XX-D The Original-Rod Relational Lapse: Composite Formula and Product Domains A Native Classical Construction, Compact Wick Realizations, and a Conditional Noncompact One-Contraction Extension

We derive the complete kinematic second jet of the original-rod relational lapse on the power-law Einstein–scalar background reproduced in XX-C. The calculation retains every metric component, both second-jet fields, the clock embedding, the induced spatial metric, and the original slice-elliptic rod equation. The second spatial rod does not enter the quadratic lapse.

The unsmoothed classical coefficient is a continuous polynomial on a spatial Schwartz configuration domain. Its Hessian has noncompact dipole tails. After a compact source cutoff, that Hessian is microcausal: its equal-time off-diagonal covectors occur with opposite signs. Nevertheless its raw internal Hadamard contraction fails the sufficient product criterion at coincidence. We give both a Hadamard-subtracted one-contraction prescription and a transverse Taylor extension of the corresponding punctured numerical kernel.

A conditional theorem constructs the native, cutoff-free Wick symbol through quadratic perturbation order and first order in Planck's constant, together with products against compact regular probes. The theorem states separately the infrared hypotheses for internal and external contractions. Those hypotheses have not been verified for a selected XX-B/XX-C representative. Products requiring two noncompact rod legs need additional control and cannot be assigned a universal inverse-square separation bound.

The formal classical symbol recovers the printed classical coefficients exactly. This is not an expectation-value limit. The requested native quantum Q3 gate remains open wherever its noncompact contraction hypotheses remain unverified; no universal extension no-go is claimed.

The requested deliverable concerns the native unsmoothed insertion, not merely a numerical finite part, a compact approximation, or a classical functional. These objects are separated throughout this manuscript.

The supplied registered records distinguish the original-rod lapse from other relational prescriptions and explicitly leave its quantum product audit incomplete CheungEm02E,CheungXIXA,CheungXIXD. Their statements are documentary scope controls; the calculations below use the equations reproduced here.

XX-C, supplied in the present exchange XXC, does not select a covariance with verified global infrared estimates. Therefore a complete unconditional quantum Q3 certificate cannot be deduced from its Hadamard class alone.

The outputs considered here are:

1. a native classical polynomial, with the exact original Newton inverse;

2. compact-source-cutoff Wick symbols in the free algebra, without spatial ultraviolet smoothing;

3. a native quantum Wick symbol and its stated products, conditional on explicit noncompact contraction estimates.

Only an unconditional realization of the third output for an actually specified covariance and product domain would close the quantum Q3 gate.

orders. All formulas are taken modulo $s^3$ and ${\hslash }^2$. The parameter $s$ grades the kinematic Taylor jet; it is not silently identified with a gravitational coupling. First order in $\hslash$ means one free contraction here, not a complete interacting one-loop contribution inventory.

jets. The second-jet fields $k,\psi$ are retained as independent smooth prescribed jets. They are central parameters in the free Wick construction. If they are instead defined as interacting operator-valued responses, their own construction and cross-contractions must be added. No such identification is made.

references. The compact Wick-kernel framework and local subtraction templates are those of Brunetti–Fredenhagen and Hollands–Wald BF,HWlocal,HWexistence. The noncompact Newton estimates and native coefficient calculations are proved below. Those references do not automatically admit a nonlocal lapse.

Use signature $(-,+,+,+)$ and the action

Fix $M>0$, $p>1$, $t_{\ast }>0$, and

The spacetime is

The background solves the Einstein–scalar Euler equations: substitution uses $A^2=2p{M}^2$, $H=p/t$, and $V(\bar{\varphi })=M^{2}p(3p-1)/t^2$.

Retain the factorial jet

The coordinate clock is the monotone relabeling

The lapse itself remains

It is not replaced by ${\left(-{g_s}^{-1}(\mathrm{d}{T}_s,\mathrm{d}{T}_s)\right)}^{-1/2}$. In particular,

All output tests use the fixed background measure

No field-dependent volume factor is inserted.

