X-A-ST: Supported classical covariance of a fixed-ramp relational scalar A local Grade S transfer through second coupling order on a prescribed conical spacetime

We give a supported transfer of one fixed classical relational-coordinate prescription to a regular laboratory of the prescribed I-B conical spacetime. The calculation is performed on a declared Minkowski completion, with signature $(+,-,-,-)$, and then restricted to the laboratory; the host retains its independently prescribed signature $(-,+,+,+)$, worldsheet defect, and future relation. No global conical inverse is assumed.

At fixed ramp width $\epsilon >0$, the divergence-form coordinate problem has unique formal coefficients through order ${\kappa }^2$. On the gauge algebra fixed before the calculation, we prove the complete second-order coordinate intertwiner, off-shell invariance of the label-smeared relational scalar, and covariance of the switched scalar solution with unchanged free input. A separate varied-action derivation displays every switch contribution. An explicit edge-zone parameter has the strictly negative integrated preflight defect $-{{\int }_0}^{\epsilon }{\sigma }^{\prime }(s)\alpha (s)\mathrm{d}s$; this disproves an unrestricted extension of the preflight identity without changing the lower-order prescription. Pure-gauge and sourced diagnostics verify the retained quadratic terms.

The contribution is a precise supported realization and sign, support, and operator audit of Cheung's X-A result, not a new claim of priority for its classical covariance. Its potential usefulness is to provide a reproducible classical baseline that distinguishes a successful plateau-restricted identity from regulator-induced breaking. In particular, smoothness is proved for observation-smeared first-derivative kernels, not for the unsmeared two-point wave kernel. A finite destructive-clock instrument and reference-preserving regular transport encode supplied evaluations without acquiring a continuum field. The release is local Grade S only: no time-ordered product, normalized retarded map, sharp-width limit, quantum Ward identity, or spacetime-emergence theorem is constructed.

and attribution{ The classical target is constructed through ${\kappa }^2$ at each fixed $\epsilon >0$, on the explicitly frozen plateau algebra. The source result is attributed to N. Cheung's X-A manuscript.

The conical transfer is a supported presentation on a declared flat completion. It is not an intrinsic construction on the entire regular conical product. The finite Enter interface reads installed encodings of evaluations already defined mathematically. It does not establish a field-to-qubit acquisition map.}

Relational coordinates can remove a simultaneous coordinate change of fields only when the coordinate prescription itself transforms correctly. Initial data, switching, and regularization are therefore part of the statement, rather than ancillary notation. The field-dependent-coordinate literature provides useful context, but does not prove covariance of the particular switched and regulated problem considered here BFR,FrobLima.

The starting classical prescription is Cheung's X-A: a compact switch multiplies the complete curved-minus-flat scalar density before variation, whereas an independent smooth time ramp regulates the coordinate density XA. We retain this distinction. In particular, neither substituting $gh$ into a metric nor convolving metric arguments defines the prescription below.

The operational background is the Enter calculus, in which destructive readout consumes the measured quantum names and returns a classical record with any unmeasured residual CheungEnter,I. We use ordinary finite Born instruments and prove the channel identities needed in this manuscript. The operational characterization of the Born strategy is not a premise of the classical proof.

The philosophy of Enter emphasizes distinctions between readout and recovery, between an input–output channel and a thermodynamic implementation, and between an analogy and an identified physical model EnterPhilosophy. Here those distinctions prevent three unwarranted conclusions:

1. a finite clock encoding does not define the continuum time coordinate used by the Cauchy problem;

2. a finite record of a classical evaluation does not constitute a measurement of that functional on a field;

3. a prescribed spacetime supporting a protocol is not spacetime derived from the protocol.

For this manuscript, a local Grade S package consists of:

1. a prescribed host with a supported regular laboratory;

2. classical objects with explicit domains, equations, initial data, and transformation laws;

3. one finite destructive-clock factor and a stated evaluation dictionary;

4. complete finite instruments, including their untouched-reference extensions;

5. a scheduled-locality theorem and a nonempty reference-preserving route whose physical dependency containment is separately certified.

Grade E, metric emergence, and a quantum field construction are not included in this definition.

No companion-sheet theorem is needed as a hidden premise: the host choices and analytical prescription used here are stated explicitly, and the classical identities are proved below.

Fix $0<{\delta }_{\mathrm{c}}<2\pi$, ${\alpha }_{\mathrm{c}}=1-{\delta }_{\mathrm{c}}/(2\pi )$, and $c>0$. The regular host is

Its stipulated completion has singular set

Thus the defect is a worldsheet, not a worldline. The time orientation is increasing $T$. These are I-B-type inputs IB.

