One declared classical-limit package for a relational observable, and the stops it does not close Em06-A: a scalar-only coherent limit of the first plateau coefficient

We construct a controlled classical limit for one precisely weakened relational observable: the first plateau-compatible scalar coefficient with a prescribed classical metric coefficient and only the free scalar quantized. The classical functional, its restricted gauge domain, and its dependence support are specified and proved before quantization. An exact compact scalar test represents the functional. For a declared family of scaled canonical-commutation-relation representations and coherent states, its mean equals the named classical value for every positive classicality parameter, while its variance is exactly that parameter times a finite one-particle norm. Its spectral law converges to the classical point mass in the quadratic Wasserstein metric. A nonzero example and a preparation-response diagnostic are evaluated.

The outcome is a repaired, scalar-only classical-limit theorem, not quantization of the metric-dependent insertion as a joint gravitational functional. The metric jet, Minkowski causal structure, scalar equation, switch, initial surface, and preparation are inputs. No metric reconstruction, Einstein dynamics, interacting gravitational product, holographic map, or inherited quantum-response coefficient is obtained. The separate product and completion stops remain untouched.

The object studied here is the first-order plateau relational coefficient associated with the construction described in Cheung's Manuscript VII-A, specifically its thm:first CheungVIIA. We retain its first-order architecture: a compactly switched scalar response, unswitched zero-Cauchy harmonic coordinates, and gauge parameters supported inside the switch plateau and away from the initial surface. All definitions and the required classical identity are reproduced and proved below.

The quantum question is deliberately narrower. The metric coefficient is fixed as an external classical parameter. Only the scalar is quantized, and only the first relational coefficient is represented. This restriction is substantial: the joint metric–scalar insertion problem of VII-B is not the scalar-only problem constructed here CheungVIIB. The distinction between external-metric, finite-mode metric, and composed stress-response objects is also explicit in Manuscript V CheungV.

outcome. In the terminology of the Em06 prospectus CheungDossier, the result is a repaired limit. The exact change from the larger proposed quantum target is:

1. the metric coefficient is classical, not a quantum field;

2. the observable is the first plateau coefficient, not the full nonlinear relational scalar or its second-order completion;

3. the quantum carrier is a free scalar Weyl representation, not an interacting gravitational polynomial algebra;

4. the limit varies the effective CCR parameter, not a switch, boundary regulator, or ultraviolet cutoff.

These restrictions are part of the theorem, not conclusions about the unrestricted problem.

The coherent-state concentration mechanism is standard Glauber,Hepp. The contribution here is its explicit application to a named first-order relational coefficient, including the density operator, sign audit, compact test, restricted gauge identification, nontrivial example, and separate fluctuation estimate.

different claims. The construction attempts only operator–symbol correspondence in this scalar-only model:

Here “operator reconstruction” does not mean boundary-to-bulk reconstruction. Holographic entropy and entanglement-wedge reconstruction require additional, different structures RT,DHW; none is an input of this proof.

discipline. Cheung references are identified by the versions and labels supplied with the problem. No conclusion about an unexamined later version is asserted. Objects described only in the supplied prospectus, including the VII-H comparison, are cited as documentary comparison material, not used as theorem premises. The mathematical proof below does not assume registration, compilation, machine verification, numerical execution, calibration, or experimental realization.

Fix

and a mass $m>0$. Set

Fix $T>0$, a real switch

and an open set $U$ with compact closure in $(0,T)\times {\mathbb{R}}^3$, such that $\chi =1$ on a neighborhood of $\overline{U}$.

The admitted gauge parameters are exactly

For $\xi \in {\mathfrak{g}}_U$, define

The metric coefficient $h$ is an arbitrary real smooth symmetric covariant tensor on $M$. The scalar input ${\varphi }_0$ is a real smooth solution of $P_m{\varphi }_0=0$. The observation density is

All these data are independent of the classicality parameter introduced later.

Indices are raised and lowered with $\eta$. Write

and

The scalar density perturbation is the differential operator

In particular, $Q_{\chi h}$, not $\chi Q_h$, is the switched operator. Derivatives of the switch are retained.

