Landmark VII-H Normalization-independent connected quantum responses: complete ambiguity, observable descent, and a nonzero free-field benchmark

We formulate prescription independence at the response level and complete the three-stage construction for one explicitly declared free-field benchmark. The source and readout are the vacuum-centred Wick square of a massless scalar on four-dimensional Minkowski spacetime. At first order in a compact external source, the complete connected vacuum susceptibility has a punctured retarded kernel $\hslash \Theta (t){\delta }^{\prime }(t^2-|\mathbf{x}{|}^2)/(4{\pi }^3)$.

A locally integrable logarithm gives an explicit retarded extension. Reality, translation covariance, agreement away from coincidence, and scaling degree at most four imply that every admissible normalization differs by exactly $\hslash c{\delta }_0$. Thus the complete response ambiguity is determined, rather than inferred from two regulators. We construct an admissible annular test whose pairing is exactly $-\hslash t_0/(2{\pi }^2)$ for every prescription, and provide a single delayed source/readout experiment with a rigorously nonzero response.

We also prove the general observable-descent criterion, its test-determined topological separation condition, reconstruction descent, and a finite-bank certificate that remains valid when the underlying ambiguity is infinite-dimensional. Calibration, minimum-variance weights, regulator smoothing, and imperfect annihilation are treated separately.

The benchmark is not the reflected triangle of VII-D and is not the unresolved gravitational characteristic coefficient. The latter construction remains open on the supplied data. Classical admission, the nonzero auxiliary remainder of XII-F, and finite instruments in VII-G are not substituted for that missing quantum response.

release{ Constructed: one connected quantum response at first source order; its complete ambiguity in a stated prescription class; an exactly evaluated nonzero invariant test; a delayed source/readout realization; and the general descent and stability results.

Explicit restriction: the affirmative result is for a local free massless scalar benchmark with fixed vacuum, calibrated field and composite normalization, and no physical spacetime boundary.

Not constructed: an embedding of VII-D's reflected triangle in a complete interacting response; the inherited gravitational characteristic quantum coefficient; an interacting gravitational Ward identity; or a measurement instrument for that coefficient.

The benchmark demonstrates the proposed logical sequence without claiming to close an unrelated analytic gate.}

The question is not whether a preferred finite part can be assigned. It is whether a response, after physical matching, has a nonzero pairing independent of every permitted prescription change:

Three distinctions govern the construction:

Local renormalization gives extension and comparison theorems under specified hypotheses BF,HW2001,HW2002. Finite-renormalization frameworks compare interactions and observables together BDF,DF. These results do not identify the ambiguity of an unspecified nonlocal response.

The supplied corpus does not contain a complete construction of the proposed reflected-triangle or gravitational connected response. In particular, the supplied XIV-I and XIV-J statements stop before the required full characteristic composition CheungXIVI,CheungXIVJ.

We therefore do not announce a completion of that response. Instead, we select a different model for which all three arrows in (eq:programme) can be proved in one manuscript:

Its first source derivative is already quantum, singular, connected, and normalization dependent at coincidence. Its full allowed ambiguity can be classified and annihilated without deleting its nonlocal signal.

This restriction is made before any calculation. It is not a reinterpretation of VII-D's coefficient.

VII-E provides an observation framework and, in the displayed version, a conditional section entitled “A dictionary for distributional observations” CheungVIIE. The supplied descriptions that both deny and attribute that dictionary cannot be used simultaneously. We use only its limited interpretation: observations may be blind to declared differences, but blindness does not construct the response.

Reconstruction descent is a separate condition, stated in Em01-A and XIV-J CheungEm01A,CheungXIVJ. The finite separation argument also has a classical prototype in Landmark VI CheungVI. All mathematical criteria needed here are proved below.

VII-G's finite instruments distinguish selective maps, outcome effects, and implementation comparison CheungVIIG. That distinction is retained: constructing a scalar response distribution does not by itself construct an instrument measuring it.

All spaces in this section are real locally convex spaces. Complex responses can be treated by adjoining their real and imaginary components.

Fix:

1. a physical model, preparation, source/readout calibration, and physical boundary problem;

2. a perturbative order $N$;

3. an admitted test space $\mathcal{E}$;

4. a nonempty class ${\mathfrak{P}}_0$ of physically matched prescriptions;

5. a constructed response ${{\mathcal{R}}_{\mathcal{N}}}^{\mathsf{P}}\in {\mathcal{E}}^{\prime }$ for every $\mathsf{P}\in {\mathfrak{P}}_0$.

