VI-B: Local harmonic Einstein–scalar descent and finite Enter certificates A supported-presentation adaptation on a prescribed conical spacetime

We give a self-contained local adaptation of Manuscript VI to the regular part of I-B's prescribed conical spacetime. On an explicitly supported developing presentation, we construct a second-order Einstein–scalar coefficient space and prove descent of a finite-time harmonic relational scalar under formal diffeomorphisms fixed near the initial surface. A constraint-compatible sourced family makes the construction nonempty and gives a strictly positive second-order relational coefficient.

The principal obstruction is retained rather than hidden by the adaptation. Replacing the scalar coefficient by a compactly switched retarded response produces an exact first-order gauge defect. A nested-support example has a normalized smeared gap at most $-1/2$, although the switch equals one near the entire observation. No correction beginning at second coupling order, or at positive $\hslash$-order, removes that classical gap while the lower-order prescription and identification are held fixed.

The additional contribution is an explicit finite certificate interface: three descended evaluations and the separate plateau-gap witness are encoded in finite diagonal matrices, read by a complete destructive-clock instrument, and transported with untouched references on a target-adapted causal route. Fixed-table approximation has a proved complete-channel limit. This combination may be useful as a reproducible separation between classical descent, switched compatibility, and operational readout; it is not a measurement construction for an unknown continuum field.

The local background, external schedule, initial-data prescriptions, and transport permissions are inputs. The regular cone is not globally hyperbolic, and the sourced example is not claimed to have compact metric Cauchy data. No intrinsic conical Green operator, quantum Ward identity, radiative response, or quantum renormalization separator is constructed. The quantum certificate remains RG-open.

scope{ The positive result is local Grade S, defined below. It is not Grade E: records do not determine the metric, lapse, deficit, or causal cone.

The classical descent and compact-switch obstruction are adaptations of the findings of VI, not priority claims over that manuscript. The finite encoding is an interface for already specified classical evaluations, not a finite-qubit field instrument.}

The present problem has three different maps:

The first can descend even when the second fails to intertwine the proposed representative identification. The third can record that failure without repairing it.

Manuscript VI identifies this distinction for a second-order Einstein–scalar expansion CheungVI. We reprove the required classical statements on a supported local domain. No proof of a companion manuscript is an unprinted premise.

Relational perturbation theory and higher-order invariant observables have substantial antecedents BFHPR,Frob. In particular, neither relational observables nor gauge cancellations in previously constructed quantum correlation functions are claimed as new here. The question addressed here is compatibility of one fixed finite-time prescription with one fixed compact switch, followed by a precisely limited finite operational interpretation.

Cheung's Enter calculus treats measured quantum names as linear resources: destructive readout consumes those names and returns a classical record, while explicitly unmeasured factors may survive CheungEnter,CheungI. The introductory philosophy emphasizes distinctions among readout, quantum recovery, physical implementation, and geometric identification CheungPhilosophy.

We adopt the specified finite Born instrument and prove its complete channel formula directly. We do not rederive the measurement characterization from an abbreviated list of axioms, and we do not extend it to a quantum field apparatus.

I-B supplies the model of a finite Enter interface on a prescribed conical product CheungIB. Its geometric time coordinate remains after clock consumption. Accordingly, a phrase such as “consuming the fourth dimension” is, in this paper, only an operational analogy for consuming a finite encoding calibrated to an external time coordinate.

A local Grade S package consists of:

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

2. named classical coefficient equations and proved descent statements on that presentation;

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.

This definition makes no claim that records generate the background or its equations.

The supported-laboratory discipline is comparable with the Grade S sheets III-B and V-B CheungGradeS. Their results are not substituted for the proofs below.

Fix

Prescribe

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

The singular locus in the completion is

It is a worldsheet. No smooth metric chart or ordinary pointwise Einstein tensor is asserted on $\mathcal{W}$.

The metric (eq:cone) has signature $(-,+,+,+)$. Choose the future orientation defined by ${\partial }_T$. For a nonzero future-causal tangent $a{\partial }_T+w$,

At a static port, $\mathrm{d}\tau =\mathrm{d}T$.

On a sufficiently small lifted angular interval,

is a local isometry to Minkowski space. The global conical gluing remains nontrivial.