For orientation, this is the coefficient in the Euler equation of the declared first-order scalar action

Its equation is $P_m\varphi +\kappa Q_{\chi h}\varphi =0$. The parameter $\kappa$ labels the classical metric jet; it is not the classicality parameter.

Let ${G_m}^{\mathrm{R}}$ and ${G_m}^{\mathrm{A}}$ be the retarded and advanced Green operators of $P_m$. For compact smooth sources,

With $D_m=\sqrt{-\Delta +m^2}$, the chosen signs are

These formulas give the Green identities and finite propagation. They also fix the signs independently of differently normalized Green operators in comparison manuscripts. Standard general Cauchy and Green-operator results may be found in BGP.

Define $w[h]$, for $t\ge 0$, by the componentwise massless Cauchy problem

This problem has a unique smooth solution on every finite time slab. For arbitrary smooth $h$, it is understood locally through finite propagation; no spatial Fourier transform of an arbitrary nondecaying $h$ is required.

The formal metric is $g_{\kappa }=\eta +\kappa h$. Indeed,

Consequently, ${X_{\kappa }}^{\mu }=x^{\mu }+\kappa w[h{]}^{\mu }$ obeys the harmonic coordinate equation through first order. Its inverse has first coefficient $-w[h]$.

The switched first scalar coefficient is

Its source is compact and lies strictly to the future of ${\Sigma }_0$, so it has zero Cauchy data there.

The observable of Em06-A is exactly

It is the first coefficient of the relational scalar obtained by composing the switched scalar solution with the inverse harmonic coordinate map and integrating against the fixed density in relational-coordinate space.

This definition names a coefficient, not an approximation whose uncontrolled remainder is silently discarded in the classical limit. The theorem below concerns (eq:observable) itself. It asserts no $O({\kappa }^2)$ estimate for a full finite-$\kappa$ observable.

Later $g_{\mathrm{cl}}$ denotes the prescribed metric one-jet

The scalar preparation must also be specified: a relational scalar coefficient is not a functional of the metric alone. We therefore write the classical target honestly as

If ${\mathcal{O}}_f[g_{\mathrm{cl}}]$ is used as shorthand, the fixed scalar preparation, switch, and initial surface are suppressed inputs. None is reconstructed from quantum data.

For every smooth $\xi$,

Thus, on a free scalar solution,

Moreover, $Q_h$ is formally self-adjoint for real $h$.

Proof. The trace of $K\xi$ is $2{\partial }_{\mu }{\xi }^{\mu }$. Differentiating (eq:C) gives (eq:CK).

Expanding (eq:Q) gives

Also,

Substitution proves (eq:QK). The on-shell identity follows from $P_m{\varphi }_0=0$. Formal self-adjointness follows by integration by parts in the divergence-form expression (eq:Q).

For every datum in Specification spec:data and every $\xi \in {\mathfrak{g}}_U$,

This is invariance under the declared additive first-order gauge action, not unrestricted diffeomorphism invariance.

Proof. Since all derivatives of $\xi$ are supported where $\chi =1$, $\chi K\xi =K\xi$. Lemma lem:gaugeidentities gives

The last equality holds because $X_{\xi }{\varphi }_0$ is compactly supported and vanishes near ${\Sigma }_0$.

Likewise, (eq:CK), the vanishing initial data of $\xi$, and uniqueness of (eq:w) imply

The two terms in (eq:observable) therefore cancel exactly.

This reproduces the needed first-order content of VII-A's thm:first within the explicitly printed conventions CheungVIIA. No second-order source theorem is imported.

Let $K_f=\mathrm{supp}f$, and let

This is compact. The functional ${\mathcal{O}}_f$, regarded as a functional of smooth $h,{\varphi }_0$ before restricting ${\varphi }_0$ to the free shell, has dependence support contained in ${\mathcal{D}}_f$.

Proof. The retarded scalar response observed in $K_f$ uses sources only in $J_{\eta }^{-}(K_f)$. Its source is restricted by $\chi$ to $0<t<T$. The coordinate response in $K_f$ uses $C(h)$ only between ${\Sigma }_0$ and $K_f$, again inside $J_{\eta }^{-}(K_f)$. The direct scalar factor in the coordinate term is evaluated in $K_f$. Differentiation does not enlarge these supports.

