their imaginary parts above threshold are

Proof. At $w=0$, use $J_n(0)=1/(2n+1)$ in (eq:F2closed)–(eq:F0closed) and $a=4m^2$. In the scalar combination,

At $w=1$, $J_1=1$, $J_2=1/3$, and $J_3=1/5$, which gives (eq:thresholdchecks).

The imaginary term in (eq:J0above), together with the recursion, gives the stated imaginary parts; equivalently use Lemma lem:dispersion.

These are analytic checks, not reported computer execution.

For either bounded density $r_i$, the momentum-space integral

exists in ${\mathcal{S}}^{\prime }({\mathbb{R}}^4)$. Its inverse Fourier transform is supported in ${\overline{V}}_{+}$.

Proof. For fixed $u>0$, the first multiplier is the Fourier transform of the retarded Klein–Gordon fundamental solution of mass $\sqrt{u}$. The second is the transform of $-{\delta }_0/u$, so their difference has future causal support.

We verify convergence on Schwartz tests. On $|p{|}_{\mathrm{E}}\le \sqrt{u}/2$, the denominator is bounded away from zero, and subtraction gives a bound $C|p{|}_{\mathrm{E}}^2/u^2$. Its test pairing is therefore $O(u^{-2})$.

On the complementary region, the poles occur at $p^0=\pm \sqrt{u+|\mathbf{p}{|}^2}$. Partial fractions reduce them to one-dimensional principal values and delta terms, with prefactor $1/(2\sqrt{u+|\mathbf{p}{|}^2})$. The principal-value estimate obtained by subtracting the test value at the pole uses only a test and its first derivative. After the rescaling $p=\sqrt{u}k$, Schwartz decay on $|k{|}_{\mathrm{E}}\ge 1/2$ dominates the Jacobian and these finite-order distribution bounds. Choosing a sufficiently high, fixed Schwartz seminorm makes this region $O(u^{-2})$ as well. Thus the whole pairing is bounded by $C{u}^{-2}{p}_N(\psi )$ for some finite Schwartz seminorm $p_N$, uniformly for large $u$.

Since $r_i$ is bounded, integration converges in ${\mathcal{S}}^{\prime }$. Its finite spectral integrals have future causal support, and that support property is preserved by distributional convergence.

Equation (eq:referencePi) defines a real, tempered, future-causal tensor distribution. It agrees with (eq:Kubo) on $M\setminus \{0\}$ and has

Proof. Lemma lem:retardedexistence gives the reduced retarded distributions. Applying the constant-coefficient polynomial differential operators corresponding to ${\mathsf{Q}}_2,{\mathsf{Q}}_0$ preserves causal support and temperedness.

The retarded-minus-advanced difference is, by (eq:disc) and (eq:Wspec),

It is therefore a retarded splitting of the prescribed commutator. Away from the origin, causal support and the splitting identity give (eq:Kubo). Possible equal-time derivative terms from representing the stress tensor by polynomial differential operators are supported at the origin and are part of the extension prescription, not omitted nonlocal terms.

The boundary-value relation ${F_i}^{\mathrm{R}}(-p)=\overline{{F_i}^{\mathrm{R}}(p)}$, with real even tensor polynomials, proves reality.

Finally, the explicit form factors are bounded by $C\log (2+|p{|}_{\mathrm{E}})$. Thus

Under $x\mapsto \lambda x$, its pairing with a fixed Schwartz test is bounded by $C_{\psi }{\lambda }^{-8}(1+|\log \lambda |)$. Multiplication by ${\lambda }^{\omega }$ therefore gives zero as $\lambda \downarrow 0$ for every $\omega >8$. This is the claimed scaling-degree bound.

The standard scaling-degree extension framework is discussed in BF; the wavefront and product distinctions are reviewed in BDH. Here the spectral construction supplies the actual distribution before any extension comparison is made.

The improper spectral integral is a definition and convergence proof within an already specified free continuum field theory. It is not a regulator-removal theorem for a family of interacting gravitational cutoff theories.

A permitted response is a tempered retarded extension agreeing with (eq:Kubo) off the origin, with scaling degree at most eight, and satisfying:

1. real position-space coefficients;

2. parity-even Lorentz covariance;

3. symmetry in each stress pair and under exchange of the index pairs;

4. transversality in both stress slots.

The mass and stress insertion are fixed. This is an extension class for the selected two-stress response, not a classification of all gravitational counterterms.

Every permitted response differs from ${{\Pi }^{\mathrm{R}}}_{\mathrm{ref}}$ by exactly a tensor of the form

The three tensors are linearly independent. Here $\alpha$ has mass dimension two and $\beta ,\gamma$ are dimensionless in the stated four-dimensional convention.

Conversely, every contact (eq:contacts) gives a permitted extension.

