X-E-ST: Einstein–Hilbert normalization and one matter-induced tree A supported fixed-width transfer on the inherited I-B host

We give a supported fixed-width realization of Cheung's Landmark X-E on the conical host and Minkowski completion already retained by X-A-ST, X-C-ST, and X-D-ST. The host is not reconstructed or evolved.

Starting from the Palatini identity, we derive the Einstein–Hilbert quadratic form, its positive transverse-traceless kinetic sign, the Feynman-weight de Donder operator ${\mathsf{P}}_{\mathrm{gr}}=-\frac{1}{4}{\mathsf{P}}_{\mathrm{tr}}\square$, and its retarded inverse. The switched total action has an explicit Noether defect. Its invariant two-jet equations require $({\partial }_{\mu }g)T^{\mu }{}_{\rho }[{\varphi }_0]=0$. For every nonconstant fixed switch, an open dense set of smooth free-scalar inputs violates a pointwise necessary condition; their coupled invariant-jet fibres are empty, although the gauge-fixed retarded response exists.

We construct the isolated scalar-sourced tree by inserting $-2{\mathsf{P}}_{\mathrm{tr}}{{\mathsf{E}}_0}^{\mathrm{R}}(g:T[\mathbf{\varphi }]:)$ into the affine second-metric slot of the completed scalar observable. An explicit three-slot distribution proves microcausal admission of the resulting Wick cubic. We distinguish its distributional kernel from its smooth contracted tests, correct the support bound to $C^3$, and prove fixed-support continuity. Its coordinate response is

With the free graviton held fixed, the complete one-tree expression is invariant through order $\kappa$, but has the order-${\kappa }^2$ defect $-{\hat{A}}_1[{\mathcal{L}}_{\xi }\mathbf{r}]$. A supported free-solution witness proves that this defect is not an equation-zero representative.

The principal action and tree constructions are credited to X-E. The contribution of this transfer is an explicit host-compatible proof and a corrected support, topology, jet-order, and truncation audit. These distinctions provide a usable baseline for subsequent tree completions without mistaking one admitted scalar cubic for an interacting quantum metric or a loop construction. A finite supplied evaluation dictionary has a complete destructive-clock instrument and reference-preserving scheduled transport. The release is local Grade S only.

corrections made before the proof{ The full tree kernel is a compact distribution, not a smooth test in ${\mathcal{D}}_C$. Its smooth partial contractions do belong to ${\mathcal{D}}_C$.

The scalar-response part can have its first argument in $C\setminus K$, so the general two-point support is $C\times S$, not $K\times S$.

The coefficient of the tree is denoted $\tau$; its contribution is ${\kappa }^2\tau$, counted once.

The nonzero background defect is a defect of the specified one-tree truncation. It is not a no-go theorem for additional trees or a different gauge completion.}

Landmark X-E fixes an Einstein–Hilbert action, a flat gauge-fixed graviton operator, and one scalar-induced retarded tree XE. This manuscript transfers those constructions to the supported presentation retained by X-A-ST, X-C-ST, and the supplied X-D-ST XAST,XCST,XDST.

The action, its normalization, the scalar-sourced tree, and the one-tree defect are attributed to X-E. The proofs below are written on the completion rather than asserted by analogy with the host. In particular, the principal variation, free operator, cubic admission, and nonzero witness are supplied explicitly.

The identifier X-E-ST denotes this proposed source. It is not evidence of repository installation, external registration, submission, acceptance, or independent verification.

Enter's relevant distinction is between destructive classical readout and the return of a quantum system. The measured clock names are consumed from the returned quantum interface; the complete output retains the classical code and the unmeasured residual EnterI,EnterPhilosophy,En09.

For this paper the operative separations are:

1. a supplied mathematical evaluation versus acquiring an unknown continuum field;

2. a complete instrument versus its classical outcome law;

3. preservation of a reference versus generation of entanglement by gravity;

4. a finite protocol on a prescribed spacetime versus derivation of that spacetime or its equations.

I-B supplies the prescribed-host interpretation, calibration distinctions, and target-adapted transport pattern IB. None of these operational statements derives the action used below.