Write $z=(h,\phi )$ and $\zeta =(k,\psi )$. The field bundle is

The integrated pairing is

Define the action Hessian by

where

The gauge generator, gauge condition, and native gauge-fixed operator are

With four-dimensional trace reversal

retain

The principal differential part of $\mathcal{L}$ is ${\square }_{\bar{g}}\mathrm{id}$. Thus its Cauchy theory and causal inverses are those of a normally hyperbolic bundle operator BGP.

The contraction representative, when specified, obeys

Here

The factor of $\hslash$ is not included in $W$; a free contraction is $\hslash W$.

Write

It is real and symmetric in a real field trivialization. No positivity is assumed.

The complete auxiliary and ghost multiplet is retained. The kinematic lapse has zero functional derivatives with respect to those components. Hence they contribute no direct free internal contraction to the insertion. This observation does not eliminate ghost or auxiliary contributions to an interacting calculation.

Set

Matrix inversion gives

For

the exact coefficients are

Consequently

On the displayed background,

Thus no shift component has been dropped.

Let the embedding of the $T_s=t$ slice be

The clock equation is equivalently ${\varphi }_s\circ {\iota }_s=\bar{\varphi }(t)$. It gives

Expanding the exponential clock directly gives

and its inverse yields the same (eq:u). This verifies the retained calibration.

Write

The complete induced coefficients are

The last line is the embedding contribution. Its last term uses ${\bar{g}}_{00}=-1$.

Let

Expanding the original elliptic density gives

where

There is no lapse multiplier in this density.

The rods satisfy

For

retain the spatial inverse

It follows that

Define the local rod-source map

Then $B_1=\text{calCz}$, and

After integration by parts,

The kernel $P_i$ is locally integrable and has spatial scaling degree two.

The spacetime rod label is obtained by transporting $R_s$ from its clock slice. Let ${\alpha }_s$ be the inverse clock-and-rod map and write

At first order,

At second order the clock equation gives

The relational lapse

has coefficients

In particular $Y_2$ does not occur.

Proof. Taylor expansion gives

Because $N_0$ is spatially homogeneous, ${{\alpha }_2}^i{\partial }_i{N}_0=0$. Substitute the first spatial displacement and the displayed temporal second displacement. The spatial terms combine to $-2{Y_1}^i{\partial }_i(N_1+{\dot{N}}_0{u}_1)$. This proves (eq:composition) and the stated absence. No quantum substitution was used.

This derivation reproduces the composition structure in the supplied XIX-A passage CheungXIXA (locators :lapse-composition, :n1, :n2).

Define the local linear map

Then the complete result is

where

The spatial metric enters through the rod term, not through an artificially restricted scalar metric subclass.

Let

with seminorms

Use the same domain for the prescribed second jet $\zeta$. Spatially compact smooth configurations are included.

For $z\in \mathcal{X}$, $\text{calRz}$ is smooth and

uniformly on $J⋐I$. The map is continuous in the stated seminorms. It is the unique decaying solution of its rod equation. The same conclusions hold for uniformly compact smooth sources.

Proof. The source $\text{calCz}$ is spatial Schwartz, with smooth bounded time coefficients on $J$. For compact sources the derivative bounds follow directly from (eq:P) outside twice the source radius.

For Schwartz sources split the integral into $|\mathbf{y}|\le |\mathbf{x}|/2$ and its complement. In the first region the differentiated kernel has the claimed decay and the source has uniform finite moments. In the second region the source decreases to arbitrary order. Near $\mathbf{y}=\mathbf{x}$, use local integrability of $P$, transferring derivatives onto the smooth source. Away from that region differentiate the kernel directly. These estimates use finitely many Schwartz seminorms for each desired bound, proving continuity.

