VII-A-ST: Plateau intertwiners on a prescribed conical spacetime Supported classical descent, cutoff comparison, and finite destructive-clock certificates

We construct a supported local spacetime adaptation of the plateau intertwiner of Manuscript VII-A. The physical background is the regular part of I-B's prescribed conical Lorentzian product, not global Minkowski spacetime. A developing chart supports a precisely specified switched scalar action, unswitched finite-time harmonic coordinates, and formal diffeomorphisms supported inside a switch plateau. We prove the complete second-order scalar and coordinate intertwiners, including the density transformation and every switch derivative, without imposing a field equation on the metric coefficients. The construction restricts to an explicitly defined, nonempty Einstein–scalar coupling jet.

The repair is deliberately relative. For general compact gauge supports we derive the exact first-order residual ${\mathsf{E}}_m{{\mathcal{Q}}_{(1-g)\mathcal{K}\xi }}^{(m)}{\varphi }_0$, construct a strictly negative smeared witness, and prove that no higher coupling coefficient or positive $\hslash$-order removes the fixed classical gap. We also prove causal dependence on the truncated observation past and a unique Cauchy-data-preserving comparison cocycle between switches. An explicit example shows why equality on an observation plateau alone is insufficient.

The additional contribution is a finite operational realization that records both the repaired coefficients and the unrepaired gap. A sourced massless family supplies a nontrivial evaluation table; a complete destructive Enter instrument reads its finite encoding, admits a quantitative complete-channel approximation bound, and transports the payload and its classical control record along a prescribed causal route. This separation of classical covariance, preparation memory, and finite readout may provide a useful reproducible interface for later analytic work. It is not field acquisition or a quantum-gravity measurement construction. The classical plateau gate is closed on the stated domain; no quantum product, quantum Ward identity, radiative coefficient, or counterterm separator is constructed.

boundary{ This is a local Grade S construction on supplied geometry. The classical intertwiner and its larger-support obstruction are attributed to VII-A and are reproved here. The spacetime adaptation and finite evaluation interface do not derive the metric, lapse, deficit, or cone from records.

The regular cone is not globally hyperbolic. All analytical inverses below belong to a supported developing presentation. The quantum certificate remains RG-open.}

The problem is to keep three statements separate:

The last statement must neither erase the failure nor supply an unconstructed quantum field observable.

Manuscript VII-A proves a classical repair for a specified action-level switch, fixed harmonic Cauchy prescription, and restricted gauge group CheungVIIA. Its repair does not extend to every compact gauge generator. We give the local proofs needed to place that construction in the conical laboratory of I-B CheungIB. VI supplies a documentary comparison for the unrestricted obstruction, not a premise of our argument CheungVI.

Relational observables and their perturbative expansions have established antecedents BFHPR,Frob. In particular, neither the existence of relational constructions nor previously computed gauge-invariant quantum correlation functions is claimed as a discovery of this paper. Our object has its own finite-time boundary data, switch prescription, and gauge domain.

Cheung's Enter calculus treats runtime quantum names as access rights to tensor factors. A destructive readout consumes the measured names and returns a classical record; unmeasured factors may remain CheungEnter,CheungI. The philosophical distinction is between an output interface and an unrestricted claim about the fate of all implementing physical degrees of freedom. Destructive readout, quantum recovery, a thermodynamic implementation, and a geometric identification are different tasks CheungPhilosophy.

We adopt the specified finite Born instrument and prove its complete channel formula. We do not rederive the Enter characterization from an abbreviated axiom list or promote it to a field-instrument theorem.

I-B attaches a finite clock, local rods, and causal transport permissions to a prescribed conical spacetime CheungIB. Consuming that clock does not consume the geometric time axis. The manifold, worldsheet defect, and future cone remain input data.

A local Grade S package specifies and establishes:

1. a prescribed Lorentzian background, local rods, and a supported analytical laboratory;

2. named classical equations and proved descent identities on that laboratory;

3. a finite destructive clock with a complete recorded instrument;

4. an explicit finite encoding of selected classical evaluations;

5. reference-preserving transport whose complete dependencies respect the prescribed causal relation.

Grade E would additionally require a construction determining geometric data from operational records. It is not asserted.

The supported-presentation pattern is comparable with III-B and V-B CheungGradeS. Their theorems are not substituted for the arguments below.