Retain the following X-D-ST/X-A-ST instance without modification:

1. the regular time-oriented conical product $({\mathcal{M}}^{\circ },G)$, its stipulated worldsheet $\mathcal{W}$, and its supplied future relation;

2. the supported injective developing chart

   $$\iota :\mathcal{L}⟶{\mathcal{M}}^{\circ }$$

   and its retained compact reserve, disjoint from $\mathcal{W}$;

3. the independently declared analytical completion

   $$\hat{M}={\mathbb{R}}^{1+3},\eta =\mathrm{diag}(1,-1,-1,-1),M_{T_{\ast }}=[0,T_{\ast }]\times {\mathbb{R}}^3,\Sigma =\{t=0\};$$

4. the signature and time ledger

   $${\iota }^{\ast }G=-\eta ,t=c_{\mathrm{host}}(T-T_{\Sigma });$$

   the host retains signature $(-,+,+,+)$, and the analysis retains $(+,-,-,-)$;

5. the observation region

   $$\mathcal{O}⋐(0,T_{\ast })\times {\mathbb{R}}^3,D_{\ast }:=J_{\eta }^{-}(\overline{\mathcal{O}})\cap M_{T_{\ast }}⋐\mathcal{L}.$$

   Compact laboratory tests are transported and extended by zero on $\hat{M}$.

The regular conical product is not treated as a globally hyperbolic Cauchy spacetime. All global free fields and Green operators in this paper are on $\hat{M}$.

The conical singular set remains a worldsheet. There is no substitution of a worldline, no global conical inverse, and no host-metric update. The inherited non-global-hyperbolicity diagnosis is not used as a failure of the well-posed flat problems on the completion.

The release consists of:

1. the inherited supported host;

2. named action identities and response kernels with proved normalization, support, and transformation laws;

3. a finite installed dictionary of supplied evaluations;

4. a complete destructive-clock instrument, including untouched references;

5. scheduled locality and a nonempty regular reference-preserving route under declared finite operational permissions.

It does not include Grade E, metric emergence, or Einstein dynamics derived from records.

Fix

Let $U$ be a nonempty relatively compact open set with

The formal group ${\mathcal{G}}_p$ is generated through second order by $\kappa \xi +{\kappa }^2\zeta$, with $\xi ,\zeta \in {\mathfrak{g}}_p$. Set

Choose $\delta >0$ so that

Fix $0<\epsilon <\delta$, once and for all, and use

where $\chi$ is the smooth predecessor profile, zero on $(-\infty ,0]$, one on $[1,\infty )$, and flat at both endpoints. No width-removal theorem is sought.

The backgrounds $h,k$ are fixed smooth real symmetric tensors on the completion. No equations or spatial decay are imposed on them. Write

Formal identities for observables are understood modulo ${\kappa }^3$. Variations of the exact action are derived before truncation; negative powers of $\kappa$ in its normalization are not manipulated in a nilpotent ring.

All unspecified raising and lowering uses $\eta$. Let $H={\eta }^{\mu \nu }{h}_{\mu \nu }$, and define

Thus, with $J_q=\sqrt{-\det q}$,

Let

Use the positive retarded and advanced inverses ${{\mathsf{E}}_{\mathsf{m}}}^{\mathrm{R}},{{\mathsf{E}}_{\mathsf{m}}}^{\mathrm{A}}$ of $P_m$, and ${{\mathsf{E}}_0}^{\mathrm{R}},{{\mathsf{E}}_0}^{\mathrm{A}}$ of $\square$. In particular,

Their Cauchy, support, and continuity properties are used only on the completion BGP.

For a smooth symmetric tensor $l$, write

The second coordinate coefficient solves

with zero position and velocity at $\Sigma$. Retarded expressions for nondecaying backgrounds are defined locally by the finite-time Cauchy problem. Every retained dependence lies in $D_f$.

The compactly switched Euler coefficients are

Each divergence retains its $\partial g$ terms.

Define

These are the completed tests of X-C-ST XCST,XC. They are restated to fix the insertion target, not reconstructed as additional response maps.

There is a symmetric bilinear test map $\mathcal{Q}[h_1,h_2]$ such that

For fixed smooth backgrounds,

Their dependence on the backgrounds near the retained causal reserve and on fixed-support observation tests is continuous in the smooth topologies.