Proof. Two extensions agreeing off zero differ by a distribution supported at zero. A point-supported distribution is a finite sum of delta derivatives. The scaling-degree bound restricts its derivative order to at most four, so its Fourier transform is a tensor polynomial of degree at most four.

Parity-even Lorentz covariance and the pair symmetries give the general polynomial form

where $A,B$ have degree at most two in $s$, $C,D$ at most one, and $E$ is constant. Contracting with $p^a$ gives the independent equations

Writing $C=c_0+c_{1}s$, $D=d_0+d_{1}s$, these require

There are therefore exactly three free real coefficients.

The degree-two solution is proportional to $\mathsf{E}$. At degree four the independent spin-two and spin-zero solutions are ${\mathsf{Q}}_2,{\mathsf{Q}}_0$. Their independence follows on a timelike rest frame: ${\mathsf{Q}}_2,{\mathsf{Q}}_0$ are independent projectors multiplied by $s^2$, while $\mathsf{E}$ has degree two. This proves (eq:contacts).

Each displayed contact is real, local, transverse, pair-symmetric, and of scaling degree at most eight. Its support at zero is compatible with retarded support, proving the converse.

Transversality is a normalization requirement of this extension class. An arbitrary unnormalized extension can contain additional nontransverse local terms. The theorem neither counts them as physical parameters nor claims that the Ward condition holds automatically for every such extension.

Among the permitted extensions, exactly one satisfies

It is ${{\Pi }^{\mathrm{R}}}_{\mathrm{ref}}$. Equivalently, calibration sets the degree-two and degree-four Taylor tensors to zero.

This conclusion uses $m>0$. No $m\to 0$ limit is taken; it does not discharge XVI-N, produce a massless susceptibility, or match VII-H.

Proof. The gap $a>0$ makes $F_i(z)$ analytic near zero. Since $F_i(0)=0$, each term ${\mathsf{Q}}_i(p)F_i(p^2)$ begins at momentum degree six. Thus the reference satisfies (eq:zeronorm).

For a different extension, the degree-two term is $\hslash \alpha \mathsf{E}$, forcing $\alpha =0$. Independence of ${\mathsf{Q}}_2,{\mathsf{Q}}_0$ then forces $\beta =\gamma =0$ at degree four.

The massless Wick-square susceptibility in the supplied VII-H record has a different insertion, quantum order, and contact classification CheungVIIH. Even a future massless limit here would require a connecting theorem before it could be compared with that benchmark. Choosing $m>0$ is not a completion of the massless special-family problem in XVI-N CheungXVIN.

normalized kernel. For the zero-momentum prescription, the completed response is

Equations (eq:F2closed)–(eq:J0above) evaluate its two scalar functions completely.

Every extension in Theorem thm:contacts satisfies

Let

for a compact smooth vector field $\xi$. The unprepared bilinear response satisfies

These are complete linear identities for the printed kernel, all compact tests, and the stated contact family. They are not interacting gravitational BRST/BV identities.

Proof. Lemma lem:transverse annihilates each polynomial tensor in the reference and contact formulas. Multiplication by the scalar boundary values preserves the polynomial zero identities. Fourier inversion proves (eq:ward). Compact integration by parts in either source slot gives the two bilinear identities.

Oai282-A/B provide documentary examples of finite free-scalar Ward audits and homogeneous stress coverage. Oai282-B's improvement scalar $L=-Q/2$ on its ordered Wick algebra is not any of the three coefficients here; its two probes do not cover unrestricted tensors CheungOai282A,CheungOai282B.

Conservation of a conformal-Killing current does not by itself prove Ward action on every field. No free conformal action, unrestricted stress classification, or verification of Family 282 is imported Catalogue. The supplied Oai069-A finite free-scalar Ward-defect summary is a second comparison only, not this kernel's identity or a chiral-to-four-dimensional bridge CheungOai069A.

For $p^2>a$,

This discontinuity is unchanged by every permitted contact. It is nonzero, so no permitted contact can make the response local or identically zero.

Proof. The advanced form reverses the boundary value, while the real local polynomial is unchanged. Lemma lem:dispersion gives (eq:absorptive). The positive densities and nonzero rest-frame projectors show nonvanishing.

A distribution supported only at zero has polynomial Fourier transform and hence no spectral discontinuity. Thus the nonzero discontinuity excludes locality and cannot be cancelled by the polynomial family.

On positive-energy momenta, the norm in Proposition prop:detector can equivalently be read from

This relation identifies spectral detection; it does not construct an instrument.

The contact-independent discontinuity is a scheme-stable feature of this benchmark, not a renormalization-group residue, physical beta function, or the graph-specific curvature-span statement in XVI-M. VI's RG-open quantum certificate and XVI-M's separate scale-matching boundary remain documentary comparisons CheungVI,CheungXVIM.

The closed-time-path construction concerns the same vacuum and two-stress insertion. It is not an all-orders source-evolution theorem.