Every response must be expressed in the same calibrated test coordinates. Its complete difference space is

No finite-dimensionality is assumed.

For formal truncated responses, the definition is applied coefficientwise. Equivalently, collect the finitely many coefficients into a finite direct sum of distribution spaces. An identity through order $N$ is an identity of those coefficients, not an uncontrolled numerical estimate of the omitted orders.

Set

The closure used for continuous observations is

where $\sigma ({\mathcal{E}}^{\prime },\mathcal{E})$ is the weak topology generated by the actual admitted test pairings.

Fix a reference response ${\mathcal{R}}_{\ast }$ in the matched family.

1. The pairing $⟨\mathcal{R},F⟩$ is independent of every prescription in ${\mathfrak{P}}_0$ if and only if $F\in {{\mathfrak{A}}_{\mathfrak{N}}}^{\perp }$.

2. There exists $F\in {{\mathfrak{A}}_{\mathfrak{N}}}^{\perp }$ with $⟨{\mathcal{R}}_{\ast },F⟩\ne 0$ if and only if

   $${\mathcal{R}}_{\ast }\notin {\overline{{\mathfrak{A}}_N}}^{\sigma ({\mathcal{E}}^{\prime },\mathcal{E})}.$$

3. Every invariant test defines a continuous linear functional on the weak Hausdorff quotient

   $${\mathcal{E}}^{\prime }/{\overline{{\mathfrak{A}}_N}}^{\sigma ({\mathcal{E}}^{\prime },\mathcal{E})}.$$

Proof. For (i), prescription independence is exactly the vanishing of the pairing on every generating difference in (eq:complete-ambiguity). Linearity extends it to their span.

If a test annihilates ${\mathfrak{A}}_N$, weak continuity makes it annihilate its weak closure. A nonzero pairing with ${\mathcal{R}}_{\ast }$ therefore implies (eq:detectability).

Conversely, a weak neighbourhood separating ${\mathcal{R}}_{\ast }$ from the closure is determined by finitely many tests $F_1,\dots ,F_m$. Define

Then $T({\mathcal{R}}_{\ast })\notin T({\mathfrak{A}}_N)$; otherwise an ambiguity would match all those pairings and belong to the separating neighbourhood. The linear subspace $T({\mathfrak{A}}_N)\subset {\mathbb{R}}^m$ is closed. Choose $w$ orthogonal to it with $w^{\mathsf{T}}T({\mathcal{R}}_{\ast })\ne 0$. The test $F={\sum }_i{w}_i{F}_i$ has the required properties.

The pairing is continuous in the weak topology and vanishes on the closed subspace. The universal property of the quotient proves (iii).

An algebraically nonzero class need not be detectable by the admitted tests. Replacing the weak closure in (eq:detectability) by a stronger closure requires a separate duality theorem. No reflexivity or completeness assumption is hidden in Theorem thm:descent-observation.

Let $a,b\in {\mathcal{E}}^{\prime }$ be linearly independent and consider

The derivative at $\theta =0$ detects only $a$, but the span of all finite differences contains both $a$ and $b$. A test annihilating $a$ alone need not give a prescription-independent value.

Thus the definition (eq:complete-ambiguity) is used even when prescription changes have a nonlinear parametrization.

Let $P:\mathcal{U}\to {\mathcal{E}}^{\prime }$ be a defined continuous linear map and let $u^{\mathsf{P}}\in \mathcal{U}$. If

then

The observation condition is $⟨Pr,F⟩=0$, not $⟨r,F⟩=0$ unless the two objects have actually been identified.

More generally, if

the ambiguity is contained in the span of the images of the allowed joint changes $({c_{\nu }}^{\mathsf{P}}-{c_{\nu }}^{\mathsf{Q}}{)}_{\nu }$. It equals a proposed image span only when those changes exhaust its generators.

Proof. The first assertions are linearity. For the last assertion, subtract the two response formulas. Independent variation of every catalogue entry is not implied by the formula; correlations among allowed changes must be retained.

If the propagation map also changes, the exact comparison is

The second term cannot be omitted.

