Em06-B: Robust classical concentration and quantum fluctuations of scalar-only plateau relational observables Finite families, displaced states, and controlled classical inputs

We prove a quantitative extension of the scalar-only first plateau coefficient studied in the supplied Em06-A draft. The metric coefficient remains a prescribed classical input. We independently derive the density variation, zero-coordinate-time Cauchy prescription, compact-test representation, and restricted gauge identity. On common compact supports we prove an explicit, nonoptimal estimate using six derivatives of the observation test and metric coefficient and three derivatives of the switch.

For displaced Fock-normal states with finite residual number expectation $n_{ϵ}$, the condition $ϵn_{ϵ}\to 0$ gives uniform mean, variance, mean-square, and spectral Wasserstein concentration on bounded input families. Observable-specific second moments give the exact necessary and sufficient condition, so the number condition is sufficient but not necessary. Finite commuting families possess joint spectral laws; general families are treated by ordered Weyl expectations, not by an assumed joint probability distribution.

Normalized fluctuations retain the declared, possibly degenerate CCR form. Their laws are exactly the laws of the undisplaced fields in the residual states, and concentration alone does not determine a fluctuation limit. We prove input-stability estimates and evaluate nonzero success, bias-failure, amplified-variance, and nonconvergent-fluctuation examples. An explicitly defined additive-noise POVM separates intrinsic from measured concentration.

The release comprises the scalar-only gates B1–B5. The optional finite-mode metric gate B6 is not entered. No interacting gravitational product, regulator removal, backreaction, metric reconstruction, Einstein dynamics, holographic identification, or inherited gravitational coefficient admission is obtained.

The question is:

Under which explicit preparation and input-control assumptions do families of scalar-only plateau relational coefficients concentrate on their named classical values, and what survives at the normalized fluctuation scale?

The observable is the first relational coefficient, not a full finite-coupling nonlinear observable. Its architecture agrees with the first-order plateau construction described in Cheung's VII-A thm:first CheungVIIA and with the supplied Em06-A draft Em06A. Every identity needed here is rederived in the conventions printed below. No second-order theorem is imported.

The coherent-state mechanism is established mathematics Glauber,Hepp. Our contribution is the explicit quantitative package for this named scalar-only coefficient: compact-test estimates, finite families, residual-state assumptions, input control, and the separation between concentration and quantum fluctuations. We do not claim a new general coherent-state or semiclassical-limit principle.

The Wigner-measure literature provides a broader bosonic framework ANphase,ANgeneral. Its nonlinear propagation results have their own hypotheses ANpropagation. They are not premises of the fixed-input estimates proved here. Likewise, dynamical fluctuation theorems such as BenArous are comparisons, not substitutes for our exact displacement calculation.

precise weakening is retained. Only the scalar is quantized. The metric coefficient is fixed before the compact scalar test is passed to the quantum representation. We do not construct a quantized metric-dependent insertion, an interacting BRST/BV complex, or an interacting relational product. This retains the distinction emphasized in V and in the sharp joint insertion problem of VII-B CheungV,CheungVIIB.

attached XVII-A is a different project. The supplied file XVII-A.tex declares a finite-matrix ${\mathrm{AdS}}_2$ boundary reconstruction benchmark XVIIA. Its geometry, matrices, boundary algebra, and correspondence $U$ are not inputs here. No identification between its effect and the present Minkowski relational coefficient is asserted.

comparisons are not verified premises. Cheung citations refer to the supplied versions and excerpts. The added descriptions of VII-I and VII-J include theorem labels without the full corresponding proofs. We therefore use those descriptions only as comparison material Dossier, not as independently audited theorems or premises. In particular, a summary calling a landmark “verified” does not verify its signs, domains, or compatibility with the present model.

Fix

a mass $m>0$, and $T>0$. Set

Fix compact sets

for some $R>0$, where $B_R$ is the spatial ball of radius $R$. Fix an open set $U$ compactly contained in $(0,T)\times B_R$.

The allowed observation tests and switches are

with $\chi =1$ on a neighborhood of $\overline{U}$. We assume this switch class is nonempty.

The metric coefficient $h$ is any real smooth symmetric covariant tensor on $M$. Its relevant control region is

The admitted gauge parameters are exactly

All support sets, the initial surface, $m$, and $T$ remain fixed.