The equation follows from $\Delta {\mathsf{G}}_{\mathrm{sp}}=\mathrm{id}$. A difference of two smooth decaying solutions is an entire harmonic function tending to zero; the maximum principle makes it zero.

The products in (eq:gamma2) and (eq:B12) are Schwartz. The product $B_1\partial Y_1$ is also Schwartz because $B_1$ is Schwartz and $Y_1$ has bounded derivatives. Thus $Y_2$ is defined on this domain as well.

For $f\in {C_c}^{\infty }(\mathcal{M})$, set

Equation (eq:Fclass) is a continuous polynomial on $\mathcal{X}\times \mathcal{X}$. As $f$ varies it defines a distribution-valued polynomial section, without spatial smoothing or a replacement of the original Newton inverse.

Proof. Every local term uses finitely many derivatives and products of Schwartz functions. Lemma lem:Newton controls the sole nonlocal factor. On the compact support of $f$, all these operations are continuous in finitely many seminorms. The dependence on $f$ is continuous in the ordinary test topology.

For distributions, the following are sufficient operation rules.

1. Multiplication. The product $uv$ is defined by the usual wavefront criterion if no $(x,\xi )\in \mathrm{WF}(u)$ and $(x,-\xi )\in \mathrm{WF}(v)$ occur simultaneously.

2. Fixed pullback. For a smooth map $j$, $j^{\ast }u$ is defined if

   $$\mathrm{WF}(u)\cap N_j=\varnothing ,N_j=\{(j(x),\eta ):(\mathrm{d}{j}_x{)}^{\mathrm{T}}\eta =0\}.$$

   A fixed diffeomorphism has $N_j$ empty away from the zero section.

3. Kernel composition. Products of kernels on the intermediate space must satisfy D1, and the subsequent pushforward must have proper support or an independently proved convergent noncompact integration.

4. Newton inversion. Local elliptic distributional inversion does not by itself impose the global decaying boundary condition. We use Lemma lem:Newton, or a separately stated weighted domain, for that condition.

5. Extension. Failure of D1 or D2 does not prove that every extension is impossible. A punctured distribution with finite transverse scaling degree can instead be extended, subject to its normalization and symmetry conditions.

These are the microlocal operation principles used in the analytic frameworks cited above BF,HWlocal; their hypotheses are checked for the particular kernels below.

The coefficients in Section sec:classical come from a classical Taylor expansion of $N_s\circ {\alpha }_s$. If a classical configuration is small enough that the clock-and-rod map is invertible, its fixed inverse ${\alpha }_s$ is an ordinary diffeomorphism. Distributional pullback by that fixed map is allowed by D2.

For a fixed spacelike slice embedding, the conormal covectors are timelike. A free normally hyperbolic solution with null wavefront set can therefore be restricted to that slice. The corresponding simultaneous equal-time restriction of $W$ also exists.

Neither assertion defines

A field-dependent quantum substitution requires the products appearing in its expansion. In this manuscript its retained jet is represented only by (eq:ncomplete) with the specified Wick and extension rules.

Put

Let $b(z,w)$ be its symmetric polarization:

Thus

Define $K_f$ by

Then

with respect to the free first jet $z$.

All component kernels may be written relative to coordinate measures by absorbing the smooth $a^3$ factors. This does not change wavefront sets or scaling degrees.

Choose $\chi \in {C_c}^{\infty }({\mathbb{R}}^3)$, equal to one near the origin, and set

Define

The derivative includes the cutoff derivative. Equivalently, (eq:P) has source ${\chi }_R\text{calCz}$. Let $F_{f,R}$ and $K_{f,R}$ replace $\mathcal{R}$ by ${\mathcal{R}}_R$.

This is a compact source approximation. It is not the native rod prescription at finite $R$, and it is not ultraviolet smoothing: the equal-time delta and Newton singularity are unchanged.