A finite-time Minkowski causal past of a compact set has bounded spatial extent on the closed slab $0\le t\le T$, proving compactness. The conclusion follows from finite propagation.

For a compact $\xi$ supported to the future of ${\Sigma }_0$, but not necessarily in the plateau, the same calculation yields

There is no reason for this expression to vanish. No step below enlarges ${\mathfrak{g}}_U$.

Fix $h$ as a classical parameter. Put

and define

The function $q_{f,h}$ is real, smooth, and compactly supported strictly to the future of ${\Sigma }_0$. For every smooth free solution,

More precisely,

For fixed $h,\chi$ and fixed compact observation support $K$, the map $f\mapsto q_{f,h}$ is continuous from ${C_K}^{\infty }$ into ${C_{K\cup \mathrm{supp}\chi }}^{\infty }$.

Proof. Transpose the retarded Green operator and then the density operator:

The integrations by parts have no boundary contribution because $\chi h$ is compactly supported away from ${\Sigma }_0$. For the coordinate term,

This proves (eq:representation) with the sign in (eq:q).

Smoothness follows from smooth Green evolution and (eq:w). The first term has support in $\mathrm{supp}\chi \cap J_{\eta }^{-}(K_f)$; the second has support in $K_f$. Both sets are compact and separated from $t=0$.

For continuity, on the finite time interval containing the source and observation supports, (eq:advanced) gives continuous dependence of every derivative of $u_f$ on finitely many derivatives of $f$. This can be seen by commuting spatial derivatives through the sine propagator and applying Sobolev estimates on an enlarged compact spatial set; time derivatives follow from $P_m{u}_f=f$. Multiplication by the fixed smooth coefficients and application of the displayed finite differential operators then give the asserted fixed-support continuity.

There is also a useful exact gauge identity at the level of tests.

For $\xi \in {\mathfrak{g}}_U$,

Thus the tests define the same class modulo $P_m{C}_c^{\infty }(M)$.

Proof. Use $\chi K\xi =K\xi$, $w[K\xi ]=\xi$, and $P_m{u}_f=f$:

Writing $d_{\xi }={\partial }_{\mu }{\xi }^{\mu }$, (eq:QK) gives

The remaining divergence is $d_{\xi }f+X_{\xi }f$, proving (eq:q-gauge). Compactness follows from the compact support of $\xi$.

The compact test $q_{f,h}$ is obtained {after fixing $h$}. It does not establish joint microcausal admission of the original metric–scalar kernel. In particular, the metric dependence through the zero-Cauchy map $w[h]$ can retain the initial-surface structure which is central to VII-B's obstruction CheungVIIB. A smooth scalar test with a classical parameter is a different typed object from a distributional kernel with a quantized metric slot.

Let

The inner product is linear in its second argument. For a real compact smooth scalar test $F$, define

The map $F\mapsto k_F$ is continuous from each fixed-support $C^{\infty }$ test space into $\mathfrak{h}$. Also

Proof. Integrating by parts spatially gives, for every integer $n\ge 1$,

Since $m>0$, already a suitable fixed $n$ gives an integrable square on ${\mathbb{R}}^3$. On a fixed compact support, the $L^1$ norm is controlled by finitely many $C^{\infty }$ seminorms. This proves the norm estimate and continuity.

The exponential in (eq:k) solves $P_{m}v=0$. Integration by parts against a compact $v$ proves $k_{P_{m}v}=0$.

Choose once and for all $z_{\mathrm{cl}}\in \mathcal{S}({\mathbb{R}}^3;\mathbb{C})$. The corresponding real smooth free scalar is

It satisfies

Real compact smooth Cauchy data are included in this preparation class. With the unitary spatial Fourier transform, their amplitude is

On $\mathcal{H}={\mathcal{F}}_{\mathrm{s}}(\mathfrak{h})$, let $a(v)$ be antilinear in $v$, with

Define

For $0<ϵ\le 1$, the scaled scalar field is

Equivalently, on the underlying real space of $\mathfrak{h}$, set

Its Weyl relation is

Thus $ϵ$ is a dimensionless effective Planck parameter, with a reference unit of action fixed. It is not $\kappa$, not an edge width, and not a cutoff.