For spacetime multi-indices, define

The input-control topology is $p_6$ for fixed-support observation tests, $c_3$ for fixed-support switches, and the local seminorm $H_6$ for smooth metric coefficients. The preparation amplitude will be controlled in its one-particle Hilbert norm.

These are explicit local input controls, not a claim about convergence in a boundary-sensitive kernel topology or a topology of interacting gravitational functionals.

Indices are raised and lowered with $\eta$. For $\xi \in {\mathfrak{g}}_U$, write

Set

The switched scalar action through first order is

Its Euler equation is

In particular, the operator is $Q_{\chi h}$, not $\chi Q_h$. Derivatives of the switch are retained. The metric-jet parameter $\kappa$ is independent of the classicality parameter $ϵ$.

Let ${G_m}^{\mathrm{R}}$ and ${G_m}^{\mathrm{A}}$ be the retarded and advanced Green operators of $P_m$. With $D_m=\sqrt{-\Delta +m^2}$, our signs are

For compact smooth $F$,

We use their standard finite-propagation and Cauchy uniqueness properties BGP. The estimates needed for this model are proved in Appendix app:PDE.

For $t\ge 0$, define the unswitched coordinate coefficient by

These are coordinate-time Cauchy data. They are not replaced by a metric-dependent normal-derivative prescription.

For a real smooth solution $P_m{\varphi }_0=0$, define

The source is compact and separated from ${\Sigma }_0$, and the response has zero initial data.

The named classical functional is

For the formal metric $g_{\kappa }=\eta +\kappa h$,

Thus $x^{\mu }+\kappa w[h{]}^{\mu }$ is the harmonic coordinate map through first order, and its inverse has coefficient $-w[h]$. Equation (eq:observable) is precisely the corresponding first scalar coefficient after composition with that inverse map.

The theorem concerns this coefficient itself. It does not infer a controlled full finite-$\kappa$ observable from a formal first-order expansion.

The following identities hold:

For real $h$, $Q_h$ is formally self-adjoint.

Proof. Expanding the divergence in (eq:Q) and using

gives (eq:Q-expanded). The trace of $K\xi$ is $2{\partial }_{\mu }{\xi }^{\mu }$, and direct differentiation gives (eq:CK). Finally,

Substituting $h=K\xi$ into (eq:Q-expanded) proves (eq:QK). Formal self-adjointness follows by integration by parts in the divergence form, including its real zeroth-order term.

Put $u_f={G_m}^{\mathrm{A}}f$ and

Then $q_{f,h,\chi }$ is real, smooth, and compactly supported, with

For every smooth free scalar solution,

The dependence support in the classical $h,{\varphi }_0$ variables is contained in

which is compact.

Proof. Transposition of the retarded Green operator gives

The switched coefficient is compactly supported, so the last integration by parts has no Cauchy-surface contribution. The other term obeys

This proves the formula and its sign.

The advanced support property bounds the first term by $L\cap J_{\eta }^{-}(K)$; the second is supported in $K$. Both sets are separated from $t=0$. Although $w[h]$ is defined here for $t\ge 0$, its product with $f$ extends smoothly by zero near and below $t=0$.

Finite propagation for the scalar response and coordinate Cauchy problem gives dependence only in ${\mathcal{D}}_K$. A compact observation set has bounded spatial causal past on a finite closed time slab, so ${\mathcal{D}}_K$ is compact.

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

At the test level,

Proof. Since $\xi$ and all its derivatives are supported where $\chi =1$, $\chi K\xi =K\xi$. On the free scalar shell,

Retarded inversion gives ${G_m}^{\mathrm{R}}{P}_m(X_{\xi }{\varphi }_0)=X_{\xi }{\varphi }_0$, because this smooth function is compactly supported away from the initial surface. Equation (eq:CK) and Cauchy uniqueness similarly give $w[K\xi ]=\xi$. The two variations in (eq:observable) cancel.

For the stronger compact-test identity, $P_m{u}_f=f$ and (eq:QK) yield

The multiplier $\xi$ makes $X_{\xi }{u}_f$ compact.

This reproduces the required first-order plateau statement within the present conventions CheungVIIA. It proves no unrestricted diffeomorphism theorem.