Finite renormalizations of an interacting theory require precisely such attention to interactions and observable identifications BDF,DF. The present proposition does not assert that the needed maps exist for the inherited characteristic graph.

Let $\mathsf{Rec}:\mathcal{P}\to \mathcal{K}$ be a surjective linear presentation map. A linear map $\tilde{\mathsf{M}}:\mathcal{P}\to Y$ factors uniquely as

if and only if

If $\mathcal{K}$ has the quotient topology and $\tilde{\mathsf{M}}$ is continuous, the induced map is continuous.

Proof. Factorization annihilates every null presentation. Conversely, two presentations of the same kernel differ by an element of $\ker \mathsf{Rec}$, so (eq:reconstruction-condition) makes their images equal. Set $\mathsf{M}(\mathsf{Rec}p)=\tilde{\mathsf{M}}(p)$. Surjectivity gives uniqueness, and the quotient topology gives continuity.

This is the criterion stated in Cheung's reconstruction manuscripts CheungEm01A,CheungXIVJ. It is not the free-shell equation quotient and not prescription equivalence.

For a one-dimensional coordinate $q$,

Indeed,

Tensoring with a nonzero distribution in other variables leaves the identity unchanged. A rule extracting a highest polynomial degree from the two summands need not descend to their zero reconstruction. This is the elementary mechanism used in XIV-J's null-presentation test CheungXIVJ.

Set

There is no physical reflecting or absorbing boundary.

Let $\phi$ be the real massless free scalar in its Minkowski vacuum ${\omega }_0$, with

Equivalently,

The parameter $\hslash >0$ is retained explicitly.

Define the calibrated vacuum-centred Wick square

Its field-degree coefficient is one and ${\omega }_0(\mathcal{Q}(x))=0$. For real compact tests,

The external source enters with Hamiltonian density

The initial state remains ${\omega }_0$. At first source order, the retarded readout is

where, away from coincident source/readout points,

The sign follows by expanding source evolution with ${\mathcal{H}}_{\mathrm{src}}=-\lambda h\mathcal{Q}$.

The response is connected: its two insertions are joined by vacuum contractions, and the disconnected subtraction ${\omega }_0(\mathcal{Q}(x)){\omega }_0(\mathcal{Q}(y))$ is zero. The selected perturbative order is $N=1$ in $\lambda$. No bulk self-interaction is present.

Take

The response kernel in relative coordinates is $\mathcal{R}\in {\mathcal{D}}^{\prime }(M;\mathbb{R})$. Its joint kernel is the pullback

The difference map $(x,y)\mapsto x-y$ is a submersion, so this pullback exists.

For $H\in \mathcal{D}(M\times M;\mathbb{R})$, define

This is a compact smooth test, and

In particular, for $H=f\otimes h$,

The following data are fixed throughout the benchmark.

A multiplicative change of $\mathcal{Q}$, a source rescaling, a new vacuum, or a mass shift is not a normalization convention in this comparison.

If a different normal-ordering representation writes $\mathcal{Q}$ as a Wick square plus a smooth c-number, that c-number must be included so that the same calibrated operator and state are represented. It is not legitimate to keep the changed symbol and call the resulting preparation unchanged. The distinction between product data and a physical state is also explicit in XII-A CheungXIIA.

The source-local contact freedom concerns the convention for coincident differentiated composite insertions. Formally, an addition

to a connected source-functional normalization has mixed second derivative $\hslash c\int fh$. It does not change the free field equation or the zero-source composite.

If an apparatus calibration also fixes this coincident response, then $c$ is fixed and the admissible class becomes smaller. The invariant experiment constructed below remains valid in that smaller class. No equivalence of unmatched overlapping source/readout calibrations is claimed.

For the calibrated Wick square,

Consequently the punctured retarded response is

Proof. Vacuum normal ordering removes self-contractions at each insertion. There are exactly two pairings connecting the two fields at $x$ to the two fields at $y$. Each contributes ${\hslash }^{2}W(x-y{)}^2$. Terms with an uncontracted normal-ordered field have zero vacuum expectation. This proves (eq:wick-square); subtraction in the commutator gives (eq:punctured-response).

Thus the first-order calculation contains the complete two-insertion vacuum sum. There is no omitted tadpole, disconnected graph, action insertion, or scalar-sourced gravitational slot in this model.

Products of positive-frequency two-point distributions and the Wick-square construction belong to the usual microlocal Wick framework BF,HW2001. The next lemma checks the subsequent retarded operation rather than assuming it.