In the two field slots $u=(t_u,\mathbf{y})$, $v=(t_v,\mathbf{x})$, the nonlocal kernel, before transferring differential operators to the slots, has the form

together with its transposed term. For finite $R$, multiply it by ${\chi }_R(\mathbf{y})$.

For finite $R$, $K_{f,R}$ is a compactly supported distributional kernel satisfying

The same exclusion is locally valid for the native kernel $K_f$, but $K_f$ is generally not compactly supported.

Proof. Away from $\mathbf{x}=\mathbf{y}$, the Newton kernel is smooth. The only wavefront covectors from (eq:Kform) are

They have opposite time orientations.

At coincidence, use difference coordinates $(t_u-t_v,\mathbf{y}-\mathbf{x})$. The singular factors are a time delta and a spatial distribution in those independent variables. Their wavefront covectors, transferred to the two slots, have the form

They cannot be simultaneously future or simultaneously past causal. The local terms are derivatives of a full diagonal delta and have the same opposite-slot property.

Differentiation and smooth multiplication do not enlarge these wavefront upper bounds. Both slots are compact for finite $R$. For the native kernel only the local slot is confined by $f$; the other slot has the dipole tail of $P$.

An instantaneous off-diagonal timelike conormal is not by itself a microcausal exclusion: its sign pattern matters. Here the signs in the two field slots are opposite. The obstruction to directly importing the compact Wick theorem for $K_f$ is global noncompactness, not an asserted same-sign wavefront pair.

The kernel contraction

is not justified merely by (eq:microcausal). At coincidence $K_{f,R}$ has opposite-slot covectors with arbitrary spatial component, including null ones. The Hadamard kernel has opposite-slot null covectors. These can cancel in the product criterion D1.

Similarly, the diagonal pullback of $W$, or of its differentiated entries, meets the conormal of the diagonal. Local Wick subtraction is required.

Thus a compact normal-ordered quadratic element can exist even though its raw internal contraction is undefined by the ordinary product criterion. This distinction is retained in the compact Wick framework HWlocal.

Fix the native bundle-valued Hadamard subtraction germ of XX-C, with the action normalization ${\mathcal{A}}^{-1}$ in the source slot. Only its following property is used:

This is a transport-parametrix normalization statement, not a conclusion from wavefront upper bounds alone.

Let $H_{\mathrm{sub}}$ be a real symmetric extension of the symmetric part of that germ, using a smooth symmetric cutoff equal to one in a neighborhood of the diagonal relevant to the compact support of $f$. Its extension away from that neighborhood is part of the prescription.

Define

Then $D$ is smooth near the full diagonal.

For the equal-time map

all differentiated entries of $D$ used below have smooth pullbacks: near $\mathbf{x}=\mathbf{y}$ this follows from full-diagonal smoothness; away from coincidence, the two points on this homogeneous slice are spacelike separated, so neither $W$ nor the chosen near-diagonal subtraction has a null singularity there.

Derivatives in the two spacetime slots are taken before equal-time restriction.

For components, let $D_{00,00}$ denote the $h_{00},h_{00}$ entry, and similarly for the other displayed field components. A prime indicates the second slot.

The local one-contraction correction is

These are all the monomials of (eq:qloc); no state positivity is needed to form these smooth diagonal values.

Define the differentiated equal-time matrix

with the index $a$ summed when paired with $P_i$. The rod correction is

The integrand is locally integrable at $\mathbf{y}=\mathbf{x}$, because $d^{ia}$ is smooth there and $P_i$ is locally integrable.

The unit and linear fields are unchanged. In the $W$-normal-ordered symbol coordinates, define the finite-$R$ insertion by

This fixes a subtraction choice. Additional finite changes, if authorized, must be stated separately; none are silently used.

For every finite $R$, (eq:IR) is a well-defined free Wick symbol through the retained orders. Its quadratic field kernel is $K_{f,R}$, and its scalar correction is exactly (eq:Gamma). All terms in the clock-and-rod jet (eq:ncomplete) are represented.