Fix

The physical smooth spacetime is

where $r>0$ and $\varphi \in \mathbb{R}/(2\pi \mathbb{Z})$. The completion has singular locus

It is a worldsheet, not a worldline. No smooth Einstein tensor is asserted on $\mathcal{W}$.

The metric (eq:cone) has signature $(-,+,+,+)$. Choose the orientation for which ${\partial }_T$ is future directed. A nonzero future-causal vector $a{\partial }_T+w$ satisfies

Static observers have proper-time increment $\mathrm{d}\tau =\mathrm{d}T$.

On a sufficiently small lifted angular interval,

is a local isometry to Minkowski space. Its three spatial coordinates are local rods.

Proof. The diagonal metric matrix has one negative and three positive entries. Its causal inequality is $-c^2{a}^2+|w{|}_{\mathrm{spatial}}^2\le 0$. A nonzero causal vector cannot have $a=0$, and the chosen orientation selects $a>0$. The static proper-time formula follows by setting $w=0$. Finally,

The angular deck transformation changes the developed angle by $2\pi -\mu$. In the inverse-deck convention the gluing is rotation by $\mu$. Thus local flatness does not remove the global conical identification.

The spacetime $M$ has a noncompact causal diamond.

Proof. Fix $r_0>0$, one angle, and $z=0$. Take events $p,q$ at that spatial point with times $-2r_0/c$ and $2r_0/c$. For any $r_n\downarrow 0$, the events $(0,r_n,\varphi ,0)$ belong to $J^{+}(p)\cap J^{-}(q)$, by radial causal travel. They have no convergent subsequence in $M$, because the apex is excluded.

No global Cauchy theorem is inferred from local flatness. Global Green-operator results have additional hypotheses BGP.

Choose a regular spatial chart

whose image is convex and contains ${\overline{B}}_R$ in its interior. Translate the spatial coordinates if necessary, and set

For $0<b<R$, use the shrinking domain

Its open positive-time part is denoted ${\mathcal{D}}_{+}^{\circ }$. Fields are smooth up to the initial surface on neighborhoods of the portions used.

The analytical data consist of:

1. the chart, $\mathcal{D}$, ${\Sigma }_0$, and the fixed initial data prescriptions;

2. compact switches, tests, and gauge generators in ${\mathcal{D}}_{+}^{\circ }$;

3. a relatively compact open plateau $U⋐{\mathcal{D}}_{+}^{\circ }$, with its closure separated from ${\Sigma }_0$;

4. compact truncated past cones for all evaluations, with positive margin inside the developing laboratory.

Solutions are determined from their stated initial data, with no independent incoming lateral-boundary data.

Supportedness is a dependence condition. It does not require the metric coefficients themselves to have compact support or compact Cauchy data.

The domain $\mathcal{D}$ is backward-cone closed relative to ${\Sigma }_0$: if $(s,z)$ lies between $0$ and a point $(t,x)\in \mathcal{D}$, with $|z-x|\le t-s$, then

This is the property used by all local inverses.

For $m\ge 0$, define

There is a unique smooth solution of $P_{m}u=F$ on $\mathcal{D}$ with prescribed smooth Cauchy data. The zero-data inverse ${\mathsf{E}}_m$ is locally represented by

It has finite propagation and is continuous in local smooth topologies on compact truncated cones.

If $u$ has zero Cauchy data, then

For general $u$,

Proof. For compact spatial data and sources, spatial Fourier transformation reduces the equation to forced oscillators. Solving them gives (eq:Em) and the displayed homogeneous solution.

Finite propagation and smooth uniqueness follow from the energy identity on a truncated backward cone: the outward lateral flux is nonnegative, and zero data and zero source give zero energy. Differentiated energy estimates and local Sobolev estimates give smooth dependence.

For a compact target in $\mathcal{D}$, cut off and extend the required data and source from its truncated past to compact spatial data. Different extensions agree on the target by the preceding uniqueness argument. Exhaustion by compact targets defines the smooth local solution. Subtracting $H_m[u]$ proves (eq:leftinverse) and (eq:homogeneous-subtraction).

The flat Fourier formula is an auxiliary realization of the specified local problem, not a declaration that all of ${\mathbb{R}}^{1,3}$ is physical spacetime. After construction the result pulls back to the regular conical laboratory. No intrinsic global conical inverse is selected.

Work in $C^{\infty }(\mathcal{D})[[\kappa ]]/({\kappa }^3)$, without factorials in the coefficient expansion:

The prescribed background $\bar{g}$ and the formal unknown $\gamma$ have different roles. The former supplies the laboratory and apparatus cone. The latter is an input to the coefficient equations. No record-dependent metric update is introduced.