Proof. The metric matrix has one negative and three positive diagonal entries. The causal inequality is $-c^2{a}^2+|w{|}_{\mathrm{spatial}}^2\le 0$, and a nonzero causal tangent cannot have $a=0$. For $w=0$, the proper-time formula follows directly.

Differentiation of the developing map gives

A deck shift $\varphi \mapsto \varphi +2\pi$ changes the developed angle by $2\pi -\mu$. In the inverse-deck convention this is rotation by $\mu$. Thus local flatness does not eliminate the conical identification.

The regular spacetime $M$ has a noncompact causal diamond.

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

Consequently local flatness alone does not justify a global conical Cauchy or Green theorem. Standard global results require appropriate global hypotheses BGP.

Choose a regular spatial developing chart

with $V$ convex and containing a closed Euclidean ball ${\overline{B}}_R$ in its interior. Set

Fix $0<b<R$, and define the shrinking Cauchy domain

All coefficient fields are real and smooth on neighborhoods of the portions of this domain where they are used.

An admitted analytical packet fixes:

1. the chart, ${\Sigma }_0$, $b$, and ${\mathcal{D}}_{R,b}$;

2. compact observation tests, compact interaction switches, and compact gauge generators in the interior $0<t<b$;

3. the coefficient Cauchy data and every initial constraint preparation used;

4. a compact collection of truncated past cones containing all dependencies of the selected evaluations, with a positive margin inside the developing laboratory.

Wave solutions are determined from their stated initial data, without independent incoming lateral-boundary data.

“Supported” refers to this dependence contract. It does not require every coefficient or every metric Cauchy datum to be compactly supported. In particular, the sourced family of Section sec:sourced retains an explicit auxiliary Newton preparation, whose smooth restriction is admitted initial data.

Any finite collection of compact past cones in ${\mathcal{D}}_{R,b}$ remains away from the worldsheet after pullback. No global identification across angular gluing is performed.

Let

For a smooth source on ${\mathcal{D}}_{R,b}$, the wave and Klein–Gordon equations with prescribed smooth initial data have unique smooth solutions on that domain. On every compact target they depend only on the data and source in its truncated past. The solution maps are continuous in the local smooth topology.

Proof. For compact spatial data and sources, spatial Fourier transformation gives the cosine and sine evolution, with

For example, zero-data solutions of $Pu=F$ are

The multiplier has a continuous massless value at zero frequency.

Finite propagation follows by integrating the usual wave energy on a truncated backward cone. For the homogeneous difference of two solutions, the lateral flux has the outward nonnegative sign, and zero initial energy gives zero energy throughout the cone. The same argument includes the nonnegative mass term.

For a compact target in ${\mathcal{D}}_{R,b}$, its initial shadow lies strictly inside $B_R$. Extend its required data and source smoothly with compact cutoffs, solve the compact problem, and restrict. Two choices agree by the cone uniqueness just proved. An exhaustion by compact targets gives a compatible smooth solution on the full domain. Differentiated energy estimates and Sobolev estimates on compact cones give continuous dependence in every smooth seminorm.

Write

For compact sources supported after $0$, the Klein–Gordon retarded inverse is

These operators are defined on the developing presentation and then pulled back. They are not an intrinsic global conical prescription.

The prescribed $\bar{g}$ controls the local presentation and the finite apparatus schedule. The formal metric $\gamma (\kappa )$ below is an unknown of stipulated coefficient equations. We do not identify the apparatus cone with a nonperturbatively constructed causal cone of that unknown metric.

All indices in coefficient formulas are raised with $\eta$. For a symmetric tensor $a$, put

For a vector $\xi$,

Set

One has

Proof. The mixed derivatives of ${\partial }_{\mu }{\xi }^{\mu }$ cancel in $D(K\xi )$. Substituting this result in (eq:G1) proves the pure-gauge identity, and taking a divergence proves the linearized Bianchi identity.

Finally,

which is $-L_{K\xi }u$.

Use the action convention

The Ricci convention is the one yielding (eq:G1). Compactly supported inverse-metric variation gives

where

Indeed, the coefficients of the metric variation are $2G_{\mu \nu }/{\kappa }^2-T_{\mu \nu }/2$. Thus the source factor $1/4$ is fixed, not freely suppressed.