Fix the prescribed $h_{\mathrm{cl}}$, and abbreviate

The quantum representative of the named observable is the section

The classical symbol ${\mathcal{O}}_f[h_{\mathrm{cl}};\cdot ]$ is fixed while its CCR representation varies. This is the states-or-representations interpretation of the Em06 acceptance pair.

For every $v\in \mathfrak{h}$, $\Phi (v)$ is essentially self-adjoint on the finite-particle domain ${\mathcal{D}}_{\mathrm{fin}}$. Coherent vectors have finite moments of every order of its self-adjoint closure.

Proof. On vectors with at most $r$ particles, the creation and annihilation estimates give

The analytic-vector series converges for every fixed sufficiently small positive argument, indeed for every argument. The dense finite-particle domain therefore consists of analytic vectors. The analytic-vector criterion gives essential self-adjointness Nelson.

For completeness, if

the exponential-vector expansion yields the unitary displacement identity

The vacuum has finite moments of every order, by the same finite-particle estimates. Applying (eq:displacement) to the vacuum proves the coherent-vector assertion.

All field powers and expectations below refer to these closures and their finite coherent-vector moments. No formal expectation of an undefined unbounded insertion is used.

Define

Each ${\Psi }_{ϵ}$ is a normalized vector. Its restriction to the represented Weyl algebra is consequently a positive normalized state. Explicitly,

To verify the formula, normal ordering gives

and therefore $⟨\Omega ,e^{\mathrm{i}t\Phi (v)}\Omega ⟩=e^{-t^2\Vert v{\Vert }^2/2}$. Conjugating by the displacement in (eq:states) then gives (eq:characteristic). This construction is the familiar coherent-state mechanism Glauber,Hepp, with its normalization printed here.

Fix Specification spec:data, $h=h_{\mathrm{cl}}$, and the preparation $z_{\mathrm{cl}}$ of (eq:classicalfield). For the observable (eq:observable), the quantum section (eq:quantumobservable), and the states (eq:states), the following statements hold.

1. Mean. For every $0<ϵ\le 1$,

   $$\boxed{{\omega }_{ϵ}({\hat{\mathcal{O}}}_{f,ϵ})={\mathcal{O}}_f[g_{\mathrm{cl}};{\varphi }_{\mathrm{cl}}].}$$

   In particular, the mean converges in the ordinary real topology to the named classical value.

2. Variance. For every $0<ϵ\le 1$,

   $$\boxed{{\mathrm{Var}}_{{\omega }_{ϵ}}({\hat{\mathcal{O}}}_{f,ϵ})=ϵ\Vert k_{q_{f,h_{\mathrm{cl}}}}{\Vert }^2⟶0.}$$

3. Spectral topology. Let ${\mu }_{f,ϵ}$ be the spectral probability law of ${\hat{\mathcal{O}}}_{f,ϵ}$ in ${\Psi }_{ϵ}$, and put $o_f={\mathcal{O}}_f[g_{\mathrm{cl}};{\varphi }_{\mathrm{cl}}]$. Then

   $${\mu }_{f,ϵ}=\mathcal{N}(o_f,ϵ\Vert v_f{\Vert }^2),$$

   with the point-mass interpretation if $v_f=0$, and

   $$W_2({\mu }_{f,ϵ},{\delta }_{o_f})=\sqrt{ϵ}\Vert v_f\Vert .$$

   Thus the spectral laws converge in the quadratic Wasserstein metric, not merely by a cutoff estimate.

4. Restricted gauge independence. Replacing $h_{\mathrm{cl}}$ by $h_{\mathrm{cl}}+K\xi$, $\xi \in {\mathfrak{g}}_U$, leaves the represented observable, its classical value, and its spectral law unchanged.

5. Fixed-support uniformity. For any bounded subset $\mathcal{B}\subset {C_K}^{\infty }((0,T)\times {\mathbb{R}}^3;\mathbb{R})$,

   $$\underset{f\in \mathcal{B}}{\sup }{\mathrm{Var}}_{{\omega }_{ϵ}}({\hat{\mathcal{O}}}_{f,ϵ})\le ϵC_{\mathcal{B}},\underset{f\in \mathcal{B}}{\sup }{W}_2({\mu }_{f,ϵ},{\delta }_{o_f})\le \sqrt{ϵC_{\mathcal{B}}},$$

   where $C_{\mathcal{B}}<\infty$ is independent of $ϵ$.