Raise coefficient indices with $\eta$. Define

Writing

matrix inversion and determinant expansion give

Indeed,

and ${\rho }_{\gamma }=\exp (\frac{1}{2}\mathrm{tr}\log ({\eta }^{-1}\gamma ))$; multiplication proves (eq:density-expansion).

Fix

The test is real $f\in {C_c}^{\infty }(U)$. Let

The admitted formal transformations are

They are formal jets of orientation-preserving diffeomorphisms generated inside $U$. Their group is denoted ${\mathcal{G}}_U$.

The Lie bracket preserves ${\mathfrak{g}}_U$, and the truncated Baker–Campbell–Hausdorff formula preserves the class (eq:formal-group). Including $\zeta$ is necessary for this convenient composition-stable parametrization. Every admitted actual flow is the identity outside a compact subset of $U$, preserves $g$, and is the identity near ${\Sigma }_0$.

Define the switched coefficients

Our scalar action and Euler operator are

The action is variationally defined with compact variations; its numerical value need not be finite on every smooth input. Its Euler output represents a scalar density relative to ${\mathrm{d}}^{4}x$.

For a symmetric tensor $H$, put

Then

where

In particular,

The quadratic metric vertex has one factor $g$, not $g^2$. The iterated first-order response below contains two switch insertions for a different reason.

Fix a real smooth free scalar $P_m{\varphi }_0=0$ with declared Cauchy data. Define

Both positive-order coefficients have zero Cauchy data.

The expression (eq:scalar-coeff) uniquely solves

with the stated leading data and zero positive-order data. It is defined for arbitrary smooth $h,k$.

Proof. The two coefficient equations are

Apply Lemma lem:Cauchy successively. Substituting the first solution gives (eq:scalar-coeff).

Define

The harmonic labels are four scalar functions

satisfying the unswitched equation

and data ${X_{\gamma }}^{\mu }{|}_0=x^{\mu }{|}_0$, ${\partial }_t{X}_{\gamma }^{\mu }{|}_0={\partial }_t{x}^{\mu }{|}_0$.

The unique coefficients are

Proof. At first order the equation is ${\square }_{\eta }{w}^{\mu }={\mathcal{C}}^{\mu }[h]$. At second order it is

Both have zero data, so Lemma lem:Cauchy applies.

The index $\mu$ labels the four scalar functions. It does not make the labels the components of a globally defined conical vector field.

Their formal inverse is

Substitution in $X_{\gamma }(Y_{\gamma }(y))$ proves the formula and uniqueness order by order.

Define

The smearing uses the fixed label density, not ${\rho }_{\gamma }{\mathrm{d}}^{4}y$. No global numerical invertibility of $X_{\gamma }$ is required.

Taylor expansion gives

Write

In particular, the complete second coefficient is

All five displayed contributions are retained.

For a vector field $\xi$, let

Here $D_{\xi }$ is the Lie derivative on scalar densities represented in the fixed chart.

For smooth $\xi ,u$,

If $\xi$ vanishes near ${\Sigma }_0$, then $w[K\xi ]=\xi$. On $P_m{\varphi }_0=0$,

Proof. The trace of $K\xi$ is $2{\partial }_{\rho }{\xi }^{\rho }$; the mixed derivative cancels in $\mathcal{C}[K\xi ]$. The zero-data left-inverse identity then gives $w[K\xi ]=\xi$.

For the second formula,

Its divergence is $-{\square }_{\eta }{\xi }^{\nu }$. Thus

This is $D_{\xi }{P}_{m}u-P_m{X}_{\xi }u$.

For every smooth $h$, every free ${\varphi }_0$, and $\xi \in {\mathfrak{g}}_U$,

Consequently ${A_1}^{(g)}$, and hence $C_1[g]$, is invariant. No metric field equation is required.

Proof. All derivatives of $\xi$ have support in the plateau. Linearity and (eq:QK-shell) give

The last equality uses the vanishing near ${\Sigma }_0$, not an unrestricted inverse identity. The coordinate identity follows from Lemma lem:linear. Substitution in (eq:A12) cancels the two variations.