The regular product (eq:host) has a noncompact causal diamond and is not globally hyperbolic.

Proof. Fix a regular spatial point of radius $r_{\ast }>0$, and two events there separated by more than $2r_{\ast }/c$. At an intermediate time, points on the same radial ray with radii decreasing to zero lie in the causal diamond between the two events. They have no subsequence converging in the regular product.

Choose an injective product developing chart

whose relevant compact closure stays off $\mathcal{W}$. Set $t=c(T-T_{\Sigma })$ on its image. In developed coordinates,

Choose a relatively compact open observation region $\mathcal{O}⋐(0,T_{\ast })\times B_R({\mathbf{x}}_{\ast })$ such that

Only tests $f\in {C_c}^{\infty }(\mathcal{O};\mathbb{R})$ are used for the supported observable.

All nested retarded calculations for such a test are retained in $D_{\ast }$. No independently adjustable lateral boundary datum is introduced.

The chart can be chosen with enough regular margin to contain the compact supports of the diagnostics below. For those diagnostics the same retained-shadow condition is imposed before their tests are selected.

The analytical completion is the separate Minkowski space

All coordinate and scalar Cauchy problems are solved on this completion. Smooth fields are defined on a neighbourhood of $M_{T_{\ast }}$, without decay assumptions. A compact laboratory test is transported to $\hat{M}$ and extended by zero. The initial surface is the complete flat $\Sigma$, not a claimed Cauchy surface of ${\mathcal{M}}^{\circ }$.

This makes explicit the completion pattern of IV-B IVB. It is not a global developing map.

The two signature conventions are related on the chart by

but are not silently substituted for one another. All density expansions, Euler signs, and Cauchy inverses below use $\eta$ and

The host's unit-lapse stationary proper time is $T$; the developed comparison variable is $t=c(T-T_{\Sigma })$. The formal metric jet introduced below is not a command that evolves $G$.

Proof. The displayed chart metric gives the first identity. The remaining statements specify which objects enter the analytical equations and which enter physical port placement. Reversal of the overall signature leaves their common null directions unchanged, but does not justify changing an Euler or Green-kernel sign.

The developing chart does not supply a global Minkowski representation, an intrinsic Newton inverse on the cone, or a global field-controlled unitary.

Proof. Parallel transport around the missing conical axis has nontrivial rotational holonomy for $0<{\delta }_{\mathrm{c}}<2\pi$. A global isometry into Minkowski space would identify this connection with one having trivial holonomy, which is impossible.

A local isometry also does not fix nonlocal inverse boundary conditions. For example, two flat-completion sources agreeing in a chart but differing by a nonnegative bump outside it can have different Newton potentials inside it. Finally, a chart supplies neither a field representation nor a coupling Hamiltonian or controlled unitary on such a representation.

The nonimplication, rather than a theorem prohibiting every future intrinsic construction, is the point of the last two assertions.

Work over

An equality modulo ${\kappa }^3$ is coefficient equality, not a numerical error estimate.

The configuration variables are

where $h,k$ are arbitrary smooth real symmetric tensors and ${\varphi }_j$ arbitrary smooth real scalars. There is no Einstein equation, free-shell quotient, gauge fixing, or decay condition on this off-shell configuration space.

Indices are raised using $\eta$, and parentheses include a factor $1/2$. Write

Set

Polarization is normalized by $a_2[h,h]=a_2[h]$ and $b_2[h,h]=b_2[h]$.

For $v_q=\sqrt{-\det q}$ and ${A_q}^{\mu \nu }=v_q{q}^{\mu \nu }$,

Proof. The inverse is

Expanding $\exp \{\frac{1}{2}\mathrm{tr}\log (I+\kappa {\eta }^{-1}h+{\kappa }^2{\eta }^{-1}k)\}$ gives $v_q$. Multiplication gives $A_q$.

Define

A direct calculation gives

Fix $m\ge 0$, $0<\epsilon <T_{\ast }$, and

For the supported realization, the retained interaction support is in the laboratory. Contributions outside a given $D_f$ will in any event not enter its evaluation.

Define

It is smooth and flat at both endpoints; ${\sigma }^{\prime }(t)>0$ on $0<t<\epsilon$, and $\sigma =1$ for $t\ge \epsilon$.

Choose a nonempty relatively compact open set $U$ with

Set ${\mathfrak{g}}_p={C_c}^{\infty }(U;T\hat{M})$. Its elements are extended by zero to the completion.

