X-F-ST: Three fixed-width retarded trees and the background identity A supported transfer with a variational, support, and equation-ideal audit

We give a supported realization of Cheung's Landmark X-F on the conical host and independently declared Minkowski completion retained by X-A-ST, X-D-ST, and X-E-ST. At fixed positive ramp width, background de Donder gauge is applied to the quantum fluctuation, while the classical metric coefficients remain unconstrained.

The mechanism is a typed commutator identity:

It yields a retarded second-order response whose background variation is ${\mathcal{L}}_{\xi }\mathbf{r}$ modulo the free graviton equation ideal. The mixed tree consequently cancels the one-tree defect of X-E-ST. Together with the matter and self-interaction trees, the completed observable is background invariant through order ${\kappa }^2{\hslash }^0$, with both free fields held fixed. The exact Einstein-density covariance identity retains its off-shell Lie term; no linearized Einstein constraint is imposed on the gauge-fixed free graviton.

We construct all three kernels, prove compact support and microcausal admission by a three-slot sum-zero argument, and exhibit the graviton-slot equation representative needed for the cancellation. The matter kernel retains support $C^3$; the mixed and self kernels can reach the initial-layer causal reserve. Smooth scalar-slot contractions lie in ${\mathcal{D}}_C$, whereas the complete kernels are distributions in a polynomial carrier.

A necessary correction concerns interpretation: source switching defines an equation-level retarded response, not automatically a variational Møller map. A local formal-adjoint test shows why that stronger identification cannot be asserted for the unchanged off-shell source law. Thus the direct tree identity survives without claiming a switched-action or Bogoliubov construction.

The principal tree mechanism is credited to X-F. The contribution here is its supported proof and explicit type, support, topology, and variational audit. This provides a reproducible tree-level baseline for successors without identifying it with loop admission or spacetime emergence. A finite supplied-evaluation dictionary has a complete destructive-clock instrument and regular reference-preserving transport. The release is local Grade S only.

fixed before the calculation{ The response constructed here is the unique retarded solution of a prescribed source equation. “Variational Møller map” is not used as a synonym.

The two Lie derivatives in the commutator have different tensor types.

The kernel before onset multiplication need not have compact support in its source argument. Compactness is proved after multiplication by $g$, $\sigma$, or compact generator coefficients.

Only contractions leaving the scalar slot necessarily lie in ${\mathcal{D}}_C$. A contraction leaving a graviton slot can have support in the larger causal reserve.

Each tree is a coefficient. The observable multiplies their sum by ${\kappa }^2$ exactly once.}

The analytical source is N. Cheung's X-F, version 1, which replaces the flat interaction gauge of X-E by a background de Donder gauge and supplies the mixed and self trees in addition to the matter tree XF,XE. The supplied X-E-ST and X-D-ST are predecessor source records XEST,XDST. Their labels do not by themselves assert registration or independent verification.

This paper states the inherited definitions needed by the proof. It does not assume the parent all-tree theorem as a premise. The direct commutator, admission, and equation-ideal arguments are supplied below.

The supplied X-G correction distinguishes retarded solution maps from variational Møller maps XG. That distinction is incorporated explicitly. It does not change the three source expressions or the direct cancellation proved here.

The identifier X-F-ST denotes this proposed manuscript. No repository installation, external registration, submission, acceptance, or successful compilation is asserted.

The relevant Enter distinction is destructive classical readout versus quantum return. Measured quantum clock names are consumed; the complete returned interface contains a classical record and the unmeasured residual EnterI,EnterPhilosophy,En09.

We retain four separations:

1. a supplied mathematical evaluation is not acquisition of unknown continuum fields;

2. agreement of record probabilities is weaker than agreement of complete instruments;

3. preservation of an untouched reference is not generation of entanglement by gravity;

4. a protocol on a prescribed spacetime is not a derivation of that spacetime or its dynamics.

I-B supplies the prescribed-host clock and transport interpretation IB. It supplies neither the Einstein–Hilbert action nor the three trees.

The finite operational results below use explicitly specified Born instruments and prove their channel identities directly. No abbreviated reconstruction of Enter's characterization theorem is attempted.

Retain the X-A-ST instance already used by X-D-ST and the supplied X-E-ST:

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 analytical equations retain $(+,-,-,-)$;

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 used as a globally hyperbolic Cauchy spacetime. Every global Green operator and every free field in this paper belongs to $\hat{M}$.