The distribution (eq:punctured-response) is defined on $M\setminus \{0\}$, has scaling degree four at the omitted point, and equals

Here ${\delta }^{\prime }$ denotes differentiation with respect to the scalar argument $s$.

Proof. The nonzero wavefront covectors of $W$ have one time orientation. There is no opposing pair within $W\cdot W$, so this square is a defined distributional product. The same holds for $W(-z{)}^2$.

On the nonzero future null cone, $t>0$ and $\mathrm{d}s\ne 0$. The one-dimensional boundary-value identity

therefore pulls back to $s$. Using (eq:W-boundary) gives

Substitution proves (eq:cone-kernel).

Away from the origin, the commutator is supported on the nonzero null cones, where the time cut is locally smooth. At $t=0$ and $\mathbf{x}\ne 0$ it is locally zero. Hence the time cut creates no additional punctured product problem.

Under positive dilation, $s$ has degree two and ${\delta }^{\prime }(s)$ has degree minus four. The punctured kernel is a nonzero homogeneous distribution of degree minus four, giving scaling degree four.

The wavefront assertions use the product and pullback calculus Hormander. Normal-topology continuity theorems apply only on compatible cones and support-proper stages BDH; no unverified characteristic product is introduced.

For $\mu >0$, let

This is a locally integrable function. Its definition is not a product of an undefined distribution with a characteristic function.

Define

The square denotes two applications of $\square$, hence four derivatives in total.

The distribution (eq:reference-response) is real, invariant under proper orthochronous Lorentz transformations, retarded, and extends ${\mathcal{R}}^{\circ }$. Its scaling degree is four. For every compact $K\subset M$, there is a finite constant $C_{K,\mu }$ such that

where $p_{j,K}$ is the maximum of the derivatives through order $j$ on $K$.

Proof. Logarithmic singularities are locally integrable transverse to the nonzero cone. At the tip, radial integration also gives a finite integral of a logarithm. Reality, Lorentz invariance, and retarded support follow from the defining function and the operator ${\square }^2$.

We verify the constant using the retarded Riesz family. In dimension four its normalization is

on its locally integrable domain, continued holomorphically as a distribution. The identities

are the standard Riesz identities BGP (Section 1.2).

On the punctured future cone, put $Y_{\beta }(u)=u_{+}^{\beta -1}/\Gamma (\beta )$, continued in $\beta$. Since $Y_{-1}={\delta }^{\prime }$ and $1/\Gamma (\alpha /2)=\alpha /2+O({\alpha }^2)$,

On the other hand,

Differentiate ${\square }^2{\mathsf{R}}_{\alpha +4}={\mathsf{R}}_{\alpha }$ at zero. The derivative of $C$ contributes only a multiple of ${\delta }_0$, and the logarithmic term contributes

It follows that (eq:reference-response) restricts to $\hslash \Theta (t){\delta }^{\prime }(s)/(4{\pi }^3)$, as required.

For the estimate, weak differentiation gives

The absolute logarithmic integral on $K$ is finite, and $\left|{\square }^{2}F\right|\le 16p_{4,K}(F)$.

Finally, positive dilation of this formula gives the exact logarithmic scaling identity proved below. It implies scaling degree at most four. The nonzero punctured homogeneous restriction makes the degree exactly four.

An admissible scalar response prescription is a real distribution ${\mathcal{R}}^{\mathsf{P}}=\hslash K^{\mathsf{P}}$ on $M$ satisfying:

1. $K^{\mathsf{P}}{|}_{M\setminus \{0\}}={\mathcal{R}}^{\circ }/\hslash$;

2. $\mathrm{supp}{K}^{\mathsf{P}}\subset J^{+}(0)$;

3. proper orthochronous Lorentz invariance; its associated joint kernel depends only on $x-y$;

4. ${\mathrm{sd}}_0(K^{\mathsf{P}})\le 4$.

The physical data and calibrations of Section sec:model are fixed. If a regulator is used, its limit must exist and satisfy these conditions. An unsuccessful regulator does not become a prescription merely by being named.

These are axioms for the selected scalar vacuum response at this order, not a classification of all higher time-ordered products. The allowance for a contact term is deliberate; no extra coincident-response calibration has been imposed.