Let $\xi \in {C_c}^{\infty }({\mathcal{D}}_{+}^{\circ };T\mathcal{D})$, without the plateau support restriction. Under the unchanged-free-input comparison $(h,{\varphi }_0)\mapsto (h+K\xi ,{\varphi }_0)$,

Proof. The difference is

The zero-data identity and (eq:QK-shell) give $X_{\xi }{\varphi }_0=-{\mathsf{E}}_m{{\mathcal{Q}}_{\mathcal{K}\xi }}^{(m)}{\varphi }_0$. Linearity in the tensor argument proves (eq:defect).

For comparison with an expanded retarded formula, write ${\overline{H}}^{\mu \nu }=H^{\mu \nu }-\frac{1}{2}{\eta }^{\mu \nu }{\mathrm{tr}}_{\eta }H$. Then

Indeed, expand ${\mathcal{Q}}_{gK\xi }$, and use ${\mathsf{E}}_m{P}_m(g{X}_{\xi }{\varphi }_0)=g{X}_{\xi }{\varphi }_0$. This independently reproduces the classical mechanism compared with VI CheungVI.

For every $m\ge 0$, an admitted packet exists with $f\ge 0$, $\int f=1$, and a compact larger-support generator $\xi$, constant near $\mathrm{supp}g$, such that

but

The gap is independent of the baseline $h$.

Proof. Choose an interior event $(t_{\ast },0)$ and free scalar data at $t=t_{\ast }$ equal to $x^1$ near zero, with zero time derivative, and smoothly compactly supported in space. The resulting free solution satisfies ${\partial }_1{\varphi }_0(t_{\ast },0)=1$. Choose a small plateau neighborhood $U$ where ${\partial }_1{\varphi }_0\ge 1/2$, and a unit-integral nonnegative test $f$ there. Choose $g=1$ near $\overline{U}$.

A compact smooth ${\chi }_{\xi }$, supported after $0$, can be chosen equal to one near the entire $\mathrm{supp}g$. Set $\xi ={\chi }_{\xi }{\partial }_1$. All derivatives of $\xi$ vanish near $\mathrm{supp}g$, so $gK\xi =0$. The scalar response has zero variation, while the coordinate response changes by $\xi$. Therefore

The cutoff of $\xi$ occurs outside the plateau containing the observation. Its support is not contained in $U$.

With ${A_1}^{(g)}$, the free input, and the larger-support identification fixed, no regular addition beginning at ${\kappa }^2$ makes the observable invariant under all those compact generators. Terms of positive $\hslash$-degree cannot cancel the nonzero $\kappa {\hslash }^0$ residual either.

Proof. Equality of formal series with nonnegative powers is coefficientwise. Extract the coefficient of $\kappa {\hslash }^0$ and use (eq:negativegap).

This is an existential obstruction to extending the theorem to the full compact class. It is not a claim that every generator outside the chosen $U$ produces a nonzero residual. For example, additional generators supported in another part of $\{g=1{\}}^{\circ }$ may also satisfy the cancellation.

The defect is classical, not a loop anomaly. Restricting the gauge domain does not refute the unrestricted obstruction.

Expansion of the metric pullback gives

Holding $k$ fixed while changing $h$ by $K\xi$ is generally not this transformation.

For an admitted actual diffeomorphism $F$, and hence coefficientwise for (eq:formal-group),

The density pullback is $F_{\mathrm{dens}}^{\ast }r=(\det DF)(r\circ F)$.

Proof. The inverse-metric density ${\mathcal{A}}_{\gamma }$ is a contravariant tensor density, and ${\rho }_{\gamma }$ is a scalar density. Their coordinate transformation laws follow by matrix inversion and the determinant chain rule.

Because $F^{\ast }g=g$, the discrepancy between $F^{\ast }{\mathcal{A}}_g[\gamma ]$ and ${\mathcal{A}}_g[F^{\ast }\gamma ]$ is

Inside $U$, $1-g=0$. Outside $U$, $F$ and its derivative are the identity. The discrepancy therefore vanishes. The same argument applies to ${\rho }_g$.

For completeness, divergence of a vector density is natural: pair it against a compact scalar test, integrate by parts, and change variables by $F$. The transformed integral is exactly the integral of the pulled-back divergence. Applying this fact to ${\mathcal{A}}_g[\gamma {]}^{\mu \nu }{\partial }_{\nu }u$ and adding the mass density proves (eq:plateaucov). Formal expansion gives the coefficient identity.

This lemma concerns the full divergence-form operator. It does not delete the terms in (eq:switch-derivatives), and by itself it is not the scalar intertwiner.