Use the unitary spatial Fourier convention

Let

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

Write

For $z\in \mathcal{S}({\mathbb{R}}^3;\mathbb{C})$, define the real smooth free scalar

Direct integration gives

The target therefore depends on the scalar preparation as well as the prescribed metric coefficient. Writing $o[g_{\mathrm{cl}}]$ without naming the preparation would suppress an essential input.

On a fixed compact support $S\subset M$,

Proof. Spatial integration by parts gives

The integral

is finite: $m>0$ controls the origin, and the radial integrand decays as $r^{-3}$ at infinity. The $L^1$ norm is bounded by a fixed-support constant times $p_2(F)$.

The exponential in (eq:k) is annihilated by $P_m$. Compact integration by parts proves $k_{P_m\psi }=0$.

There is a finite constant $C$, depending only on the fixed support geometry, $m,T,R$, and a fixed spatial cutoff used in Appendix app:PDE, such that

The derivative orders are sufficient and are not claimed optimal. Consequently the map to $v$ is continuous in the stated input controls, uniformly on bounded families.

Proof. The detailed Cauchy and advanced estimates are proved in Appendix app:PDE. They bound $u_f$ through four derivatives by $C{p}_6(f)$, and $w[h]$ on $K$ through three derivatives by $C{H}_6(h)$. Applying the product rule to (eq:q) gives the first estimate. Lemma lem:mass-shell, with common compact support $K\cup L$, gives the second.

By Proposition prop:gauge and Lemma lem:mass-shell,

This is exact descent into the free scalar representation. It is not a theorem about a joint gravitational quantum ideal or elimination of interacting auxiliary fields.

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

Let $N$ be the number operator and

Initially these operators act on the finite-particle domain ${\mathcal{D}}_{\mathrm{fin}}$.

For every $v\in \mathfrak{h}$, $\Phi (v)$ is essentially self-adjoint on ${\mathcal{D}}_{\mathrm{fin}}$. Its closure, denoted by the same symbol, satisfies

and the quadratic-form inequality

Proof. The creation–annihilation bounds are

Using $\Vert x+y{\Vert }^2\le 2\Vert x{\Vert }^2+2\Vert y{\Vert }^2$ proves (eq:number-bound) on ${\mathcal{D}}_{\mathrm{fin}}$.

For a vector supported in particle sectors at most $r$,

Its analytic-vector series converges. The analytic-vector criterion Nelson gives essential self-adjointness. Particle truncation converges in the graph norm of $N^{1/2}$; the displayed bound and closedness then extend the estimate and domain inclusion.

We always mean these self-adjoint closures. Commutator identities for unbounded fields are stated on the invariant common core, or as the corresponding quadratic-form identities, not as everywhere-defined products.

Define the unitary displacement and scaled Weyl operators by

Their conventions give

These follow by the scalar commutators of the generators, first on analytic vectors and then for the bounded unitary groups. They are the standard displacement/Weyl identities Glauber, with the signs fixed by our inner-product convention.

Let ${\rho }_{ϵ}$ be positive trace-class operators with

For a declared $z\in \mathcal{S}({\mathbb{R}}^3;\mathbb{C})$, put

For $v=v(f,h,\chi )$, the represented relational coefficient is

The parameter $ϵ$ is a dimensionless effective CCR parameter relative to a fixed reference unit of action. It is neither $\kappa$ nor a smoothing width. The classical symbol is fixed while the representation section in (eq:O) varies.

of unbounded traces. If $\rho ={\sum }_{\ell }{p}_{\ell }|{\psi }_{\ell }⟩⟨{\psi }_{\ell }|$ has finite number expectation, its positively weighted vectors with $p_{\ell }>0$ lie in $\mathrm{Dom}{N}^{1/2}$. We define

The first moment is the absolutely convergent associated sum. These are spectral moments, independent of the decomposition. Notation such as $\mathrm{Tr}(\rho \Phi (v{)}^2)$ below means this quadratic moment; it does not assume that the formal product $\rho \Phi (v{)}^2$ is an already-defined trace-class operator.