The class in Definition def:prescriptions is nonempty. For every fixed reference scale ${\mu }_0>0$, all its members are exactly

Every real $c_{\mathsf{P}}$ is allowed. Consequently,

For the joint kernel, the difference is

Proof. Lemma lem:reference supplies a member. The difference between any member and the reference is supported at the origin. A point-supported distribution is a finite sum of derivatives of ${\delta }_0$. The scaling degree of ${\partial }^{\alpha }{\delta }_0$ in dimension four is $4+|\alpha |$. The order bound therefore permits only $c{\delta }_0$. Reality makes $c$ real.

Conversely, adding a real multiple of $\hslash {\delta }_0$ preserves every condition in Definition def:prescriptions. Thus the catalogue is exhaustive, not merely an upper bound. Evaluation of ${\delta }_0$ proves the annihilator formula, and the joint statement follows from the difference-map pullback.

This is the point-extension mechanism of microlocal renormalization BF (Theorem 5.3). Cheung's XVI-J has the same type of one-dimensional extension freedom in its stated narrowed numerical sector CheungXVIJ. No identification of those two sectors is needed or asserted here.

For positive scales $\mu ,\nu$,

For $F_{\ell }(z)={\ell }^{-4}F(z/\ell )$, $\ell >0$,

Scale conversion is additive and every test in (eq:actual-annihilator) is insensitive to it.

Proof. The Riesz identity at $\alpha =4$ gives

Subtract the logarithms in (eq:reference-response) to obtain (eq:scale-conversion). For (eq:dilation), change variables $z=\ell w$ in the weak integral. Four differentiations give the factor ${\ell }^{-4}$, while $\log ({\mu }^{2}s(\ell w))=\log ({\mu }^{2}s(w))+2\log \ell$. The same delta identity completes the calculation.

The logarithm in (eq:scale-conversion) is a normalization conversion for this response. It is not a physical beta function or a gravitational Wilson coefficient.

Choose $t_0>a>0$, and let

Set

Then

The second equality uses the evenness of $\rho$.

Define the real compact test

It is smooth because its spatial support avoids $\mathbf{x}=0$. Its support avoids $z=0$, although it crosses the nonzero future null cone.

For every admissible prescription in Definition def:prescriptions,

This value is unchanged by every finite normalization change in the complete class, not only by a selected regulator conversion.

Proof. Since $F_{\ast }(0)=0$, Theorem thm:ambiguity removes every allowed difference. The common value can be evaluated with the punctured cone kernel.

For fixed $t>0$, integrate spatially in polar coordinates and put $u=t^2-r^2$. Near $u=0$,

The ${\delta }^{\prime }$ pairing therefore gives

Multiply by $\hslash /(4{\pi }^3)$, integrate in $t$, and apply (eq:bump-moments).

The class of ${\mathcal{R}}_{{\mu }_0}$ in

is nonzero and is detected by the continuous test $F_{\ast }$.

Proof. A response in the ambiguity subspace would pair to zero with $F_{\ast }$, contrary to (eq:main-value).

The result is quantum: it is the connected response of a quantum composite in a declared vacuum and has coefficient $\hslash$. It is not an auxiliary classical metric variation.

Choose any $v\in \mathcal{D}(M;\mathbb{R})$ with $\int v=1$, and set

This is compact and smooth; (eq:joint-relative) gives $F_{H_{\ast }}=F_{\ast }$. Hence

This is an exact joint-test certificate. A general joint weight is not automatically one product $f\otimes h$. The next subsection provides a single product experiment instead.

The invariant can be observed without coincident source/readout support and without treating an arbitrary joint test as a single experiment.

Let $G$ be a smooth compact test supported at positive times away from the origin. On the cone,

If its cone sampling lies in $t\in [d,D]$, $d>0$, then

Proof. For each $t,\mathbf{n}$, the substitution $u=t^2-r^2$ changes $r^2\mathrm{d}r$ into $r|\mathrm{d}u|/2$. The negative derivative at zero of $rG(t,r\mathbf{n})/2$ is

This proves (eq:annular-pairing). Use $\mathrm{area}(S^2)=4\pi$ to obtain the bound.

Put $q=t_0-a>0$ and

Choose a positive number $\delta$ satisfying

Let

and define

The source test is nonnegative and has integral one. The source lies in $\delta <y^0<3\delta$, while the readout lies at $x^0>2\delta +q>3\delta$. They are strictly time ordered and disjoint.