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

In coefficient form,

The free input ${\varphi }_0$ is unchanged.

Proof. Lemma lem:plateaucov shows that the pulled-back scalar solves the transformed equation. The pullback is the identity near ${\Sigma }_0$, so all its prescribed Cauchy data agree with those of the transformed scalar assignment. Proposition prop:scalar proves equality.

The unswitched massless density equation is natural by the same change-of-variables argument, without the switch discrepancy. Each pulled-back harmonic label has the prescribed initial data. Proposition prop:coordinates therefore identifies it with the transformed coordinate solution. Expansion proves (eq:intertwiner-coeff).

Let $V,W$ and $V^{\prime },W^{\prime }$ be the operators before and after (eq:metric-transform). Put

Expanding (eq:plateaucov) gives the off-shell identities

These retain the density divergence in $D$ and the terms proportional to $P_m$.

One can verify the second scalar law without appealing further to a covariance slogan. Set $a=u_{1,g}$, $b=u_{2,g}$, and $b_{\mathrm{cand}}=b+Xa+Y{\varphi }_0$. Using $P_{m}a=-V{\varphi }_0$, $P_{m}b=-Va-W{\varphi }_0$, and (eq:operator-audit),

The candidate has zero Cauchy data because the gauge generators vanish near ${\Sigma }_0$. Uniqueness gives the second scalar law.

For arbitrary smooth $h,k$, the pointwise relational scalar and its fixed-label smearing satisfy

under ${\mathcal{G}}_U$. In particular, $C_1[g]$ and $C_2[g]$ descend under this restricted formal action.

Proof. The coordinate intertwiner implies

Composing with the scalar intertwiner gives

The fixed label density gives the smeared identity. Appendix app:direct checks the second coefficient directly.

For a pure-gauge pullback of the flat metric,

the first and second relational corrections vanish. The metric family $\eta +\kappa K\xi$ with $k=0$ does not generally have this property: it omits part of the pullback.

The off-shell intertwiner does not require gravitational equations. To state its intended on-shell restriction, fix the normalization

This is VII-A's normalization. A coefficient such as the $1/4$ used in another action convention must not be transferred without rescaling the coupling.

Define

The differential definition is given in Appendix app:curvature. The leading stress tensor is

Let ${{\mathcal{J}}_g}^2$ consist of smooth tuples

with $P_m{\varphi }_0=0$, the scalar assignment (eq:scalar-coeff), and

No zero-data condition is imposed on $h$ or $k$. Their gravitational constraints are not replaced by zero-data wave evolution.

The metric equations are unswitched at this order, whereas the scalar equation is switched. On the plateau the scalar operator equals the unswitched curved density operator, so these are the Einstein–scalar coefficient equations there. Outside it, no nonlinear completion or switched gauge-fixed gravitational action is claimed.

A nonzero ${\varphi }_0$ is a coupling-expansion coefficient. It is not an assertion that $(\eta ,{\varphi }_0)$ is an exact nonzero-scalar Minkowski solution at nonzero coupling.

The transformations (eq:metric-transform) and (eq:intertwiner-coeff) preserve ${{\mathcal{J}}_g}^2$. The relational coefficients therefore descend on its ${\mathcal{G}}_U$-orbit set.

Proof. The scalar equations and data are preserved by Proposition prop:intertwiners.

For the metric, pullback preserves the Levi–Civita connection because it preserves torsion freedom and metric compatibility. It therefore preserves the curvature and Einstein tensor. Taking coefficients gives

The last term vanishes on (eq:metric-equations). The leading scalar remains ${\varphi }_0$, so its order-two source is unchanged. The final assertion follows from Theorem thm:second.

For $m=0$, let $a_{\mu }\ne 0$ be constant and put

Define

Together with the assigned switched scalar coefficients, this is a nonzero-source member of ${{\mathcal{J}}_g}^2$.

Proof. The leading scalar is free and its stress is the displayed constant nonzero tensor. Direct linearized curvature gives

Here

Thus $G^{(1)}[k]=T$. The scalar coefficients exist by Proposition prop:scalar; $u_{1,g}=0$ and $u_{2,g}=-{\mathsf{E}}_0{{\mathcal{Q}}_{gk}}^{(0)}{\varphi }_0$.

This polynomial local example imposes neither asymptotic flatness nor compact metric Cauchy data. It proves nonvacuity of the stated equations, not Einstein dynamics derived from records.

For a nonzero test $f$, define its truncated flat past