The metric jet and causal structure are prescribed inputs. The theorem asserts neither metric reconstruction nor a derivation of Einstein dynamics.

Proof. By (eq:displacement),

The vacuum mean of $\Phi (v_f)$ is zero. Equations (eq:classicalpairing) and (eq:representation) identify the constant in (eq:shiftedoperator) with $o_f$. This proves the mean statement separately.

For the variance, center (eq:shiftedoperator). Since $\Phi (v_f)\Omega =a^{†}(v_f)\Omega$,

Lemma lem:k proves that the norm is finite. This proves the vanishing-variance statement without inferring it from the mean.

The characteristic function of the spectral law, obtained from (eq:characteristic) with $v=t{v}_f$, is

This identifies the stated Gaussian law. Every coupling to the point mass ${\delta }_{o_f}$ has second coordinate $o_f$, so

For gauge independence, Proposition prop:testgauge and Lemma lem:k give

Hence the quantum representatives agree as operators. Classical agreement is Proposition prop:plateau.

Finally, the continuous linear maps of Proposition prop:test and Lemma lem:k send a bounded fixed-support test set to a bounded subset of $\mathfrak{h}$. Taking

proves both uniform estimates.

For every named $f$,

Proof. This is the centered norm evaluated in the proof of Theorem thm:main.

The convergence is state-relative and spectral. It is not operator-norm convergence of unbounded fields, and it is not uniform over all states. Even for the bounded Weyl functions, if $v\ne 0$, the spectrum of $\Phi (v)$ is $\mathbb{R}$, and

The theorem therefore makes no norm-continuous identification of the entire quantum algebra with a classical algebra.

The acceptance pair should not be confused with concentration of the zero operator. The following admissible data give a nonzero observable and an explicit classical target.

Choose

Let

be nonnegative and nonzero, and let $\rho \in {C_c}^{\infty }({\mathbb{R}}^3)$ be nonnegative with $\int \rho {\mathrm{d}}^{3}x=1$. Set

Choose $\chi$ equal to one on a neighborhood of

Such a compact switch separated from $t=0$ exists.

Choose compact smooth scalar Cauchy data equal to $(1,0)$ on a ball containing the initial causal dependence region of ${\mathcal{D}}_f$. Then their amplitude belongs to $\mathcal{S}({\mathbb{R}}^3)$, and

In particular $v_f\ne 0$.

Proof. Finite propagation and uniqueness imply ${\varphi }_{\mathrm{cl}}(t,x)=\cos (mt)$ throughout ${\mathcal{D}}_f$. For the stated metric coefficient,

The unswitched coordinate solution is consequently

On the observation dependence region, the switch is one whenever $h$ is nonzero. The scalar density source on the free shell is

The retarded solution therefore gives

in the observation region. Subtracting the coordinate term yields

Integrate the first term by parts. Its endpoint terms vanish. The trigonometric identity

gives

Integrating against $f$ proves (eq:witness).

For $s\in (t_1,t_2)$, $t\in (t_3,t_4)$, both $\cos (ms)$ and $\sin (m(t-s))$ are strictly positive. The integral is therefore strictly negative. If $v_f=0$, (eq:classicalpairing) and (eq:representation) would make the same value zero, a contradiction.

A response diagnostic must specify what is varied. Here it is a classical scalar Cauchy amplitude, not a gravitational source law.

For $\zeta \in \mathcal{S}({\mathbb{R}}^3;\mathbb{C})$, let

Write ${\varphi }_{\zeta }$ for the real free scalar given by (eq:classicalfield) with amplitude $\zeta$.

For the same observable as in Theorem thm:main,

The equality holds for every $ϵ>0$; hence the response has the stated classical limit.

For the data of Proposition prop:witness, take $\zeta =z_{\mathrm{cl}}$. The response is precisely the strictly negative number in (eq:witness).