Work modulo ${\kappa }^3$, without factorials:

Define $G^{(2)}[h,h]$ by

An explicit differential specification of $G^{(2)}$ appears in Appendix app:G2.

Let

One has

where

Proof. Matrix inversion and the determinant expansion give

Their product is

Insert this and the inverse density in

The linear term is $L_h$, the part linear in $j$ is $L_j$, and the remaining quadratic term is

which equals (eq:M).

Let ${\mathcal{J}}_{R,b}$ be the real smooth tuples

on the supported presentation satisfying

Each tuple includes its coefficient initial data. No assertion is made that arbitrary unconstrained initial metric data extend to this space.

These are the coefficient equations of (eq:equations). They are assumptions defining the model, not equations derived from the Enter records.

The exact zero-scalar background is $(\eta ,0)$. A nonzero ${\varphi }_0$ in (eq:expansion) is a leading coupling-expansion coefficient. It is not asserted to be an exact nonzero Minkowski Einstein–scalar background at nonzero coupling. This distinction is also essential in VI CheungVI.

Let

and define formal pullbacks

The generators vanish near ${\Sigma }_0$ and near the lateral boundary. Their formal compositions again have compact generators, by the truncated Baker–Campbell–Hausdorff formula.

The action (eq:gauge) is

It preserves ${\mathcal{J}}_{R,b}$ and the coefficient initial data.

Proof. Expand the exponential and collect powers of $\kappa$.

The Levi–Civita connection of a pullback metric is the pulled-back connection: it is torsion free and compatible with that metric, which characterizes it uniquely. Its curvature and Einstein tensor therefore pull back. The gradient-divergence definition of the scalar wave operator gives, by the chain rule,

The stress tensor has the same covariance by its displayed tensor formula. Taking coefficients proves preservation of (eq:jet1)–(eq:jet5). All changes vanish on a neighborhood of the initial surface, including their initial time derivatives.

The scalar changes in (eq:gauge-coefficients) are part of the action. Omitting them gives a different identification.

Define four scalar coordinate functions, indexed by an external label $\mu =0,1,2,3$:

Require

The label $\mu$ does not make ${X_{\gamma }}^{\mu }$ the components of a spacetime vector.

For arbitrary smooth $h,j$, the unique coefficient solution of (eq:coordinate-data) is

The coefficients and their first time derivatives vanish at $0$. They depend continuously on $h,j$ on compact truncated pasts.

Proof. Since $\square x^{\mu }=0$ and $L_h{x}^{\mu }=-D(h{)}^{\mu }$, the first coordinate equation is

The second is

Lemma lem:wave proves existence, uniqueness, locality, and continuity for their zero-data problems.

Under (eq:gauge),

In particular,

Proof. Scalar covariance shows that $F^{\ast }{X_{\gamma }}^{\mu }$ solves the harmonic equation for $F^{\ast }\gamma$. Because $F$ is the identity near ${\Sigma }_0$, it has precisely the data (eq:coordinate-data). Coefficientwise uniqueness in Proposition prop:coordinates identifies the solutions. Expansion gives (eq:coordinate-transform).

The formal inverse coordinate map is

Substitution in $X_{\gamma }(Y_{\gamma }(X))$ verifies this identity directly.

For real $f\in {C_c}^{\infty }({{\mathcal{D}}_{\mathcal{R},b}}^{\circ })$, interpreted as a test in label coordinates, define

The measure is the fixed label density. It is not a fluctuating spacetime volume element.

The coefficients of (eq:observable) are

Each is invariant under (eq:gauge-coefficients). Consequently they descend to the orbit set ${\mathcal{J}}_{R,b}/\mathcal{G}$ of the stated local formal gauge group.

Proof. Taylor expansion using (eq:inverse) gives all displayed terms. No factor $1/2$ is attached to ${\varphi }_2$, because (eq:expansion) has no factorials.

Lemma lem:coordinate-covariance implies

Together with ${\phi }^{\prime }=\phi \circ F$, this gives

The same $f(X){\mathrm{d}}^{4}X$ is used on both sides, so equality is coefficientwise. All integrals are finite, and the required coordinate coefficients are defined on compact causal pasts.