The singular set remains a worldsheet, not a worldline. No second cone, global conical inverse, worldsheet gluing, or host-metric evolution is introduced IB,XAST.

The package consists of the inherited supported host; named identities and response kernels with proved normalization, support, and transformation laws; a finite installed dictionary of supplied evaluations; a complete destructive-clock instrument including untouched references; and scheduled locality with a nonempty regular reference-preserving route under declared finite operational permissions.

Grade E, Einstein dynamics from records, and spacetime emergence are not included.

Fix

Choose a nonempty relatively compact open plateau $U$ with

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

Reserve $K$ for the Killing operator

For observation support write

These sets are compact.

Fix $\delta >0$ so that

Fix $0<\epsilon <\delta$, and set

where $\chi$ is smooth, zero for $t\le 0$, one for $t\ge 1$, and flat at both endpoints. The width remains positive throughout.

The classical coefficients $h,k$ are arbitrary smooth real symmetric tensors near the closed slab. No equations or spatial decay are imposed. They are not the host metric $G$.

Let

Use 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 $\square$. Thus $P_m{{\mathsf{E}}_{\mathsf{m}}}^{\mathrm{R}}=P_m{{\mathsf{E}}_{\mathsf{m}}}^{\mathrm{A}}=1$ and ${{{\mathsf{E}}_{\mathsf{m}}}^{\mathrm{R}}}^{\mathrm{T}}={{\mathsf{E}}_{\mathsf{m}}}^{\mathrm{A}}$. All existence, finite-propagation, and continuity statements are applied on the completion BGP.

The fluctuation bundle is $E={\mathrm{Sym}}^2{T}^{\ast }\hat{M}$. Its Euler derivatives are sections of $E^{\vee }={\mathrm{Sym}}^{2}T\hat{M}\otimes |{\Lambda }^4{T}^{\ast }\hat{M}|$. The invariant pairing is integration of their contraction.

Write

They act respectively on $E$ and $E^{\vee }$, and

whenever the integrand has compact support. An expression $[{\mathcal{L}}_{\xi },Q]$, for $Q:E\to E^{\vee }$, always means ${{\mathcal{L}}_{\xi }}^{d}Q-Q{{\mathcal{L}}_{\xi }}^c$.

Indices without an indicated metric are moved by $\eta$. For $H={\mathrm{tr}}_{\eta }l$, define

These are the coefficients of $J_q{q}^{\mu \nu }$ and $J_q=\sqrt{-\det q}$ for $q=\eta +\kappa h+{\kappa }^{2}k$.

The zero-data coordinate coefficients are

Their positions and velocities vanish at $\Sigma$. For nondecaying data these equations are defined locally by finite-time Cauchy theory and finite propagation, not by an unjustified integral over an infinite past.

The scalar Euler coefficients are

Every switch derivative remains inside these divergences.

Set

These are the X-C-ST insertion representatives XCST,XC. They are actual off-shell tests before any quotient is taken.

There is a symmetric bilinear test map $\mathcal{Q}$ such that

For fixed smooth data, $d_1[l],d_2[h,k]\in {\mathcal{D}}_C$, continuously in the smooth data near the retained causal reserve.

Proof. The $k$-linear part of the second coordinate equation is the first coordinate equation with input $k$. Uniqueness gives $v[h,k]=v[h,0]+w[k]$. The $k$-part of $c_2$ is therefore $c_1[k]$; the $k$-part of $L_2$ is $L_1[k]$. This proves the affine identity, with polarization of the remaining quadratic expression.

The bare representatives are supported in $K_f$. Every $L_j$-term is supported in $\mathrm{supp}g$, and advanced propagation restricts it to $J_{\eta }^{-}(K_f)$. Smooth Cauchy estimates and Green continuity prove the stated continuity.

The existing scalar pair $-L_2{{\mathsf{E}}_{\mathsf{m}}}^{\mathrm{A}}f+L_1{{\mathsf{E}}_{\mathsf{m}}}^{\mathrm{A}}{L}_1{{\mathsf{E}}_{\mathsf{m}}}^{\mathrm{A}}f$ is not regenerated. The new operation is only the affine insertion of a derived second-order response.

This section fixes the inherited action and records the identities needed to prevent an equation-space misidentification. It does not reopen X-E-ST's action choice XEST,XE.

Use the curvature convention

The local action is

These denote local Lagrangians with compact variations; global finiteness of the action integral is not assumed.