Proof. The displacement calculation gives the exact affine formula

Differentiate this identity. The second equality in (eq:diagnostic) follows from (eq:classicalpairing) and (eq:representation). There is no interchange of an uncontrolled derivative and limit. The final assertion follows from ${\varphi }_{\zeta }={\varphi }_{\mathrm{cl}}$ and Proposition prop:witness.

This diagnostic evaluates the named relational coefficient. It is not VII-G's instrument, an entanglement or recovery statistic, En01-B's sphere contrast, or En01-C's total-diffusion certificate CheungDossier. It does not identify a physical gravitational mechanism or supply a detector preparation and residual-error ledger.

For the frozen data and F1 hypotheses of Cheung's XI-G thm:obstruction, the supplied result is

Here $[u{]}_{\mathrm{sol}}$ is the class of compact cubic kernels with two metric slots and one scalar slot, identified by equality on the declared free solutions; ${\mathcal{F}}_{\mathrm{mc},\mathrm{pol}}$ is the predecessor's admissible microcausal polynomial target CheungXIG. We neither enlarge that target nor change the frozen class.

Theorem thm:main constructs no representative of $[u{]}_{\mathrm{sol}}$ in ${\mathcal{F}}_{\mathrm{mc},\mathrm{pol}}$, and no interacting gravitational product. It therefore provides no repair of (eq:XIG). In particular, any additional claim assigning such a representative under the unchanged XI-G hypotheses would still contradict (eq:XIG), independently of a classical-limit estimate.

Proof. The constructed map is

The metric argument has already been fixed before the first quantum map. Its output is a scalar-linear field in a free scalar representation. No step has an output with the two quantum metric slots and scalar slot of $u$; no map between those kernel classes is defined. Products of the resulting free scalar operators, when defined on their domains, are not the missing gravitational retarded products.

If a further construction produced a representative in the stated intersection, its existence would contradict (eq:XIG). An estimate on expectation values of a different, already-defined operator does not alter that set-theoretic contradiction.

The same typing distinction prevents transferring this theorem to the sharp joint insertion of VII-B CheungVIIB, or to X-G's separate variational and one-loop questions CheungDossier. The limit here is not an interacting quantum product.

The supplied prospectus describes VII-H as a free massless Minkowski Wick-square susceptibility, with contact ambiguity $\mathbb{R}\hslash {\delta }_0$ and a prescription-independent pairing

under that construction's hypotheses CheungDossier. This is documentary comparison material only. Its full construction is not assumed here.

The present coefficient is a massive scalar-linear relational response, and its variance coefficient is

It is not VII-H's pairing and is not an inherited gravitational characteristic coefficient. Even if both quantities scale with an effective Planck parameter, that common scaling does not identify their domains, kernels, states, or observables.

No counterterm is introduced in this proof. More generally, a local distributional correction and its propagated image are different objects: for a retarded inverse, propagation can enlarge support. Nothing here replaces the propagated image by the local contact itself.

The following is a scope ledger, not a list of assumptions used in the proof.

The data flow is

There is no reverse map reconstructing the background.

The construction neither uses nor derives an entropy–area dictionary. It supplies no boundary region, code subspace, bulk–boundary encoding, entanglement wedge, or reconstruction theorem of the type studied in RT,DHW. It also does not derive gravitational field equations from entanglement constraints; results of that different kind retain their own holographic hypotheses Faulkner. No Einstein equation is imposed or deduced for $h_{\mathrm{cl}}$.

Consequently the logical separation is retained:

Let $v_f\ne 0$. In the unscaled representation, every coherent state $D(z)\Omega$ has

Thus coherent displacement alone, at fixed CCR and for the unscaled observable, does not prove the absolute vanishing-variance acceptance condition.

Proof. Apply (eq:displacement). Displacement changes the mean by a constant and leaves the centered vacuum variance unchanged.

This is why $ϵ$ and its representation are specified explicitly. The theorem is not a claim that large occupation alone makes the unscaled absolute fluctuations vanish.

The chosen coherent states obey

For the smooth preparation,

Thus the scaled classical energy is finite while the expected unscaled particle number grows. No bounded-resource preparation or laboratory implementation is inferred.

Two further substitutions have been excluded:

1. The correct scalar density perturbation is $Q_{\chi h}$. Replacing it by $\chi Q_h$, or omitting its trace–mass term, would change the declared action and the off-shell test calculation.