Proof. Proposition prop:microcausal gives compact microcausal kernels. The finite-component algebraic Wick construction is therefore available, using the contractions of (eq:W); its distributional multiplication uses the positive-frequency relation and does not require a positive state representation.

The change from the declared symmetric subtraction to the $W$-symbol coordinates contracts the symmetric Hessian once with $D$. For the local terms this is (eq:gammaloc); for the rod term it is (eq:Gamma). The latter is finite by compact source support and local integrability.

Since ${F_{f,R}}^{(2)}=s^2{K}_{f,R}$, the normal-ordering change is $(\hslash /2)⟨D,{F_{f,R}}^{(2)}⟩$, giving the coefficient in (eq:IR). The central linear second-jet term generates no internal free contraction.

This theorem constructs algebraic Wick symbols. It does not assert that the indefinite gauge-fixed contraction is a physical state.

There is also an explicit extension route when one starts with the raw punctured equal-time contraction.

Let $\mathbf{r}=\mathbf{x}-\mathbf{y}$. Off $\mathbf{r}=0$, the differentiated Hadamard contraction in the rod term is smooth. Its spatial singularity has an upper bound of the form

on compact tangential sets. Indeed, the four-dimensional Hadamard germ has leading degree two, the differentiated field factor has at most two derivatives, and the dipole kernel has degree two.

The bound is conservative: cancellations or equal-time parity can lower the actual degree.

Suppose the punctured coefficient $\beta$ has the stated Hadamard bound with smooth dependence on $(t,\mathbf{x})$. It has a distributional extension across $\mathbf{r}=0$. In the class with transverse scaling degree at most six, the extension freedom is

The coefficients are not claimed to form a finite-dimensional space without further restrictions.

Proof. Choose $\theta \in {C_c}^{\infty }({\mathbb{R}}^3)$, equal to one near zero. For a compact test $\eta$, put

The remainder is $O(|\mathbf{r}{|}^4)$. Thus

converges absolutely near zero: the radial majorant is a constant times $(1+|\log r|{)}^N\mathrm{d}r$. It defines a continuous extension after specifying values on the subtracted jets.

The finite scaling-degree extension principle of BF (Theorems 5.2–5.3 and Section 6) gives the scaling-degree bound. A difference of such extensions is supported on $\mathbf{r}=0$. Its allowed normal derivative order is at most $6-3=3$, giving (eq:contacts).

This is a numerical extension, not automatically a compatible quantum insertion prescription. Exchange symmetry, reality, field-equation relations, and any gauge conditions must be imposed on it separately.

For the insertion (eq:IR), the chosen route is the Hadamard-subtracted smooth remainder, not an unannounced identification with every raw Taylor extension.

There is no additional off-diagonal light-cone extension in this internal rod contraction: equal-time distinct points are spacelike. Other graphs may have light-cone products and require their own analysis.

The diagonal subtraction removes a local singularity. It does not control the noncompact smooth remainder in (eq:drod).

For every compact output region $K⋐\mathcal{M}$ there is $ϵ>0$ such that, outside a fixed ball, the differentiated entries needed in (eq:drod) satisfy

The constants are uniform for the derivative orders required for the announced output regularity.

Because $P_i=O(|\mathbf{y}{|}^{-2})$ for compact $\mathbf{x}$, this bound makes the radial tail integrable.

Under Assumption ass:internal, ${\gamma }_{\mathrm{rod},R}$ converges with the stated output derivatives to

The limit is independent of the source cutoff profile and

is a distribution in $f$.

Proof. Local convergence uses smoothness of $d$ and local integrability of $P$. For large $|\mathbf{y}|$, the product is bounded by $C(1+|\mathbf{y}|{)}^{-3-ϵ}$. Dominated convergence applies to the output derivatives, and the same estimate makes the difference of any two admissible cutoff families tend to zero.