Let $\mathcal{E}[q]=J_q{G}^{\mu \nu }[q]$, regarded as an $E^{\vee }$-valued density. Then

The quadratic transverse-traceless spatial action is

In particular its kinetic sign is positive.

Proof. The Palatini identity $\delta R_{\mu \nu }={\nabla }_{\rho }\delta {{\Gamma }^{\rho }}_{\nu \mu }-{\nabla }_{\nu }\delta {{\Gamma }^{\rho }}_{\rho \mu }$, together with $\delta J_q=\frac{1}{2}{J}_q{q}^{\mu \nu }\delta q_{\mu \nu }$ and $\delta q^{\mu \nu }=-q^{\mu \alpha }{q}^{\nu \beta }\delta q_{\alpha \beta }$, gives the first formula. Differentiating it once at $\eta$ gives the quadratic action and its symmetric Hessian. On trace-free, divergence-free spatial fields $G_{\mathrm{lin}}[l]=-\frac{1}{2}\square l$; integration by parts gives the last display.

Here $G_2$ is the symmetric polarization of the coefficient in (eq:Eexp), not the second coefficient of a differently indexed Einstein tensor. Explicitly,

It is an unambiguous constant-coefficient quadratic differential polynomial of total derivative order two. For a finite component basis it can be written

The constants are fixed by the displayed derivative definition; no independent cubic vertex is selected.

For completeness,

Define

Then

With $g,\eta$ fixed,

A coupled invariant two-jet with free input $P_m{\varphi }_0=0$ must satisfy

For nonconstant $g$, an open dense set of smooth free inputs violates a pointwise necessary condition, so its specified-input coupled-jet fibres are empty.

Proof. Under $\delta q={\mathcal{L}}_{\xi }q$, $\delta \varphi ={\xi }^{\rho }{\partial }_{\rho }\varphi$, the curved densities vary by divergences. At fixed $\eta$,

Integrate against the switch factors and compare with the Euler variation. Using ${\partial }_{\mu }{T_0}^{\mu \nu }=(P_m\varphi ){\partial }^{\nu }\varphi$ gives (eq:Noether).

Alternatively expanding the exact Bianchi identity on

immediately gives (eq:Bianchi).

Choose $x_0$ with $\mathrm{d}g(x_0)\ne 0$. The violation at $x_0$ is a continuous quadratic polynomial in the free scalar's first jet. A free solution $v$ with $v(x_0)=0$ and $\partial v(x_0)=\mathrm{d}t$ has mixed stress $\mathrm{diag}(1/2,-1/2,-1/2,-1/2)$; such a first jet is realized by smooth Cauchy data. Its contraction with $\mathrm{d}g(x_0)$ is nonzero. For every $u$, at least one component at $u+\lambda v$ has a nonzero quadratic coefficient in $\lambda$. Arbitrarily small $\lambda$ therefore give a violation. This proves density; continuity proves openness.

The full invariant-jet family is not declared empty. This obstruction is not removed by background gauge fixing or by constructing retarded responses.

For every smooth vector field $\xi$ and smooth symmetric tensor $r$,

No field equation is required.

Proof. The Einstein density is natural: $\mathcal{E}[F^{\ast }q]=F^{\ast }\mathcal{E}[q]$. This follows either from the natural curvature construction or by differentiating the invariant Einstein–Hilbert action against compact variations.

Let $F_s$ be the flow of a compact $\xi$, and apply naturality to $q=\eta +\lambda r$. The left input is

The coefficient of $s$ is $G_{\mathrm{lin}}[K\xi ]=0$. The coefficient of $s\lambda$ is

The expressions are local differential expressions, so compact localization of $\xi$ proves the identity for arbitrary smooth germs.

The exact identity is the F1 mechanism of X-F XF (Exact second-order covariance identity). The shortcut $G_{\mathrm{lin}}[{\mathcal{L}}_{\xi }r]=-2G_2[K\xi ,r]$ requires $G_{\mathrm{lin}}[r]=0$. A gauge-fixed free solution need not satisfy that stronger equation.

Define

The inherited flat gauge fixing is ${S_{\mathrm{gf}}}^{(0)}=\frac{1}{4}\int C_{\mu }{C}^{\mu }$. Compact variation gives

Indeed, $G_{\mathrm{lin}}={\mathsf{P}}_{\mathrm{tr}}(-\frac{1}{2}\square +\frac{1}{2}KC)$; the $KC$-terms cancel.