For every admissible prescription,

has the same value and satisfies

The parameter $\delta$ is an experimental source width, not an ultraviolet regulator.

Proof. Write $y=y_0+\xi$. Formula (eq:correlation-test) becomes

It vanishes at zero because source and readout supports are disjoint. Every contact difference therefore pairs to zero.

Set $G=F_{f_{\delta },h_{\delta }}-F_{\ast }$. The mean-value theorem gives

The cone is sampled only in

Its lower endpoint is at least $q/2$, and the interval length is at most $4a$. Lemma lem:annular yields

Combine this estimate with $⟨{\mathcal{R}}^{\circ },F_{\ast }⟩=-\hslash t_0/(2{\pi }^2)$.

The operational interpretation is the first derivative of the vacuum expectation of the smeared Wick-square readout under a compact quadratic source pulse. The readout smearing is real and signed, as is permissible for a linear observable.

This interpretation does not construct a bounded finite-dimensional instrument for the unbounded Wick-square observable. Such an implementation would require its own measurement specification, as emphasized by the selective-map comparison in VII-G CheungVIIG.

Let $\mathfrak{A}\subset {\mathcal{E}}^{\prime }$ be any linear ambiguity space, possibly infinite-dimensional. Fix tests $F_1,\dots ,F_m$, and define

The finite-dimensional evaluated ambiguity is $T(\mathfrak{A})$.

Choose $a_1,\dots ,a_k\in \mathfrak{A}$ such that their evaluated vectors span $T(\mathfrak{A})$, and set

There is a weight vector $w\in {\mathbb{R}}^m$ with

if and only if $s\notin \mathrm{col}(M)$. Every such $w$ gives a test

annihilating the entire ambiguity $\mathfrak{A}$.

If the evaluated coverage $\mathrm{col}(M)=T(\mathfrak{A})$ has not been proved, the same matrix calculation certifies blindness only to the listed generators.

Proof. The orthogonal complement of $\mathrm{col}(M)$ is $\ker M^{\mathsf{T}}$. A vector in this complement has nonzero pairing with $s$ exactly when $s$ is not in $\mathrm{col}(M)$. For any $a\in \mathfrak{A}$, coverage gives $T(a)\in \mathrm{col}(M)$, so

Without coverage, this reasoning applies only to the known span.

This is the finite-certificate logic used in Cheung's earlier linear quotient analysis CheungVI. It neither constructs an absent response nor proves coverage of an infinite catalogue.

Choose $p\in \mathcal{D}(M;\mathbb{R})$ with $p(0)=1$, and set

Using the complete generator $a_1=\hslash {\delta }_0$,

The exact contrast is

This is a certificate for the complete benchmark ambiguity because Theorem thm:ambiguity already proved its coverage.

Suppose finite parameters $\theta \in {\mathbb{R}}^k$ produce response changes $L\theta$, while calibration data change by $C\theta$. After exact calibration, the residual allowed parameters are $\ker C$.

The response ambiguity remaining after this linear calibration is

An invariant test need annihilate that residual space, not physical parameter directions that have already been fixed.

Proof. Matched parameter changes have $C\theta =0$, and their response images are precisely $L\theta$.

For nonlinear finite matching, replace this formula by the actual finite differences satisfying the calibration conditions. Interaction and observable matching under finite renormalization must be performed together BDF.

Assume the complete evaluated ambiguity is $\mathrm{col}(M)$ and that the measurement covariance $\Sigma$ is positive definite. Set

and let $\Pi$ be the Euclidean orthogonal projection onto $\ker C^{\mathsf{T}}$. If $\Pi q\ne 0$, the unique weights minimizing $w^{\mathsf{T}}\Sigma w$ subject to

are

They are unchanged when $s$ is replaced by $s+M\theta$.

Proof. Put $y={\Sigma }^{1/2}w$. The constraints become $C^{\mathsf{T}}y=0$ and $q^{\mathsf{T}}y=1$, and the variance is ${\Vert y\Vert }^2$. On the permitted subspace, $q^{\mathsf{T}}y=(\Pi q{)}^{\mathsf{T}}y$. Cauchy–Schwarz gives the minimum at $y=\Pi q/{\Vert \Pi q\Vert }^2$. Finally, ${\Sigma }^{-1/2}M\theta \in \mathrm{col}(C)$, which $\Pi$ annihilates.