2. The notation ${\mathcal{O}}_f[g_{\mathrm{cl}}]$ cannot erase the scalar preparation. The exact theorem names both $g_{\mathrm{cl}}$ and ${\varphi }_{\mathrm{cl}}$, and concerns the first coefficient only.

For the fixed data printed in this manuscript:

1. the classical functional (eq:observable), its dependence support, and its plateau gauge domain are defined and proved;

2. its scalar-only compact-test representation (eq:q)–(eq:representation) is exact;

3. the quantum representative, its self-adjoint domain, and the positive normalized state family are constructed;

4. the mean and variance acceptance conditions are proved separately by (eq:mean) and (eq:variance);

5. spectral convergence holds in $W_2$, with the exact rate (eq:W2) and fixed-support uniformity;

6. a nonzero classical target and the preparation-response diagnostic are evaluated by (eq:witness) and (eq:diagnostic).

The outcome is a repaired scalar-only limit for the first plateau relational coefficient.

No quantized metric insertion, full nonlinear relational limit, interacting gravitational product, unrestricted gauge theorem, Einstein dynamics, metric reconstruction, or holographic map is released. The XI-G class obstruction is untouched. No inherited quantum-response coefficient is admitted; XVI-N's separate Route B is not discharged; O.1 is not closed; H2, I3, and J2 are not resolved by this manuscript; XV-B is not opened.

Proof. Propositions prop:plateau and prop:support prove (i). Propositions prop:test and prop:testgauge prove (ii). Lemma lem:domains and the explicit vector construction (eq:states) prove (iii). Theorem thm:main proves (iv) and (v). Proposition prop:witness and Theorem thm:diagnostic prove (vi).

The maps in the proof retain a classical metric parameter and a free scalar carrier throughout. Proposition prop:noadmission establishes the non-transfer to the frozen cubic admission problem. No map or dynamical object needed for any of the excluded claims is constructed. Those claims are therefore not additional parts of this release.

boundary. All conclusions are analytical statements under the printed hypotheses. Registration, compilation, formal verification, numerical execution, calibration, and physical realization are not asserted.

CheungVIIA

N. Cheung, Plateau-compatible relational scalars: a second-order classical intertwiner. Landmark VII-A. Manuscript VII-A, version 1, n.d. Relevant supplied statement: thm:first, lines 973–1012. Enter Calculus Companion. https://enter.grok.me/browse/VII-A. The first-order construction needed here is reproduced explicitly.

CheungVIIB

N. Cheung, Landmark VII-B: Cauchy harmonic insertions and second-order retarded products: a sharp-boundary obstruction in an off-shell polynomial domain. Manuscript VII-B, version 1, n.d. Relevant supplied statement: thm:main, lines 924–985. https://enter.grok.me/browse/VII-B.

CheungV

N. Cheung, Admissible currents and first retarded corrections for a fixed-time scalar response. Manuscript V, version 1, n.d. External-metric, one-oscillator, and Newton–wave objects are separate constructions. https://enter.grok.me/browse/V.

CheungIV

N. Cheung, Represented records for three named scalar-field models: finite-plane corrections, dense-TT corrections, and sourced curvature pairings. Manuscript IV, version 1, n.d. Relevant supplied material: Model 3 and thm:model3. https://enter.grok.me/browse/IV.

CheungIII

N. Cheung, Descent of first-order relational scalars, and the free-product obstruction. Manuscript III, version 1, n.d. Classical quotient and gauge constructions; comparison only. https://enter.grok.me/browse/III.

CheungVI

N. Cheung, Relational Einstein–scalar observables at second order: classical descent and a compact-switch obstruction. Manuscript VI, version 1, n.d. Classical descent and supplied quantum-status audit; comparison only. https://enter.grok.me/browse/VI.

CheungVIIIB

N. Cheung, A sourced spinless conical packet with a persistent three-to-six polyhedral slicing event. Landmark VIII-B. Manuscript VIII-B, version 2, n.d. Relevant supplied statement: thm:dictionary, lines 451–469. https://enter.grok.me/browse/VIII-B.