This is a separately stated hypothesis. Compact-test smoothness of $W(\cdot ,g)$ does not imply (eq:internalIR).

For $g\in {C_c}^{\infty }(\mathcal{M};\mathcal{V})$, define

The Hadamard wavefront relation implies that $w_g$ is smooth: compact testing eliminates the singular second-slot covector. It does not imply decay.

For the compact tests $g$ in the announced probe class,

with uniform seminorm bounds on bounded families of probes.

This is the sufficient one-sided covariance condition used in XX-C. It is rechecked here when used.

For polynomials whose contraction is defined, the order-$\hslash$ product is

Fix an actual representative $W$ satisfying (eq:W), a declared subtraction germ, and Assumptions ass:internal and ass:external. Then, on $\mathcal{X}$ with central smooth second jets,

is a cutoff-independent native Wick symbol. Its products on either side with compact regular probe polynomials exist through the retained contraction order.

For a linear probe, the external contraction is explicitly

Every rod in this formula is the original spatial Newton inverse.

Proof. The classical limit of $F_{f,R}$ is $F_f$ in the seminorm topology of Theorem thm:classicaldomain. Lemma lem:internal supplies the scalar normalization limit.

For (eq:external), differentiate (eq:Fclass) in the free first jet. The symmetric polarization gives the displayed formula. Local entries of $w_g$ are smooth and only sampled on $\mathrm{supp}f$. The term $\text{calRz}$ is defined by Lemma lem:Newton. Assumption (eq:externalIR) gives the same lemma for ${\text{calRw}}_g$. Therefore each term in $b(z,w_g)$ is a smooth integrable compact-output expression.

A compact regular polynomial has smooth compact derivative kernels. After fixing the remaining compact field variables, its first derivative is a finite or continuously compact-tested superposition of probes of the above type. The uniform bounds in Assumption ass:external justify that integration. The reverse ordered product uses the transposed contraction. Its causal difference, compact tested in one slot, has uniformly compact spatial support on compact time intervals, so its rod composition satisfies the compact-source version of Lemma lem:Newton.

Thus both product orders exist. Cutoff independence follows from the same bounds. No infrared bound on a two-rod state covariance has been used.

The theorem has a noncompact native conclusion, but it is conditional on two concrete estimates. The supplied XX-C class does not verify those estimates for a selected representative. Consequently (eq:nativeI) is not announced as an unconditional native quantum object.

Even for smooth $z$, the first derivative ${F_f}^{(1)}[z]$ generally has a noncompact dipole component. In adjoint notation its test profiles include

whose spatial decay is $O(r^{-2})$.

Therefore

requires an extension of the covariance to pairs of such noncompact profiles. Assumption ass:external concerns one compact-tested endpoint and does not supply that extension.

A sufficient additional domain statement is a continuous bilinear covariance pairing on

with cutoff-independent pairings and the native antisymmetric part. This property must be proved for the chosen representative; it is not imposed here as an authenticated fact.

Thus products of two native lapse insertions are left undefined outside a separately verified two-endpoint domain.

For compact $g$, the retarded and advanced responses ${G_{\mathcal{B}}}^{\mathrm{ret},\mathrm{adv}}g$ are smooth and have uniformly compact spatial support on $J⋐I$. Indeed their support lies in the causal future or past of $\mathrm{supp}g$, and the coordinate propagation radius is bounded by

on compact time intervals. Their rod sources meet Lemma lem:Newton.

This proves a Green-response tail. It is not a proof of a state-covariance tail.

The conditions (eq:W) alone do not ensure a decaying rod composition after compact testing.

Proof. In a spatially homogeneous frame all coefficients of $\mathcal{L}$ depend only on time. Spatial polynomials of degree at most one are preserved by the operator, reducing the Cauchy problem on that subspace to smooth finite-dimensional ordinary differential equations.

Choose a real solution $U$ with data at $t_0\in I$