This is a mathematical measurement-design result. It assumes a specified covariance model and does not derive that model from the field theory.

Suppose an actual evaluated prescription difference is $M\theta$ with $\Vert \theta \Vert \le C_0$. Then

If a numerical matrix $\hat{M}$ satisfies ${\Vert M-\hat{M}\Vert }_{\mathrm{op}}\le {ϵ}_M$ and ${\hat{M}}^{\mathsf{T}}w=0$, then

If each estimated response pairing has deterministic error at most ${ϵ}_i$, the contrast error is at most ${\sum }_i|w_i|{ϵ}_i$.

Proof. Use

and Cauchy–Schwarz. The matrix and pairing estimates follow from the operator norm and triangle inequality.

Without a bound on the allowed coefficients, approximate annihilation does not give a uniform prescription-independence estimate.

Let ${\rho }_{ϵ}$ be a real compact smooth approximate identity with integral one, support in a Euclidean ball of radius $ϵ$, and uniformly bounded $L^1$-norm. For a fixed compactly supported test $F$ whose support avoids the origin,

For sufficiently small $ϵ$, the contact contribution vanishes exactly, uniformly in $c_{\mathsf{P}}$. On a fixed support stage the remaining error is bounded by

where $C$ depends on a reference prescription and the uniform mollifier bound, not on $c_{\mathsf{P}}$.

Proof. Pair convolution with the reflected test convolution. The mean-value theorem bounds its difference from $F$, through four derivatives, by a constant times $ϵp_5(F)$. Apply (eq:reference-estimate) on a common enlarged compact.

The difference between any prescription and the reference is $\hslash c_{\mathsf{P}}{\delta }_0$. After convolution it is supported in the radius-$ϵ$ ball, which is disjoint from $\mathrm{supp}F$ for sufficiently small $ϵ$. Its pairing is therefore exactly zero.

This proposition smooths an already defined distribution. It is not a construction of an otherwise undefined graph product, and finite-$ϵ$ smoothing need not preserve exact causal support.

The benchmark is an exact first derivative. No nonperturbative bound on the $O({\lambda }^2)$ term in (eq:source-expansion) is claimed. Such a bound, calibration uncertainty, and apparatus noise are separate from (eq:smoothing-rate) and (eq:annihilation-error).

The benchmark has only one contact direction because its locality, fixed physical data, punctured kernel, and scaling-degree condition have been proved sufficient. That conclusion must not be transferred to another graph.

On ${{\mathbb{R}}^d}_y\times {{\mathbb{R}}^{\mathbb{k}}}_n$, the distributions

have normal order zero along $\{n=0\}$, yet form an infinite-dimensional vector space. Even if their tangential support is restricted to one point, allowing arbitrary tangential distribution order remains infinite-dimensional.

Proof. Independent smooth functions $a$ give independent distributions, as follows by pairing with a test equal to one near $n=0$. For the point-supported claim, the distributions

are independent, as monomial tests with a cutoff isolate their coefficients.

This is the distinction emphasized in Cheung's punctured normal-jet analysis CheungXVF. It is compatible with the finite-dimensional point-extension result used in the benchmark: the latter also bounds the total scaling degree.

Let $E^{\mathrm{R}}$ be the wave retarded operator and $c$ a smooth compact function. A first-order change of the wave operator by multiplication with $c$ changes its inverse by

For compact tests $f,h$, the measured change is

The pairing is defined on smooth propagated tests and may be nonzero with separated $f,h$. Choose $h$ and $f$ so their retarded and advanced responses are nonzero at one intermediate point; continuity permits a compact $c$ of fixed sign in a sufficiently small neighbourhood.

Thus external separation eliminates the benchmark's completely classified coincidence ambiguity, not an arbitrary action counterterm. Cheung's Em02-E makes the same local-to-nonlocal distinction for its spatial Green-kernel model CheungEm02E.

Choose $F\in {C_c}^{\infty }((0,\epsilon ))$, $F\ge 0$, $F\not\equiv 0$. The sources $F$ and zero agree for $t\ge \epsilon$, but their zero-data response difference is

For $t\ge \epsilon$,

This is the memory test recorded in XVI-B CheungXVIB. Source agreement, response agreement, and observable agreement are separate assertions.