Lemma lem:field implies

For a self-adjoint operator $A$ in a state $\rho$, its spectral law is ${\mu }_{\rho ,A}(J)=\mathrm{Tr}(\rho E_A(J))$. For probability laws with finite second moments, $W_2$ denotes the quadratic Wasserstein metric. In particular,

because every coupling with a point mass has constant second coordinate.

For Definition def:states, write

and let ${\mu }_{ϵ}$ be the spectral law of ${\hat{O}}_{ϵ}$ in ${\tilde{\rho }}_{ϵ}$. Then

For a fixed named $v,o$, the following are equivalent:

1. the mean tends to $o$ and the variance tends to zero;

2. $ϵM_{ϵ}(v)\to 0$;

3. $W_2({\mu }_{ϵ},{\delta }_o)\to 0$.

A sufficient, not necessary, condition is

Under that condition,

Proof. The exact self-adjoint conjugation is

Unitary covariance of spectral measures gives the mean, variance, and second-moment identities. Their moments exist by (eq:state-number). Equation (eq:pointmass) proves the Wasserstein identity.

For every probability law with a finite second moment,

Both summands are nonnegative, proving the equivalences. Cauchy–Schwarz gives $b_{ϵ}(v{)}^2\le M_{ϵ}(v)$; (eq:state-number) then proves the sufficient condition and all three estimates.

The general spectral law in this theorem need not be Gaussian. No quasifree-state assumption was used.

Let $\mathcal{C}$ be any family of data in Specification spec:data with

Let $L_{\ast }=CFH(1+X)$, with $C$ from Theorem thm:estimate. For a common residual state family, all bounds in (eq:concentration-budget) hold uniformly over $\mathcal{C}$ with $\Vert v\Vert$ replaced by $L_{\ast }$. The classical targets are the corresponding $o(f,h,\chi ,z)$.

Proof. Theorem thm:estimate gives ${\sup }_{\mathcal{C}}\Vert v\Vert \le L_{\ast }$. Apply Theorem thm:states.

This is uniform concentration around the named, possibly varying classical targets. Convergence to a single fixed target additionally requires control of their variation; that is proved in Section sec:robustness.

Fix real $f_1,\dots ,f_J$, with common $h,\chi ,z$, and write

Define the real matrices

The matrix $S$ is positive semidefinite and $A$ is antisymmetric.

For a residual state $\rho$, define its field cross moments by

These sums exist by (eq:state-number). Set

For every finite-number residual state,

The symmetric covariance in the displaced state is

On the common core,

For the displaced vacuum,

so the symmetric covariance is exactly $ϵS$.

Proof. On finite-particle vectors the weak commutator gives

Particle truncation converges in $\mathrm{Dom}{N}^{1/2}$, and (eq:number-bound) passes this identity to that form domain. Summing over the state decomposition proves the imaginary-part statement.

Displacement shifts each field by its real constant $o_j$ after scaling. Covariance is unaffected by this shift, proving (eq:cov-general). The core commutator is (eq:CCR) with the scale included. For the vacuum,

and all first moments vanish.

Thus symmetric noise covariance and antisymmetric commutation are not the same quantity.

If $A_{ij}=0$ for every $i,j$, the self-adjoint fields $\Phi (v_j)$ strongly commute. Their scaled counterparts have a joint spectral law ${{\mu }_{ϵ}}^{\mathrm{joint}}$ in every state considered above.

For general residual states,

where $o=(o_1,\dots ,o_J)$ and the metric on ${\mathbb{R}}^J$ is Euclidean.

For the displaced vacuum,

Degenerate Gaussian covariance is allowed.

If some $A_{ij}\ne 0$, the corresponding operators do not strongly commute. In that case no common joint spectral law of these operators is asserted.

Proof. When $A_{ij}=0$, (eq:Weyl) shows that their unitary groups commute for all real parameters. Commuting spectral measures follow from the spectral theorem for commuting unitary groups. The squared Euclidean distance to $o$ is the sum of the squared coordinate distances. Applying (eq:mse) to every marginal proves (eq:joint-general).

Vacuum normal ordering gives

For $V={\sum }_j{t}_j{v}_j$, displacement therefore gives the joint characteristic function

This identifies (eq:joint-vacuum).

Conversely, strong commutativity would make the two unitary groups commute for every pair of parameters. Their Weyl interchange phase is $e^{-2\mathrm{i}st{A}_{ij}}$; it cannot be one for every $s,t$ when $A_{ij}\ne 0$.