CheungVIIIC

N. Cheung, Landmark VIII-C: Certified Fixed-Packet Flow, Guarded Events, and Finite Reversible Composition. Manuscript VIII-C, version 2, n.d. Relevant supplied statement: thm:C5, lines 1512–1566. https://enter.grok.me/browse/VIII-C.

CheungXIG

N. Cheung, Manuscript XI-G, version 1, n.d. Relevant supplied statements: def:class, lines 576–589; thm:criterion, lines 1036–1069; thm:obstruction, lines 1189–1216. https://enter.grok.me/browse/XI-G.

CheungEm02E

N. Cheung, Manuscript Em02-E, version 1, n.d. Relevant supplied statements: thm:lapse, lines 758–796; thm:release, lines 1795–1846. Classical relational lapse on a declared background. https://enter.grok.me/browse/Em02-E.

CheungXVIN

N. Cheung, Manuscript XVI-N, revision r12, n.d. Relevant supplied statement: Theorem 11, thm:massless-obstruction, lines 306–312. Conditional obstruction in its declared completion class. https://enter.grok.me/browse/XVI-N.

CheungXVIO

N. Cheung, Manuscript XVI-O, revision o70-sync, n.d. Relevant supplied material: rem:three-verdicts, lines 74–97; status ledger, lines 413–446. Kernel subsector and retained NOT_CLOSED status. https://enter.grok.me/browse/XVI-O.

CheungXIVH

N. Cheung, Manuscript XIV-H, version 1, n.d. Relevant supplied statements: thm:H2, lines 1621–1664; thm:release, lines 1929–1977. https://enter.grok.me/browse/XIV-H.

CheungXIVI

N. Cheung, Manuscript XIV-I, version 1, n.d. Relevant supplied statements: thm:I2, lines 1370–1460; thm:I3, lines 1541–1630; thm:release, lines 1811–1896. https://enter.grok.me/browse/XIV-I.

CheungXIVJ

N. Cheung, Manuscript XIV-J, version 1, n.d. Relevant supplied statements: thm:J2, lines 1690–1731; cor:noadmission, lines 1890–1904; thm:release, lines 1976–2021. https://enter.grok.me/browse/XIV-J.

CheungDossier

N. Cheung, Em06 prospectus and accompanying manuscript-comparison dossier. Unpublished material supplied with the problem, n.d. Documentary descriptions of VII-G, VII-H, Em04-A, Em05-A, Em05-B, and the En01/En02/Em03 comparison families are used only to state non-transfer boundaries. This entry does not assert independent publication, registration, or verification of those described sources.

Glauber

R. J. Glauber, Coherent and incoherent states of the radiation field, Physical Review 131 (1963), 2766–2788. doi:10.1103/PhysRev.131.2766.

Hepp

K. Hepp, The classical limit for quantum mechanical correlation functions, Communications in Mathematical Physics 35 (1974), 265–277. doi:10.1007/BF01646348.

Nelson

E. Nelson, Analytic vectors, Annals of Mathematics (2) 70 (1959), 572–615. doi:10.2307/1970331.

BGP

C. Bär, N. Ginoux, and F. Pfäffle, Wave Equations on Lorentzian Manifolds and Quantization. ESI Lectures in Mathematics and Physics, European Mathematical Society, 2007. doi:10.4171/037. arXiv:0806.1036.

RT

S. Ryu and T. Takayanagi, Holographic derivation of entanglement entropy from AdS/CFT, Physical Review Letters 96 (2006), 181602. doi:10.1103/PhysRevLett.96.181602. arXiv:hep-th/0603001. External comparison, not a premise.

DHW

X. Dong, D. Harlow, and A. C. Wall, Reconstruction of bulk operators within the entanglement wedge in gauge-gravity duality, Physical Review Letters 117 (2016), 021601. doi:10.1103/PhysRevLett.117.021601. arXiv:1601.05416. External comparison, not a premise.

Faulkner

T. Faulkner, M. Guica, T. Hartman, R. C. Myers, and M. Van Raamsdonk, Gravitation from entanglement in holographic CFTs, Journal of High Energy Physics 03 (2014), 051. doi:10.1007/JHEP03(2014)051. arXiv:1312.7856. External comparison, not a premise.