Landmark X-A Classical covariance of a fixed-regulator switched relational scalar through order κ 2: an explicit regulator obstruction

We freeze a compactly switched Einstein–scalar density, a plateau-supported gauge algebra, a zero-Cauchy prescription, and one explicit fixed regulator. The regulator convolves the switched metric arguments before they enter an auxiliary harmonic-coordinate problem. The resulting relational scalar is a well-defined off-shell formal functional through order ${\kappa }^2$. Its coefficient kernels are smooth and compactly supported, and its metric dependence obeys an explicit regulator-thickened support bound.

The regulated coordinate problem does not intertwine the frozen gauge action. On a pure-gauge first coefficient, its coordinate response is $S_{\epsilon }\xi$, rather than $\xi$. We prove an exact first-order observable defect and exhibit one fixed observation test, one plateau-supported gauge vector, and one free scalar for which its smearing is strictly positive at every allowed fixed regulator. The same gap occurs when the observable is evaluated on the scalar solution selected by the switched action. The action itself satisfies the retained classical plateau Noether identity; consequently the failure is attributable to the coordinate regulator, not to omitted switch derivatives or changed Cauchy data.

We also derive the second-order coordinate and relational coefficients, the retained infinitesimal defects, and the complete second-order pure-gauge test. A separate nonzero forced scalar example checks the action-derived response. The release is a proved obstruction for this prescription on its originally declared gauge algebra. No quantum product, quantum Ward identity, sharp-regulator limit, or general no-go theorem for regulated relational constructions is claimed.

Figure fig:split-X-A-conv records one view of the same separation. Figure fig:chain-X-A-conv records one view of the same separation. Figure fig:pair-X-A-conv records one view of the same separation. Figure fig:gate-X-A-conv records one view of the same separation. The drawings add no hypothesis.

Figure fig:scope-X-A-conv records that separation. The drawing adds no hypothesis.

: proved obstructed{ The construction below is not advertised as a gauge-invariant regulated observable. Its defining problems and coefficients exist, but its first-order gauge identity fails on the gauge algebra frozen before the calculation.

The defect is retained. The gauge algebra is not reduced after the defect is found, and no second-order or quantum correction is used to cancel a nonzero first-order classical coefficient.}

The question is whether a specified regulated coordinate prescription transforms with the scalar under the same gauge action. A regulator must participate in this calculation. Classical covariance of an unregulated prescription does not establish covariance after convolution, and regularity of coefficient kernels does not establish gauge invariance.

This paper chooses the obstruction branch of that question. The regulator, switch, gauge algebra, and observation test are fixed in Section sec:sheet. The preflight calculation in Section sec:preflight already determines the release. Subsequent sections identify the coefficients and isolate the failed identity; they do not attempt a repair.

The following statements are comparison data, not lemmas about the regulated operator defined here.

-A. For its switched scalar action and its unswitched harmonic Cauchy problem, VII-A proves first- and second-order plateau intertwining. Its first-order cancellation uses

It also retains the larger-support defect. Its named metric equations are unswitched, and it does not construct a switched gravitational quantum theory VIIA.

. VI separates geometric solution-jet descent from a switched retarded assignment on unchanged free inputs. It proves a first-order compact-switch defect for that assignment and explains why a second-order term cannot change the defective first coefficient VI.

. III distinguishes an unchanged free-product quotient, free-shell tangency, and descent on a named coupled first-order jet. Those are different equation spaces. None is the off-shell domain used here III.

-B and IX-A. VII-B excludes its sharp zero-Cauchy insertion from its specified off-shell polynomial microcausal domain. IX-A places that insertion in a different classical carrier with a boundary-sensitive topology, without constructing a quantum product. Neither defines the regulated auxiliary metric used below VIIB,IXA.

tracks. No result from IX-E-2 or IX-E-3 is used. In particular, a boundary-trace certificate or a conditional support bridge is not a construction of a two-insertion product. No cone model, measurement token, finite contact matrix, or Hilbert-space model enters this paper.

Relational-coordinate constructions are comparison background Frob. They do not supply the covariance of the convolution prescription frozen here.