Proof. The $k$-linear parts of $A,B,L_2$ are respectively $a_1[k],b_1[k],L_1[k]$. Equation (eq:coordinates) and uniqueness give

The $k$-part of $c_2$ is therefore $c_1[k]$. Subtracting $L_1[k]{{\mathsf{E}}_{\mathsf{m}}}^{\mathrm{A}}f$ proves the affine assertion; polarize the remaining quadratic expression.

The bare tests are supported in $K$. Each term with an $L_j$ is supported in $\mathrm{supp}g$, and advanced support restricts it to $J_{\eta }^{-}(K)$. Finite-time smooth Cauchy estimates, Green continuity, and finite differentiation prove continuity.

scalar response is not counted again{ The term $-L_2{{\mathsf{E}}_{\mathsf{m}}}^{\mathrm{A}}f+L_1{{\mathsf{E}}_{\mathsf{m}}}^{\mathrm{A}}{L}_1{{\mathsf{E}}_{\mathsf{m}}}^{\mathrm{A}}f$ already represents the second scalar response. The new operation is the affine substitution $k\mapsto k+k_1[\varphi ]$. It adds $d_1[k_1[\varphi ]]$ once, not another copy of the scalar second-order pair.}

For a genuine Lorentzian metric $q$, use

The frozen Lagrangian is

The notation denotes local Lagrangian densities: Euler derivatives are defined using compact variations. No globally finite action integral or background decay is assumed. This is the X-E action convention XE; its sign is checked in Section sec:free.

For a compactly supported variation $\delta q_{\mu \nu }$,

Consequently

Proof. The difference of two connections is a tensor, so $\delta \Gamma$ is tensorial. In normal coordinates, and hence tensorially,

Contraction and $\nabla q=0$ give $q^{\mu \nu }\delta R_{\mu \nu }={\nabla }_{\rho }{W}^{\rho }$. Furthermore,

Substitution in $\delta (J_q{q}^{\mu \nu }{R}_{\mu \nu })$ gives (eq:palatini); compact variation removes the divergence from the Euler derivative.

Set

and write $T_0=T[\eta ,\varphi ]$.

The tensor-density and scalar-density Euler derivatives are

There are no derivatives of $g$ in $\mathcal{E}$; all of them remain in $E_{\varphi }$ and in the divergence identity below. In particular,

Proof. Varying the displayed matter density algebraically gives

There are no metric derivatives in ${\mathcal{L}}_q$. Combine this with Lemma lem:palatini. One scalar integration by parts gives (eq:E-phi), and (eq:densities) gives its expansion.

The first matter vertex is therefore

For later use, direct differentiation gives

The linearized Einstein tensor is

It satisfies ${\partial }_{\mu }{G_{\mathrm{lin}}}^{\mu \nu }[l]=0$.

To specify the nonlinear coefficient without an unspecified vertex, define

This is an explicit differential polynomial. It can equivalently be calculated from

and

followed by the trace, inverse-metric, and density expansions (eq:densities).

The invariant metric equation through its second geometric order is

Equivalently,

The coefficient of ${\kappa }^2$ in $E_h$ involves the coefficient of ${\kappa }^3$ in $J_{q}G[q]$. A covariant demand at that order requires a third classical metric jet and its transformation law. It is not a condition on an unchanged two-jet alone. We use (eq:metric-expansion) for the invariant two-jet surface, and keep the exact equation (eq:E-q) when no truncation is intended.

For a contravariant tensor density of weight one,

For every smooth pair $(q,\varphi )$,

The gravitational contribution has no switch defect. On $\mathrm{int}\{g=1\}$, the right-hand side is $E_{\varphi }{\partial }_{\rho }\varphi$.

Proof. Let $\xi$ be an arbitrary compact vector field and vary $q,\varphi$ by ${\mathcal{L}}_{\xi }q,{\xi }^{\rho }{\partial }_{\rho }\varphi$, holding $g$ and $\eta$ fixed. Naturality of $J_{q}R$ makes its variation a divergence. Naturality of the curved scalar density gives ${\delta }_{\xi }{\mathcal{L}}_q={\partial }_{\mu }({\xi }^{\mu }{\mathcal{L}}_q)$.