Given finitely many compact tests $F_1,\dots ,F_m$ on an open set, choose a nonzero smooth compact function $u$ orthogonal to their finite-dimensional span. Then

A vanishing bank therefore does not imply a zero response distribution. Theorem thm:bank tests a declared quotient on that bank; it is not a general inverse theorem.

Suppose the supplied VII-D family satisfies

as distributions on its joint test space. This is a comparison input about that numerical construction CheungVIID, not the benchmark kernel of this paper.

If $r\ne 0$, the exact relation (eq:VII-D-conversion) implies

If a fixed linear response map $P$ has been constructed, its conversion is $c_{ab}P(r)$.

Let $T$ be a differential operator of order $d$ with fixed smooth coefficients on a compact test stage. An estimate

implies

Proof. Subtract the three distributions to obtain $(c_{ab}+c_{bc}-c_{ac})r=0$, and use $r\ne 0$. The response statement is Proposition prop:propagation. For the last estimate, pair $v_R$ with $TF$ and use the compact-stage bound $p_{K,64}(TF)\le C_T{p}_{K,64+d}(F)$.

This explains the test-order loss under the numerical transfer discussed in IX-B CheungIXB. It does not factor the numerical finite part through a missing sharp binary multiplication.

If the response includes independent action, composite, preparation, or boundary-normalization changes, their actual response images must be added. Relation (eq:VII-D-conversion) alone does not prove that the complete ambiguity is $\mathbb{R}P(r)$.

The supplied IX-A statement gives a continuous coordinate separating the unchanged regular approximants from the proposed sharp insertion. The supplied IX-B statement distinguishes a ternary numerical finite part from a second sharp binary product CheungIXA,CheungIXB. The VII-C obstruction concerns its stated extension and continuity hypotheses CheungVIIC.

None of those problems is used in the benchmark. Conversely, the benchmark's explicit distribution does not construct the missing sharp insertion, change its admission topology, or contradict those results.

The supplied XII-A–F chain has the following scope:

1. XII-A constructs a free-shell carrier and ordering transport.

2. XII-B distinguishes density weighting from source weighting.

3. XII-C selects the cubic-only density modification and proves its actual Euler source.

4. XII-D constructs a smooth classical retarded response and an admitted completed insertion.

5. XII-E assembles the second metric slot with the sign ${{\Phi }_{\mathrm{D}}}^{(2)}={{\mathcal{A}}_2}^{\mathrm{sc}}-{\mathcal{C}}_{3,\mathrm{D}}$.

6. XII-F evaluates a nonzero auxiliary remainder even when the propagated companion vanishes.

These are distinct claims CheungXIIA,CheungXIIB,CheungXIIC, CheungXIID,CheungXIIE,CheungXIIF.

In particular, the supplied XII-F witness has the signed value

It is a classical variation of a named off-shell lift tested on an initial free-shell configuration. It is not the response (eq:kubo), not a normalization difference, and not a quantum Ward identity.

The affine second-slot identity already determines the minus sign. An admission theorem alone does not prove a kernel nonzero, and a second co-transformation row must not be added to a slot whose variation already contains that transformation.

The supplied XIV-H statement leaves an internal amplitude obligation open. XIV-I subsequently supplies its named internal amplitude construction at I2, but leaves the full joint characteristic map open at I3. XIV-J supplies reconstruction and an off-characteristic reference for the supported sector, but leaves the full map open at J2 CheungXIVH,CheungXIVI,CheungXIVJ.

Thus it would be inaccurate to treat the earlier H2 statement as if XIV-I had made no progress. It would also be inaccurate to replace the later missing joint map by the already defined internal tree kernels.

The remaining obligations for that inherited target are:

1. the complete joint characteristic map, with reconstruction descent;

2. a joint section and controlled complete remainder;

3. the required transverse estimates;

4. contact compatibility;

5. the output wavefront theorem;

6. admission of the actual quantum coefficient.

No step is supplied by the scalar benchmark. Abstract extension or descent interfaces are not that graph operation.

different final statuses{ The local benchmark of this paper: Constructed. Its full first-order ambiguity and a nonzero invariant quantum pairing are proved.

The inherited reflected-triangle/gravitational target: Unresolved on the supplied data. No embedding of VII-D's coefficient into a complete quantum response is claimed, and no full characteristic coefficient is admitted by this paper.}