For $\xi ,\zeta \in {\mathfrak{g}}_p$, let $F_{\kappa }$ be the time-one flow of $\kappa \xi +{\kappa }^2\zeta$. The formal pullbacks

generate ${\mathcal{G}}_p$. This is the frozen plateau algebra; it is not the Killing algebra of the cone.

In particular, $U$ is open with compact closure. It is not called a nonempty compact open subset of Minkowski space.

Every admitted $F_{\kappa }$ is the identity near $\Sigma$ and outside a compact subset of $U$, preserves the slab, and satisfies

The truncated pullbacks form a group.

Proof. A compactly supported smooth vector field has a complete flow, which fixes the complement of its support. Here the flow is supported where both $g$ and $\sigma$ are one. It is the identity near the slab endpoints. Closure of the truncated pullbacks follows by expanding their product: the order-two bracket contribution is $\frac{1}{2}[\xi ,{\xi }^{\prime }]$, again supported in $U$.

For scalars put $X_{\xi }={\xi }^{\mu }{\partial }_{\mu }$; for scalar densities put $D_{\xi }r={\partial }_{\mu }({\xi }^{\mu }r)$. The full coefficient transformations are

Deleting the quadratic terms in (eq:metriclaw) would change the problem.

The auxiliary coordinate density and its operator are

The output is a scalar density relative to ${\mathrm{d}}^{4}x$. No convolution occurs.

The four components of

are scalar functions satisfying

The component label $\mu$ does not prescribe a vector-field pullback. The switch $g$ does not enter this problem.

For the scalar dynamics define instead

Thus

The action ${S_g}^{\mathrm{sc}}={\int }_{M_{T_{\ast }}}{\mathcal{L}}_g$ is understood variationally; its numerical value on arbitrary smooth fields need not be finite. Compactly supported variations and the finite changes used below are well defined.

The sign convention is

For $\mu \ge 0$, smooth source $s$ on the slab and smooth initial data on $\Sigma$, the equation

has a unique smooth solution. Its value on a compact set depends only on the source and data in the corresponding truncated flat causal past. For zero data denote it by ${\mathsf{E}}_{\mu }s$.

On compact sets this solution map is continuous in the smooth topologies. No spatial decay is required.

Proof. For compactly supported smooth data and source, spatial Fourier transformation gives the solution using

The quotient has value $t-s^{\prime }$ at zero frequency. Differentiation of the Fourier formula gives smoothness. The usual differentiated energy estimate, or the same Fourier formula in Sobolev norms followed by Sobolev embedding, gives finite-seminorm bounds on compact time intervals.

For finite propagation, apply the energy identity with energy density

to a truncated backward cone. Its lateral flux has the required nonnegative sign. Zero source and data on that cone imply zero solution at its vertex. Adding an $u^2$ term and applying Gronwall also covers the massless low-frequency estimate.

For arbitrary smooth data, fix a compact target set, cut off the source and data on a neighbourhood of its compact truncated past, and use the compact-data solution. Finite propagation makes the result independent of the cutoff there. Such solutions agree on overlaps, giving a global smooth solution on the slab. The same argument proves uniqueness and local continuity. This is the standard Cauchy mechanism used on the declared completion, not on the non-globally hyperbolic host BGP.

For a compact interior source $a$, let ${\mathsf{E}}_{\mathrm{A},\mu }a$ be the advanced solution, with zero terminal data after the source. On the slab it is smooth, compactly supported in the truncated past of $\mathrm{supp}a$, and vanishes near $t=T_{\ast }$.

For a compact interior test $a$ and smooth source $s$,

The formula also holds for smooth compact tests flat at $t=0$, extended by zero to negative times.

Proof. Use Green's identity with $u={\mathsf{E}}_{\mu }s$ and $b={\mathsf{E}}_{\mathrm{A},\mu }a$. The terminal term vanishes because $b$ is terminally zero; the initial term vanishes because $u$ has zero Cauchy data. Spatial boundary terms vanish after a cutoff outside the compact retained past. The flat-at-zero case follows by the same identity or smooth cutoff approximation.

Write ${\mathsf{E}}_0={\mathsf{E}}_0$, ${\mathsf{E}}_m={\mathsf{E}}_m$, and ${\mathsf{E}}_{\mathrm{A}}={\mathsf{E}}_{\mathrm{A},0}$.

Set $B^{\mu \nu }={a_1}^{\mu \nu }[k]+{a_2}^{\mu \nu }[h]$. Then

where

The coordinate problem has the unique solution

Equivalently, with zero data at each positive order,

Also,

Proof. Insert (eq:coordinateoperators) and $\mathsf{Co}=x+\kappa w+{\kappa }^{2}v$ into (eq:coordinateproblem). Since $\square x^{\mu }=0$, coefficient comparison gives the displayed equations. Lemma lem:cauchy proves existence and uniqueness successively. Only the source $-{\partial }_{\alpha }(\sigma {a_1}^{\alpha \mu }[k])$ contains $k$, proving the final identity.