For a frozen prescription, the alternatives are:

1. equations, data, and covariance are constructed;

2. a nonzero defect is proved on the frozen gauge algebra;

3. a named construction or proof obligation remains unresolved.

The result below is the second alternative. It is not an unresolved choice of regulator and is not a theorem excluding every possible regulator.

Let

Coordinates are denoted $x=(t,\mathbf{x})$, and

All indices are raised and lowered with $\eta$ unless explicitly stated otherwise. Parentheses include the factor $1/2$.

The configuration space is

with its compact-open smooth Fréchet topology. Its elements are $(h,\varphi )$. Fields are defined on all of $M$, so convolution requires no extension choice at the slab boundary.

For coefficient-jet calculations we use

with elements $(h,k,\varphi )$. These are smooth off-shell configuration spaces. No equation-of-motion, gauge, or gauge-fixed quotient is taken.

The metric is formal:

or, when evaluating a coefficient jet,

There is no factor $1/2$ in $k$. No numerical smallness, convergence, or global invertibility is asserted.

We fix a massless minimally coupled scalar. The coupling $\kappa$ has the sign and normalization specified by the action in Section sec:action.

Let $x_{\ast }=(4,0,0,0)$, and use the Euclidean norm only to specify compact supports and the regulator. Put

Define

The denominator is everywhere positive. Then $c=0$ for $s\le 0$ and $c=1$ for $s\ge 1$.

Freeze

Thus $0\le g\le 1$, $g=1$ on $r\le 2$, and

In particular $g=0$ near $\Sigma$ and near $t=8$.

Freeze the open sets

The switch is background data and is not transformed.

For a vector field $\xi$, set

The frozen gauge algebra is

Each member vanishes on a neighborhood of the edge of $U$, hence vanishes there to every order. It also vanishes near both temporal boundaries.

The corresponding group is generated by flows of members of ${\mathfrak{g}}_U$. At second formal order we use

This parametrization is closed under formal composition because ${\mathfrak{g}}_U$ is closed under Lie brackets.

For the single-field metric (eq:baremetric), the infinitesimal action is

For (eq:jetmetric), finite coefficient extraction gives

The nonlinear term in (eq:infaction) and the second coefficient in (eq:kprime) are retained.

Define the real even compact mollifier

Freeze one number

All statements concern this fixed number. Define

componentwise in the displayed background coordinates.

This is a spacetime convolution, not a Lorentz-invariant or diffeomorphism-natural regulator. It regularizes the metric arguments of the coordinate functional. It does not regulate the action, the scalar field argument, or the Cauchy inverse itself.

For a metric jet define the auxiliary metric

The action uses $q$, not ${\hat{q}}_{g,\epsilon }$. This distinction is part of the prescription.

Define the nonzero real bump

Freeze

The same $f$, $g$, ${\xi }_{\ast }$, and regulator are used throughout the obstruction witness. In particular, the gauge algebra is not chosen from the kernel of the regulator defect.

contract{ Signature $+---$; mass $m=0$; one compact background switch (eq:switch); gauge algebra (eq:gaugealg); metric-argument regulator (eq:regulator); auxiliary metric (eq:auxmetric); zero coordinate coefficient data at $t=0$; one label-space test (eq:fandxi); formal coefficients through ${\kappa }^2$; no quantum order.}

Fix

Write

The unswitched geometric action is

With

variation of (eq:ESaction) gives

These equations fix the normalization; their coupled solutions are not the configuration domain of this paper.

Before switching, choose the first-derivative representative

and

Set

The density (eq:gravdensity), not an unspecified equivalence class modulo divergences, is used in this expansion.

For a symmetric tensor $a$, define

Thus

To specify the gravitational vertices without hidden integrations by parts, put

where $h^0=I$. For Christoffel-type arrays set

Then

The scalar coefficients are

Freeze

The free term is interpreted variationally with compact variations; its numerical integral need not exist on every smooth configuration. The interaction terms have compact coefficient support.