At $q=\eta$, algebraic metric variation gives

Therefore

Integrate the last term by parts and use (eq:stress-div).

The Euler form of the same variation is

Here the metric term follows by substituting

and integrating once. Equality for arbitrary $\xi$ proves (eq:noether).

This is the switch-defect mechanism of X-E XE (Noether identity), now explicitly retained on the completion. The generally noncompact set $\mathrm{supp}(1-g)$ is never replaced by a compact source.

For a diffeomorphism $F$ supported in a compact subset of $U$, the density difference

has compact support and integrates to zero. The exact Euler densities obey

The first pullback is that of a contravariant symmetric tensor density, and the second that of a scalar density.

Proof. The curved scalar and Einstein–Hilbert densities are natural. Their pullbacks have unchanged integrals under a compactly supported change of variables. Moreover, $g\circ F^{-1}=g$, since $g=1$ near the support of $F-\mathrm{id}$. The remaining flat scalar term is unchanged pointwise where $1-g\ne 0$.

Differentiate this action equality after arbitrary compact variations of $q,\varphi$. Change variables in the resulting invariant tensor-density pairings. Arbitrariness of the variations proves Euler covariance.

The density itself need not be pointwise unchanged. “Exact plateau invariance” means invariance of the local action modulo a compactly supported divergence, not equality of its coordinate density values.

For a specified free scalar input ${\varphi }_0$, let ${{\mathfrak{C}}^{[2]}}_{{\varphi }_0}$ consist of

satisfying

Coordinate data remain (eq:coordinates). No gauge-fixed equation is substituted for the first condition.

The complete two-jet transformation is

Proposition prop:covariance preserves (eq:coupled); truncating a natural equation at the geometric order used there is legitimate.

Every member of ${{\mathfrak{C}}^{[2]}}_{{\varphi }_0}$ satisfies

If $g$ is nonconstant, choose $x_0$ with $\mathrm{d}g(x_0)\ne 0$. In the real smooth free-solution space

the set

is open dense. Every corresponding coupled-jet fibre is empty.

For $m=0$, the particular input ${\varphi }_0=t$ violates (eq:bianchi) wherever $\mathrm{d}g\ne 0$.

Proof. The exact Bianchi identity gives ${\nabla }_{\mu }{G}^{\mu \nu }=0$. Expanding its density version to second order on $G_{\mathrm{lin}}[h]=0$ yields

Taking the divergence of (eq:metric-expansion) and using $P_m{\varphi }_0=0$ proves (eq:bianchi). Equivalently, it is the order-zero switch condition in (eq:noether). Thus this condition follows from the exact Noether identity, not from gauge fixing.

The pointwise expression is a continuous quadratic polynomial in the first scalar jet. It is not identically zero on ${\mathrm{Sol}}_m$: choose a free solution $v$ with $v(x_0)=0$ and ${\partial }_{\mu }v(x_0)={{\delta }_{\mu }}^0$. Such a solution follows from smooth Cauchy data, or from $\sin (m(t-t_0))/m$ when $m>0$, with the massless limit $t-t_0$. Its mixed stress has diagonal entries $1/2,-1/2,-1/2,-1/2$, so contraction with the nonzero covector $\mathrm{d}g(x_0)$ is nonzero. For any $u$, at least one component of the expression at $u+\lambda v$ is a polynomial in $\lambda$ with nonzero quadratic coefficient. Its zeros have empty interior. Arbitrarily small $\lambda$ give a violation. Continuity gives openness.

For $u=t$,

and the same diagonal calculation proves the final claim.

This is an emptiness result for specified-input fibres. The union of all fibres is not declared empty: $h=k={\varphi }_0={\varphi }_1={\varphi }_2=0$ is an immediate compatible example. No impossibility theorem for an unswitched Einstein–scalar theory follows.

Define

Then

For $\xi ,\zeta \in {\mathfrak{g}}_p$, let $Z_{\xi }=X_{\zeta }+\frac{1}{2}{X_{\xi }}^2$. The completed tests satisfy

All preimages displayed inside $P_m$ are compactly supported. The associated classical relational scalar is invariant modulo ${\kappa }^3$, including on ${{\mathfrak{C}}^{[2]}}_{{\varphi }_0}$.