The defining operator contains one derivative of $\sigma$, not a discarded ramp term. No explicit ${\sigma }^{\prime \prime }$ term is generated by expanding its single divergence. Derivatives of solutions or of coefficient identities may of course differentiate ${\sigma }^{\prime }$; the statement is about the displayed defining operators.

Let $P_m=\square +m^2$. The expansion is

with

For a fixed smooth real free input $P_m{\varphi }_0=0$, the switched scalar solution is

Equation (eq:scalarsolution) is the unique formal solution of $E_g[q]{\rho }_g=0$ with the free input's Cauchy data and zero positive-order data. Its positive-order sources are smooth and supported in $\mathrm{supp}g$, and

Proof. At orders one and two the equations are respectively

Apply Lemma lem:cauchy. Every coefficient of the source operators contains $g$ or a derivative of $g$.

No coupled Einstein solution is inferred.

If $\xi \in {C_c}^{\infty }({M_{T_{\ast }}}^{\circ };T\hat{M})$ and $\mathrm{supp}\xi \subset \{t\ge \epsilon \}$, then

In particular, this holds for every $\xi \in {\mathfrak{g}}_p$.

Proof. All derivatives of $\xi$, and hence $a_1[K\xi ]$, are supported where $\sigma =1$. Therefore

Both sides have zero Cauchy data. Uniqueness proves the assertion.

This is the preflight identity of the fixed ramp prescription XA ({:preflight}).

Let $0\ne \alpha \in {C_c}^{\infty }((0,\epsilon ))$ be nonnegative, and choose $\beta \in {C_c}^{\infty }({\mathbb{R}}^3)$ with $\int \beta {\mathrm{d}}^{3}x=1$. Set

For every $t\in [\epsilon ,T_{\ast }]$,

This parameter is outside ${\mathfrak{g}}_p$.

Proof. The source for $w[K\xi ]$ is compact, so its solution has compact spatial support on each bounded time slice. Put $W^{\mu }(t)=\int w^{\mu }[K\xi ](t,\mathbf{x}){\mathrm{d}}^{3}x$ and ${\Xi }^{\mu }(t)=\int {\xi }^{\mu }(t,\mathbf{x}){\mathrm{d}}^{3}x$. Spatial divergences integrate to zero. From

one obtains

Thus

For $t\ge \epsilon$, integration by parts gives (eq:edge). Its strict sign follows from ${\sigma }^{\prime }>0$ in the ramp and $\alpha \ge 0$, $\alpha \ne 0$. The support of $\xi$ is disjoint from the prescribed post-ramp $U$.

The formula retains the parent edge witness XA ({:edge}). It does not classify the kernel of the preflight defect on every possible ramp-supported parameter.

There are a smooth scalar ${\varphi }_0$ and a supported observation test $f$ for which

No correction starting at ${\kappa }^2$, or at positive $\hslash$-order in a formal double expansion, cancels this order-$\kappa$, order-${\hslash }^0$ coefficient while the lower-order prescription is held fixed.

Proof. At a time strictly after $\epsilon$, (eq:edge) implies that $w^1[K\xi ]$ is nonzero somewhere. Choose ${\varphi }_0=x^1$ and a nonnegative bump $f$ where that component has a fixed nonzero sign. The chart reserve may be chosen in advance to contain the compact wave support and the resulting test's truncated past.

At the flat metric the first relational variation under the simultaneous first-order field pullback is precisely (eq:smearededge). Formal coefficients of different powers cannot cancel it. A universal extension to all such parameters is therefore excluded without changing the lower-order data.

This is the coefficientwise logic illustrated in VI-B, not an identification with VI-B's different switched Einstein–scalar defect VIB.

The coordinate inverse exists algebraically:

No globally invertible numerical coordinate map is required.

Define

The label-space test $f$ is fixed, not pulled back. No $q$-volume density is inserted in this pairing. The off-shell expression depends on $g$ only through the declared gauge and evaluation contract.

Taylor expansion gives

For the dynamical evaluation, replace ${\varphi }_1,{\varphi }_2$ by ${\rho }_{1,g},{\rho }_{2,g}$. In particular, the forced scalar correction is not omitted.

For an orientation-preserving diffeomorphism, a contravariant tensor density of weight one transforms as

Consequently,

This follows from the chain rule and change of variables, or from integration against compact tests. Metric density and volume satisfy

For formal jets these identities follow coefficientwise by applying the genuine identities on each compact neighbourhood for sufficiently small numerical $\kappa$, then taking Taylor coefficients. No globally uniform numerical nondegeneracy is needed.