The proof is off shell as a formal coordinate statement. Restricting it to ${\mathcal{J}}_{R,b}$ gives the named on-shell descent. It does not establish that the orbit set is a global conical reduced phase space, or that the observable has renormalized quantum products.

Changing coordinate Cauchy data is not a transformation of the specified gauge group. If $w$ is replaced by $w+a$ with $\square a=0$, then

which need not vanish. Descent therefore does not imply independence of the relational prescription.

Pull the coefficient fields and their fixed chart data back to the regular laboratory. The background chart is an isometry, and the defining differential equations and compact evaluations are identical in developed coordinates. This establishes the same local identities in that specified presentation.

It does not assert that the chartwise coefficients $w^{\mu }$ are global vector components or that the selected inverse patches consistently across the conical gluing. Such a global statement would require additional construction.

Fix a real compact switch

It is not transformed in this section. The compact minimal scalar vertex has classical density

For $H=gh$, its scalar Hessian is

Integration by parts shows formal self-adjointness. Expanding the derivatives gives

Hence, for $P{\varphi }_0=0$,

Define the classical zero-past response

It equals the zero-data solution of $P\psi =-L_{gh}{\varphi }_0$, by (eq:QL) and uniqueness. In particular, multiplication by $g$ occurs inside the full metric operator, not merely in front of $L_h$.

Consider the candidate

under the unchanged-free-input identification

This is not the full action on ${\varphi }_1,{\varphi }_2$ in (eq:gauge-coefficients).

For every smooth free scalar ${\varphi }_0$,

where

Proof. The coordinate variation is

because $\xi$ vanishes near the initial surface.

By Lemma lem:linear and (eq:Qswitch),

Therefore

Use

The function $g{X}_{\xi }{\varphi }_0$ is compactly supported after $0$, so zero-past uniqueness gives $G_{\mathrm{R}}P(g{X}_{\xi }{\varphi }_0)=g{X}_{\xi }{\varphi }_0$. Subtract $X_{\xi }{\varphi }_0$, the dressing variation, and smear.

This is a classical retarded identity. It is not a loop anomaly, a renormalized Ward identity, or a quantum Master Ward identity.

An admitted supported packet exists with real compact $f_w,g,\xi$ and a real compact-data free scalar ${\varphi }_w$ such that

while

Both metric representatives satisfy $G^{(1)}(h)=0$.

Proof. Choose $t_{\ast }\in (0,b)$. On the auxiliary flat Cauchy slice $t=t_{\ast }$, prescribe

where ${\chi }_0$ is compact smooth and equals one near zero. Klein–Gordon evolution gives compact smooth initial data at $0$, and ${\partial }_1{\varphi }_w(t_{\ast },0)=1$. Choose a small relatively compact neighborhood $O$ of $(t_{\ast },0)$ on which ${\partial }_1{\varphi }_w\ge 1/2$. Let $f_w\ge 0$ have unit integral and support in $O$.

Take a compact smooth $g=1$ near $\mathrm{supp}{f}_w$. Choose ${\chi }_{\xi }$ compactly supported after $0$, equal to one near the entire $\mathrm{supp}g$, and put $\xi ={\chi }_{\xi }{\partial }_1$. All supports can be chosen inside ${{\mathcal{D}}_{\mathcal{R},b}}^{\circ }$ by enlarging the laboratory first if necessary.

Every derivative of $\xi$ vanishes near $\mathrm{supp}g$. Thus $gK\xi =0$, and both switched responses in (eq:gap) are zero. But $w[K\xi ]=\xi$, and $\xi ={\partial }_1$ on $\mathrm{supp}{f}_w$. This proves (eq:gap). Finally, $G^{(1)}K=0$.

The same $f_w,g,{\varphi }_w$ is used for both representatives. The strictly negative gap is not produced by changing the interaction switch between comparisons.

Suppose the first coefficient is fixed by (eq:candidate). No choice of the coefficient ${{\mathcal{A}}_{f,g}}^{(2)}$ makes

invariant under (eq:free-identification) on the full stated domain. Nor can terms of positive $\hslash$-degree cancel the fixed nonzero $\kappa {\hslash }^0$ gap in a coefficientwise formal identity.

Proof. Extract the coefficient of $\kappa$, and then, if applicable, of ${\hslash }^0$. The value $\Delta \ne 0$ is unaffected by terms starting at higher orders.