Proof. The ramp coordinate density transforms naturally on the support of $F-\mathrm{id}$, since $\sigma =g=1$ there, and is unchanged elsewhere. The transformation is the identity near $\Sigma$. Cauchy uniqueness gives $X[q^{\prime }]=X[q]\circ F$ coefficientwise. The inverse-coordinate smearing density therefore obeys $c[q^{\prime }]=F^{♯}c[q]$.

Let $E[q]=P_m+L[q]$ be the positive scalar Euler operator, and let $s={{\mathsf{E}}_{\mathsf{m}}}^{\mathrm{A}}e[q]$. Then $E[q]s=c[q]$. Plateau covariance gives

The advanced solution for the transformed source is therefore $s^{\prime }=F^{\ast }s$, by coefficientwise advanced uniqueness. Applying $P_m$ gives

Expand through second order. The generators have compact support, so the indicated preimages are compact smooth functions.

For a smooth free input $u$, pairing with $u$ annihilates these equation tests. The formal inverse coordinates give the same completed scalar functional; alternatively this is the retarded scalar recursion encoded in (eq:completed).

This reproduces the fixed-algebra representative mechanism retained in X-C-ST and X-D-ST XCST,XDST. It does not use an Einstein solution to define the scalar observable.

Up to a total derivative,

This is the quadratic coefficient of (eq:action), not a separately imported free action. For transverse-traceless spatial configurations,

Thus the physical transverse-traceless kinetic term has positive sign.

Proof. For compact $l$, set $I(s)=\int J_{\eta +sl}R[\eta +sl]{\mathrm{d}}^{4}x$. Lemma lem:palatini gives

Multiplication by $-1/(2{\kappa }^2)$, with $s=\kappa$, gives (eq:FP). Symmetry of the second variation gives its Euler derivative. For trace-free divergence-free $l$, $G_{\mathrm{lin}}[l]=-\frac{1}{2}\square l$. One integration by parts gives the displayed kinetic form. Compact variation gives the same local density result without a decay assumption.

A separate stationary normalization check agrees with attraction. With ${\kappa }^2=8\pi G_N$, a positive static density $\rho$ in the unswitched weak-field diagnostic gives

where ${\Phi }_N=-G_N\int \rho (y)/|\mathbf{x}-\mathbf{y}|{\mathrm{d}}^{3}y$. For the usual stationary nonrelativistic reduction, ${\gamma }_{00}=2{\Phi }_N$ and acceleration is $-\nabla {\Phi }_N$. This is a sign check on the frozen completion action, not a static solution produced by the compact-time experiment or a host update.

Define

One has ${{\mathsf{P}}_{\mathrm{tr}}}^2=1$, and

Consequently

For a compact source $A$,

Proof. In four dimensions trace reversal reverses the trace; applying it twice gives the identity. Substitution in (eq:Glin) gives the first line.

The gauge term varies as

It cancels precisely the $C$-terms in $\frac{1}{2}{G}_{\mathrm{lin}}$. This proves (eq:Pgr). The constant invertible trace-reversal factor gives the inverse formulas and support. Finally ${\mathsf{P}}_{\mathrm{tr}}{{\Delta }_{\mathrm{gr}}}^{\mathrm{R}}=-4{{\mathsf{E}}_0}^{\mathrm{R}}$, so differentiating proves the last identity.

Strictly, ${\mathsf{P}}_{\mathrm{gr}}$ is a constant invertible bundle factor times a normally hyperbolic operator. Its Green theory follows from that of $\square$; its principal symbol need not be called the scalar identity symbol.

Gauge fixing specifies the free operator. It does not impose $C[l]=0$ on every configuration in its unrestricted covariant free-field carrier. Indeed a retarded source with nonzero divergence generally gives $C[{{\Delta }_{\mathrm{gr}}}^{\mathrm{R}}A]\ne 0$.

If $\xi$ is a compactly supported smooth vector field and ${\mathsf{P}}_{\mathrm{gr}}K\xi =0$, then $\xi =0$. In particular,