For $F\in {\mathcal{G}}_p$,

Proof. Since $F^{\ast }\sigma =\sigma$,

The right side vanishes on $U$ because $\sigma =1$, and outside $U$ because $F=\mathrm{id}$ there. Apply (eq:naturality).

For $F\in {\mathcal{G}}_p$,

Proof. The same calculation with $g$ gives

For the volume coefficient the corresponding difference is $(1-g)(F_{\mathrm{dens}}^{\ast }1-1)=0$. Naturality proves both identities.

The switch and ramp derivatives in (eq:coordinateoperators) and (eq:scalaroperators) have not been discarded. Their coefficients are unchanged where the transformation does not act.

For every smooth metric jet and every $F\in {\mathcal{G}}_p$,

In coefficients,

Proof. By Lemma lem:plateaucoord, each component of $F^{\ast }{\mathsf{Co}}_q$ solves the transformed coordinate problem. It has the same initial data because $F=\mathrm{id}$ near $\Sigma$. Proposition prop:coordeq gives uniqueness. Expansion of ${\mathsf{Co}}_q\circ F$ gives (eq:coordinatelaws).

This is the supported form of the X-A coordinate intertwiner XA ({:coordinate}).

For $F\in {\mathcal{G}}_p$,

Thus

Proof. Lemma lem:plateaueuler makes the pulled-back solution solve the transformed equation. Its Cauchy data are exactly those of the same ${\varphi }_0$, because the transformation is initial-trivial. Coefficientwise Cauchy uniqueness proves (eq:scalarintertwiner); expansion gives (eq:rholaws).

At fixed $\epsilon >0$, for the frozen data above:

1. the coordinate and switched scalar coefficient problems have their stated unique solutions;

2. preflight is exact on ${\mathfrak{g}}_p$;

3. for every off-shell scalar jet and $F\in {\mathcal{G}}_p$,

   $${\Phi }_{g,\epsilon }[F^{\ast }q,F^{\ast }\varphi ]={\Phi }_{g,\epsilon }[q,\varphi ](\mathrm{mod}{\kappa }^3),$$

   and hence

   $${\mathcal{O}}_{f,g,\epsilon }[F^{\ast }q,F^{\ast }\varphi ]={\mathcal{O}}_{f,g,\epsilon }[q,\varphi ](\mathrm{mod}{\kappa }^3);$$

4. with fixed free input, the dynamical evaluation

   $${\mathcal{O}}_{f,g,\epsilon }[q,{\rho }_g[q;{\varphi }_0]]$$

   is invariant under the same group.

The metric is off shell throughout. No quantum object is part of this theorem.

Proof. Existence and preflight have been proved. Formal inversion of (eq:intertwiner) yields

Since $F^{\ast }\varphi =\varphi \circ F$,

Pairing with the unchanged $f$ proves the off-shell statement. Substitute (eq:scalarintertwiner) to prove the dynamical one.

No larger gauge algebra is released here, even though some individual lemmas use fewer plateau conditions.

For either the coordinate or scalar operator write $P+\kappa V+{\kappa }^{2}W$, and let primes denote the transformed coefficients. Put

The proven covariance expands to

These are identities on arbitrary smooth test scalars, not only on solutions. For the coordinates, $P=\square$ and $P{x}^{\mu }=0$; inserting (eq:operatoraudit) verifies directly that

solves the transformed second-order equation with zero data. This checks the quadratic metric law without deleting the terms proportional to $P$.

Route II gives a variational derivation of the dynamics' covariance and, applied to an auxiliary coordinate action, of the coordinate operator covariance. It does not infer observable invariance from action invariance alone: the fixed Cauchy data, uniqueness, and relational composition remain necessary.

Temporarily let $\xi ,\zeta$ be arbitrary compact interior vector fields, not necessarily plateau-supported. For a scalar density $r$, define

Then

This is the Taylor expansion of density pullback. All currents have compact support in the gauge support.

The finite variational difference satisfies

Every non-divergence term contains $1-g$ or $\partial g$.

Proof. Naturality of the curved density gives

Apply (eq:Z) and $g{\partial }_{\mu }{Z}^{\mu }={\partial }_{\mu }(g{Z}^{\mu })-({\partial }_{\mu }g)Z^{\mu }$. The compact divergence integrates to zero. The difference is finite even when the individual unvaried actions have no finite numerical value.

For completeness, define the symmetric bilinear expression

The two terms in (eq:varied) are explicitly