Use the zero-momentum contacts, or transport a common contact $C$ consistently as described below. Define

and in Fourier space

Away from coincident points, these are respectively $\mathrm{i}/\hslash$ times the time-ordered, anti-time-ordered, lesser, and greater correlators.

The minus sign in $D^{--}$ follows from its anti-time-ordering boundary value after multiplication by $\mathrm{i}/\hslash$; it is not optional. The branch signs in the quadratic functional are separate from these kernel definitions.

For sources $H_{+},H_{-}$, let

All contractions below include spacetime integration and the two tensor slots.

The four kernels satisfy

The quadratic connected CTP cumulant, with $V_H=-g\mathsf{T}(H)$, is

In particular ${{\Gamma }_{\mathrm{CTP}}}^{[2]}[H,H]=0$. For real $H_{\mathrm{d}}$, the imaginary quadratic form is nonnegative.

The construction is restricted to source degree two. The four kernels are not a closed-time-path instrument, a measurement channel, a classical record, or a decoder.

Proof. Equation (eq:disc) gives

The same expression is ${\hat{D}}^{>}+{\hat{D}}^{<}$. Subtracting the negative-energy or positive-energy Wightman component from $D^{++}$ yields the retarded or advanced boundary prescription. The contacts have been transported with opposite signs on the two diagonal branches, so the identities retain them.

The vacuum source-degree-two cumulant has the displayed two self-branch terms and the two cross-branch terms. Substitute $H_{\pm }=H_{\mathrm{c}}\pm H_{\mathrm{d}}/2$. Use the first identity to cancel the $H_{\mathrm{c}}{H}_{\mathrm{c}}$ term and $D^{>}(x-y)=D^{<}(y-x)$ with exchanged tensor pairs. The mixed term is $g^2{H}_{\mathrm{d}}{\Pi }^{\mathrm{R}}{H}_{\mathrm{c}}$; the remaining difference-source term is $\mathrm{i}{g}^2{H}_{\mathrm{d}}N{H}_{\mathrm{d}}/(2\hslash )$.

For a real test, $H_{\mathrm{d}}N{H}_{\mathrm{d}}$ equals a nonnegative vacuum norm, proving the sign. No selective maps or quantum output registers occur in this calculation.

This is a specified quadratic-source use of the CTP framework; the external literature discusses its causal role in stress response AMP. Hawk04-B's classical-record versus quantum-return distinction is only a documentary non-identification: (eq:ctp) is neither output interface CheungHawk04B.

Fix a real smooth time ramp with

It is not required to be a ultraviolet regulator. Take $h,j\in {C_c}^{\infty }(M;S^{2}TM)$ and $H={\sigma }_{\epsilon }h$. Then $H$ is compact and smooth.

Equation (eq:response) is a continuous bilinear functional of $j,h$ on each fixed compact-support test-function space. It is real for real data and retarded: only points of $H$ in the causal past of the observation test can contribute.

Proof. The map $(j,H)\mapsto F_{j,H}$ is continuous into a fixed compact-support smooth test space. Pairing with the constructed distribution is continuous. Smooth multiplication by the fixed ramp is continuous. Reality follows from Theorem thm:retarded. The support of ${\Pi }^{\mathrm{R}}$ supplies the causal statement.

No sharp-preparation limit is needed for this construction.

For a compact smooth vector field $\xi$, put

It is compact and supported in the preparation layer where the ramp derivative meets $\mathrm{supp}\xi$. The product rule gives

For every permitted contact prescription,

If $\xi$ is supported in the plateau $t>\epsilon$, the right side is zero. Thus multiplication by the fixed ramp intertwines that plateau-supported linear action, not the unrestricted action.

For every nonconstant ramp (eq:ramp) and every $T>\epsilon$, there exist real compact $\xi$ in the preparation layer and a real compact tensor test $j$ supported in $t>T$ such that the residual is nonzero. For that detecting pair its value is unchanged by all contacts (eq:contacts).

The residual concerns this prepared external source; it is not the companion's density-weighted cubic source.

Proof. By (eq:productrule), the change in the source is $K({\sigma }_{\epsilon }\xi )-B_{\epsilon }(\xi )$. The first term is annihilated by Theorem thm:ward, proving (eq:residual). On the plateau $\partial {\sigma }_{\epsilon }=0$ on $\mathrm{supp}\xi$, so $B_{\epsilon }(\xi )=0$.

It remains to prove a late nonzero detector. Choose a real nonzero $\chi \in {C_c}^{\infty }((0,\epsilon ))$ such that $b(t)={\sigma }_{\epsilon }^{\prime }(t)\chi (t)$ is nonzero. Take

where $\psi$ is real, compact, and chosen with $\hat{\psi }(0,q,0)\ne 0$ for some $q\ne 0$. Then $B_{\epsilon }(\xi )$ has only its $01$ and $10$ components, equal to $b(t)\psi (\mathbf{x})$.