One switch multiplies each retained interaction density. In particular, the second scalar vertex carries $g$, not $g^2$. The regulator does not enter (eq:switchedaction).

If ${\tilde{\ell }}_j={\ell }_j+{\partial }_{\mu }{B}^{\mu }$, then

Consequently (eq:switchedaction) is a choice of model, not merely a notation for an unswitched equivalence class.

Let ${\mathcal{E}}_h$ be the ordinary first-derivative Euler operator in the tensor variable. Since every ${\ell }_j$ depends on $h$ and $\partial h$, but not on higher derivatives of $h$,

Equation (eq:metricEuler), together with (eq:gravzero)–(eq:gravtwo), is an explicit finite definition of the switched metric Euler equation. No coupled existence assertion is inferred from it.

The scalar Euler equation is $-D_g[q]\varphi =0$, where

Its output is density valued.

Define

These are all switch-derivative terms in the displayed scalar coefficient equations. There is no mass term because $m=0$.

For a metric jet, $V_{2,g}[h]$ is replaced by

For a smooth source $s$, define

The sine quotient has the continuous value $t-u$ at zero spatial frequency. Sources without decay are handled locally by finite propagation.

For every smooth source on a neighborhood of $M_T$, the zero-data wave problem has a unique smooth solution. It depends continuously on the source on each compact truncated causal region. In particular,

For smooth $a$,

where $H_0[a]$ is the homogeneous wave solution with the Cauchy data of $a$. Thus ${\mathsf{E}}_0{\square }_{\eta }a=a$ when those data vanish.

Proof. For spatially compact sources, differentiation of (eq:Ezero) gives the equation and the data. On a finite time interval, differentiated energy estimates give

The same estimates give uniqueness. Increasing $s$ and using Sobolev embedding give smooth dependence. Local energy estimates give finite propagation, allowing compatible compact cutoffs for general smooth sources. These arguments are the standard smooth wave Cauchy theory BGP.

Finally, $a-{\mathsf{E}}_0{\square }_{\eta }a$ solves the homogeneous equation with the data of $a$, proving the last assertion.

The coordinates are scalar functions with an external label $\mu$. Define them by

with

Equivalently, (eq:regcoord) is ${\square }_{{\hat{q}}_{g,\epsilon }}{X^{\mu }}_{g,\epsilon }=0$ formally. The auxiliary metric is exactly (eq:auxmetric).

Write

Define

The unique coefficients satisfying (eq:regcoord)–(eq:coorddata) are

Both coefficients and their first time derivatives vanish on $\Sigma$.

Proof. The auxiliary density expansion is

Substitution in (eq:regcoord) gives

The sources are smooth. Lemma lem:Cauchy gives the formulas and successive uniqueness.

Convolution commutes with constant-coefficient derivatives. Therefore

The same identities hold with $J,k$.

For clarity, the nonlinear divergence in (eq:v) is

and

Equations (eq:dH)–(eq:da1w) retain the actual switch-derivative terms. No ${\partial }^{2}g$ term occurs in these displayed coefficient sources. Such terms would occur in some further differentiated presentations, but no such presentation is used here.

In particular,

is not replaced by $g{a}_2(h)$, by $S_{\epsilon }(g{a}_2(h))$, or by $a_2(gh)$. Those are different prescriptions.

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

The right-hand side has zero Cauchy data near $\Sigma$.

Proof. Since $g=1$ on a neighborhood of $\mathrm{supp}\xi$,

Constant-coefficient differentiation commutes with convolution, so

The identity $CK={\square }_{\eta }$ follows directly from (eq:C): the derivative of the divergence of $\xi$ cancels the derivative of half the trace of $K\xi$.

The support of $S_{\epsilon }\xi$ lies in the Euclidean $\epsilon$-neighborhood of $\mathrm{supp}\xi$, which is separated from $\Sigma$. Lemma lem:Cauchy therefore gives

The first-order identity required for coordinate covariance is

Its actual value is (eq:preflight). There is no claim that $L_{g,\epsilon }$ globally factors as an operator on $Ch$: the switch-derivative term in (eq:CHswitch) prevents that simplification. On the pure-gauge test, however, the failure is exactly the failure of the regulated inverse to return $\xi$.