Since ${{\mathsf{P}}_{\mathrm{tr}}}^2=1$ in four dimensions, the free retarded and advanced inverses are

These map tensor densities to covariant tensors; constant $\eta$-identifications are implicit in component notation.

The free equation is ${\mathsf{P}}_{\mathrm{gr}}r=0$, equivalently $\square r=0$. It does not impose $C[r]=0$ strongly. The compact residual statement also remains unchanged: if $\xi$ is compact and ${\mathsf{P}}_{\mathrm{gr}}K\xi =0$, then $K\square \xi =0$; the compact Killing field $\square \xi$ is zero, and wave uniqueness gives $\xi =0$.

Let $\bar{q}=\eta +\kappa h$. For a covariant fluctuation $r$, put

Only the fluctuation is gauge fixed. Classical backgrounds remain off shell and unconstrained. No extra de Donder constraint ideal is imposed.

An explicit specification of $F^{(1)}$ is useful. Set ${\bar{r}}_0={\mathsf{P}}_{\mathrm{tr}}r$, $H={\mathrm{tr}}_{\eta }h$, and

Then $F^{(1)}[h]r$ is the Euler derivative in $r$ of the fully specified quadratic density

Thus no unspecified gauge vertex is hidden in the notation.

Define

It is the fluctuation-quadratic action Hessian, and

The operator $Q^{(1)}[h]$ is formally self-adjoint and of order at most two in $r$. Its coefficients are linear in the two-jet of $h$, and

Dependence on a background coefficient ${\kappa }^{2}k$ first changes $Q$ at order ${\kappa }^2$, hence affects the observable beyond the order considered here.

Proof. Differentiate (eq:Eexp) in a fluctuation direction. The term linear in $r$ at order $\kappa$ is $G_2[h,r]$. Expansion of (eq:backgroundgauge) gives (eq:C1)–(eq:gf1), hence the other term. Both arise as second variations of local quadratic actions, proving formal self-adjointness. Their explicit local differential formulas prove order, coefficient dependence, and support.

For every smooth $\xi ,r$,

Proof. The background gauge-fixing density is natural under simultaneous pullback of $r,\bar{q}$. Differentiating its invariant compact-variation pairing gives

Its derivative at zero is the first identity. Combine it with (eq:cov2) and (eq:P). Equivalently, differentiate $Q[F_s^{\ast }\eta ](F_s^{\ast }r)=F_s^{\ast }(Q[\eta ]r)$.

This is the X-F commutator mechanism XF (Key identity). Both the Einstein-shell term and the gauge-fixing correction are retained. No causal-shadow hypothesis or extended pure-gauge representative is required.

Consider formally

Its fluctuation Euler derivative through first order is

This follows directly from Lemma lem:action, polarization, and (eq:backgroundgauge).

The background-field prescription removes the terms independent of the fluctuation from its equation: equivalently, it removes constant and linear fluctuation terms from the local action expansion. It does not remove the linear kinetic equation. The classical backgrounds $h,k$ are not solved for.

Now write the fluctuation as $r+\kappa k_q$, and multiply the order-$\kappa$ gravitational sources, after Euler differentiation, by $\sigma$. The resulting definition is

For smooth inputs this is the unique zero-past solution of ${\mathsf{P}}_{\mathrm{gr}}{k}_q=\mathrm{Src}$.

Every source in (eq:response) is past-compact: its intersection with the causal past of a compact set is compact. This follows from its vanishing for $t<0$ and flat finite propagation. No spatial decay is needed BGP. Without onset multiplication the self source of a general free field is not past-compact, and the retarded Green operator cannot be applied on that basis.

For a smooth past-compact tensor $u$,

For compact smooth tensor-density tests $A$ and past-compact smooth sources $B$,

The same equalities hold distributionally on the admissible compact kernel stages used below.

Proof. Use ${{\mathsf{P}}_{\mathrm{tr}}}^2=1$, constancy of ${\mathsf{P}}_{\mathrm{tr}}$, and ${{\mathsf{E}}_0}^{\mathrm{R}}\square u=u$ on past-compact fields. The second identity is retarded–advanced transposition. Only the compact intersection $J^{-}(\mathrm{supp}A)\cap \mathrm{supp}B$ contributes. Distributional versions follow by transposition on the same proper-support stages.