The geometric transformation requires

Those terms cancel in ${\varphi }_1-X_w{\varphi }_0$. The plateau witness instead has

Thus the switched assignment fails to supply the scalar part of the full coefficient action. It does not disprove Theorem thm:descent.

The quantified obstruction retains all four choices: the unswitched coordinate dressing, the fixed compact switch, the full compact gauge-support class, and (eq:free-identification). Changing any of them defines another problem.

VII-A records a classical repair for the restricted class CheungVIIA. We neither use that result as a premise nor claim its second-order intertwiner as a theorem of VI-B.

A chartwise subtraction of the retarded remainder in (eq:defect) changes the specified observable. It does not automatically define a tensorial subtraction on the full cone. Already

for a general compact $\xi$, so the fixed flat inverse does not commute with arbitrary spacetime pullbacks. A global replacement needs its own covariance and initial-data theorem. This is the relevant warning behind the fixed-data subtraction comparison in V-B CheungGradeS.

The metric coefficient in (eq:jet2) is an unknown, not an external tensor. We now solve a nontrivial subclass.

For compact smooth spatial $b_0$, prescribe the auxiliary flat Newton branch

The kernel is locally integrable, and derivatives can be transferred to the smooth compact source. This gives a smooth potential. The identity follows from the flux of $\nabla (1/|x|)$ through a small sphere.

This initial-data branch is additional model data. It is not an intrinsic conical inverse.

Let ${\varphi }_0$ be a real free Klein–Gordon solution with compact smooth auxiliary Cauchy data. There is a tuple in ${\mathcal{J}}_{R,b}$ with

The scalar ${\varphi }_2$ can be chosen with zero initial data. For nonzero scalar data, $Q\ne 0$.

Proof. A direct differentiation of the stress tensor gives

Thus $Q$ is conserved. Finite propagation makes it spatially compact on each bounded time slab.

At $0$, set

Prescribe the trace-reversed metric data

Solve

For example, its componentwise expression is

with $S_w(t)=\sin (t|D|)/|D|$. Local smooth existence follows also from Lemma lem:wave.

Write

The initial data give $d_{\nu }{|}_0=0$. Furthermore,

Using (eq:sourcedwave), these identities imply ${\partial }_t{d}_{\nu }{|}_0=0$. Conservation gives $\square d_{\nu }=0$, so uniqueness gives $D(j)=0$. Equation (eq:G1) now gives $G^{(1)}(j)=Q$.

Finally solve

with zero initial data. The source is smooth and is spatially compact on finite slabs because it contains derivatives of ${\varphi }_0$. All five jet equations hold.

For nonzero data,

so $Q\ne 0$.

The Newton position data in (eq:metricdata) generally have tails. They are not compact-Cauchy metric data. Their restrictions to ${\Sigma }_0$ are declared before evolution.

There is no incoming lateral data for the subsequent wave problem on ${\mathcal{D}}_{R,b}$. That statement does not turn the initial Newton prescription into a local source-acquisition operation. In particular, the construction proves no worldsheet boundary condition and no global conical solution.

For $m>0$, the sourced family contains a tuple and a nonnegative compact test $f_s$ such that

Proof. Choose $F\in {C_c}^{\infty }({\mathbb{R}}^3)$, $F\ge 0$, equal to a constant $A>0$ near zero, and prescribe

Then

The nonnegative nonzero source gives $a(0)=-2\mathcal{N}\rho (0)>0$. Trace reversal yields

Since $h=0$ and $D(j)=0$, the coordinate equations give $w=v=0$. Hence the second relational density is ${\varphi }_2$. At the origin, the spatial Hessian of $F$ is zero and ${{\partial }_t}^2{\varphi }_0(0,0)=-m^{2}A$. Therefore

The zero initial data of ${\varphi }_2$ and its equation imply

For a sufficiently small positive time, ${\varphi }_2$ is positive on an open neighborhood, while ${\varphi }_0>0$ there. Choose a nonzero nonnegative $f_s$ in that neighborhood. The three assertions follow.

This proves nonvacuity and a sourced coefficient inside the stipulated equations. It is not a derivation of Einstein dynamics or of physical backreaction from records.

For the preceding tuple, let

For every real $\lambda$ this is a jet, and