For real $t=(t_1,\dots ,t_J)$, put

Products below are ordered by increasing $j$.

For arbitrary $A$,

Its distance from $e^{\mathrm{i}t\cdot o}$ is at most

For the displaced vacuum, the exact formula is

Under $ϵn_{ϵ}\to 0$, the ordered expectations converge to the classical product, uniformly for $t$ in compact subsets and for bounded input families with fixed $J$.

Proof. Repeated use of (eq:Weyl) gives the phase $-\mathrm{i}ϵ{\sum }_{i<j}{t}_i{t}_j{A}_{ij}$ and the final Weyl direction $V(t)$. Displacement gives $e^{\mathrm{i}t\cdot o}$, proving (eq:ordered-general).

Spectral calculus and Cauchy–Schwarz imply

Together with $|e^{-\mathrm{i}ϵa}-1|\le ϵ|a|$ and (eq:state-number), this proves the bound. Vacuum normal ordering proves (eq:ordered-vacuum).

On compact $t$-sets and bounded $\Vert v_j\Vert$, both $\Vert V(t)\Vert$ and $|a(t)|$ have common finite bounds. This proves uniformity.

These are ordered quantum correlations, not a joint measurement distribution for noncommuting fields. They do not invoke the density-prepared free-product or ordering-transport statements described for VII-I Dossier, and they do not construct mixed time-ordered gravitational products.

Define

This centering uses the named classical value, not necessarily the actual mean.

For every residual state family,

Accordingly, every marginal fluctuation law is exactly the spectral law of $\Phi (v_j)$ in ${\rho }_{ϵ}$, and

On the common conjugated core,

For the displaced vacuum the fluctuation correlations are independent of $ϵ$:

When some $A_{ij}\ne 0$, their CCR algebra is noncommutative. If all $A_{ij}=0$, this selected sector is commuting and has an ordinary Gaussian fluctuation law.

Proof. Subtract the displacement constant in (eq:displacement) and divide by $\sqrt{ϵ}$. This proves the self-adjoint conjugation and hence the spectral statement. The unscaled Weyl relation gives (eq:fluctuation-Weyl); the unscaled core commutator gives (eq:fluctuation-CCR). The vacuum expectation was evaluated in Theorem thm:ordered.

In particular, vanishing unscaled variance does not mean that normalized fluctuations have become commuting.

For a residual state with finite number expectation,

Thus nonzero $A_{ij}$ prevents both normalized variances from vanishing in any such state.

Proof. The Hilbert–Schmidt vectors

have squared norms ${C_{ii}}^{\rho },{C_{jj}}^{\rho }$, and inner product ${C_{ij}}^{\rho }+\mathrm{i}{A}_{ij}$, by Proposition prop:covariance. Hilbert-space Cauchy–Schwarz proves the inequality.

Suppose, for fixed directions $v_1,\dots ,v_J$,

for a density operator ${\rho }_{\ast }$. Then all the ordered fluctuation expectations in (eq:fluctuation-Weyl) converge to those of ${\rho }_{\ast }$. Each marginal fluctuation spectral law converges weakly to the law of $\Phi (v_j)$ in ${\rho }_{\ast }$.

Trace-norm convergence alone does not imply convergence of unbounded second moments.

Proof. The operators $e^{\mathrm{i}\Phi (V(t))}$ have norm one, so the difference of their expectations is at most $\Vert {\rho }_{ϵ}-{\rho }_{\ast }{\Vert }_1$. The ordered phase is unchanged. Marginal characteristic functions therefore converge to the characteristic function of the stated probability law, which is continuous at zero. The usual characteristic-function criterion gives weak convergence. The final caution is demonstrated explicitly in Section sec:examples.

Concentration imposes only smallness of $ϵ⟨\Phi (v{)}^2{⟩}_{{\rho }_{ϵ}}$, not convergence of the undisplaced characteristic functions. It therefore does not supply the extra hypothesis of this proposition.

For two data sets with the common supports of Specification spec:data, write $v_a=v(f_a,h_a,{\chi }_a)$, $a=0,1$.

With the same fixed constant type as Theorem thm:estimate,