For the fixed $f,{\xi }_{\ast }$ of (eq:fandxi), define

Then

Proof. Translation preserves $\int {\psi }^2$. Expanding the square in (eq:positiveenergy) and using $\int {\varrho }_{\epsilon }=1$ gives

which is (eq:ceps).

For every $a\ne 0$, the inner integral is strictly positive. Otherwise continuity would give $\psi (x)=\psi (x-a)$ everywhere. Iteration would make a nonzero compactly supported function invariant under arbitrarily large translations in the direction $a$, which is impossible. The mollifier is positive on the open ball $0<|a{|}_{\mathrm{E}}<\epsilon$, a set of positive measure. Thus the outer integral is strictly positive.

The coordinate prescription (eq:regcoord)–(eq:coorddata) fails first-order gauge covariance on ${\mathfrak{g}}_U$. For the free scalar

the first-order relational variation smeared with the frozen $f$ is $c_{\epsilon }>0$.

Proof. The scalar variation is $X_{{\xi }_{\ast }}{\varphi }_0=\psi$. The coordinate variation is $S_{\epsilon }{\xi }_{\ast }$, whose action on ${\varphi }_0$ is $S_{\epsilon }\psi$. Their difference has smearing (eq:ceps), which is positive by Lemma lem:positivegap. The complete relational expansion proving the insertion sign appears in Section sec:observable.

preflight decision{ The gauge algebra ${\mathfrak{g}}_U$, the switch, the observation test, and the regulator remain unchanged. The first-order covariance gate is proved obstructed. The remaining calculations document this prescription and its defects; they do not reopen a successful covariance claim.}

The formal inverse of (eq:Xexp) is

It is obtained by substitution in $X(Y(z))=z$; no numerical global inverse is required.

Define

The test $f$ is fixed in relational-label space. It is not transformed as a spacetime-coordinate test.

For a vector $w$, write

The coefficients of (eq:observable) are

For the bare single-field functional, set $k=0$ in these formulas.

Proof. Taylor-expand $\varphi (Y(z))$ using (eq:inverse). The linear displacement gives $-X_w\varphi$. The second displacement gives $(w^{\nu }{\partial }_{\nu }{w}^{\mu }-v^{\mu }){\partial }_{\mu }\varphi$, and the quadratic Taylor term gives $\frac{1}{2}{w}^{\mu }{w}^{\nu }{\partial }_{\mu }{\partial }_{\nu }\varphi$.

Let $a_2(H_1,H_2)$ denote the symmetric polarization of $a_2(H)$. Explicitly,

Set $W(H)={\mathsf{E}}_{0}CH$, and define

Then

For the bare functional $k=0$, split

where

Put

and

where $X_{\xi }w$ means the vector with components ${\xi }^{\nu }{\partial }_{\nu }{w}^{\mu }$.

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

In particular, first-order covariance would require $a=\xi$ in the coordinate identity. That requirement is not satisfied.

Proof. Lemma lem:preflight gives

Linearity of $W{T}_{g,\epsilon }$ gives

and polarization gives

Hence

At first order,

At second order,

Direct differentiation of (eq:Jw) gives

Therefore the second-order integrand is

Using (eq:Rtwo) yields (eq:infdefect).

No modification beginning at order ${\kappa }^2$ makes the retained functional gauge invariant on ${\mathfrak{g}}_U$ while preserving its first-order coefficient.

Proof. Formal invariance requires equality of each coefficient separately. Theorem thm:preflightstop gives a nonzero coefficient of $\kappa$. A coefficient beginning at ${\kappa }^2$ cannot change it.

These are classical kernel statements. They are not a quantum-domain or product theorem.

For a tensor component label $A=(\alpha \beta )$, let $e_A$ denote the constant symmetric component basis. Define

Then

is the first coordinate kernel:

For fixed $\epsilon$, ${{\Lambda }_A}^{\mu }(z,y)$ is jointly smooth on a neighborhood of $M_T\times M$. The polarized second kernel