For $\xi \in {\mathfrak{g}}_p$, with $r,\varphi$ held fixed,

For a smooth free $r$, the second term is zero. For the free quantum field it is an equation-ideal term after admissible compact smearing.

Proof. The background dependence of the source is linear:

The second equality holds because the operator coefficients are supported in $\mathrm{supp}\xi \subset U$, where $\sigma =1$. By (eq:key),

The field ${{\mathcal{L}}_{\xi }}^{c}r$ has compact support. Apply Lemma lem:leftinverse.

For a compact smooth test $A$,

The inner test is smooth and supported in $\mathrm{supp}\xi$. The bilinear version required by the observable is proved in Section sec:ideal.

Theorem thm:jet is dynamical: it follows from the specified retarded equation. It neither constrains the classical backgrounds nor removes (eq:Bianchi). It also does not transform the free field itself by a pullback.

The source-switched mixed operator $\sigma Q^{(1)}[h]$ is not, for all smooth backgrounds $h$, the Hessian of a local action on unrestricted off-shell fluctuations.

Consequently (eq:response) cannot be identified, on the stated off-shell domain, with the first variational Møller correction of an unspecified switched action merely because its unswitched source was obtained from (eq:fluctuation).

Proof. For fixed $h$, a quadratic-action Hessian is formally self-adjoint under the tensor-density pairing. Proposition prop:Q gives

There is no general equality with $\sigma Q^{(1)}[h]$.

An explicit admissible local example is $h=\eta$. For a positive constant $c$, the four-dimensional Einstein density obeys $\mathcal{E}[cq]=\mathcal{E}[q]$. Hence $D\mathcal{E}[c\eta ]r=c^{-1}{G}_{\mathrm{lin}}[r]$. The background gauge-fixing quadratic action at $c\eta$ also scales as $c^{-1}$: its density contributes $c^2$, the lower metric $c$, and the two divergence factors $c^{-4}$. Therefore

In an open part of the ramp where ${\sigma }^{\prime }\ne 0$,

is a nonzero differential operator. It is detected by compact variations in that open set. Thus $-\sigma {\mathsf{P}}_{\mathrm{gr}}$ is not formally self-adjoint.

This is a concrete version of the distinction emphasized in the supplied X-G record XG. It does not invalidate the retarded equation or the direct identity (eq:jet). It excludes a stronger unqualified variational interpretation of the unchanged off-shell source law.

Switching a local density before Euler differentiation instead produces derivative-of-$\sigma$ terms. They are generally nonzero in the onset layer and are not silently included in (eq:response). They can propagate to later observations. What is used in the plateau variation is the separate fact that $Q^{(1)}[K\xi ]$ has support where $\sigma =1$, not a claim that all onset effects vanish.

We therefore use the phrase equation-level retarded response. No variational Bogoliubov map, Peierls-intertwining theorem, or time-ordered family is released.

Use a free scalar $\mathbf{\varphi }$ satisfying $P_m\mathbf{\varphi }=0$ and a free tensor $\mathbf{r}$ satisfying ${\mathsf{P}}_{\mathrm{gr}}\mathbf{r}=0$. These are not the classical coefficients $h,k$.

Fix scalar Hadamard bisolutions $H_m,H_0$ with one common microlocal orientation, and use zero cross blocks and the stipulated product-kernel normalization

Multiplication by a constant real matrix does not reverse a wavefront orientation. The product and commutators are defined by this frozen matrix:

Only finitely many terms occur for polynomials.

The Green inverses remain (eq:Delta). The action Hessian, equation operator, causal antisymmetric kernel, and product kernel are not interchanged by a sign convention. In particular, no additional incompatible antisymmetric-part condition is imposed on (eq:Wickmatrix). An identification with a separately signed Peierls convention requires its own convention dictionary; it is not needed for the tree proof.

This explicit choice is an algebraic free carrier, not a positive physical graviton state. No strong de Donder constraint quotient is added. The standard free microcausal Wick construction is used with its actual wavefront hypotheses FR,Hormander.

For $n\ge 2$, define

Covectors are identified by the flat completion coordinates. A compact kernel with wavefront in $Z_n$ satisfies the microcausal condition, because co-oriented causal covectors cannot have zero sum unless all are zero.

Let $\mathcal{W}$ be the resulting microcausal polynomial carrier, and $\mathcal{J}$ its closed dynamical equation ideal. It is generated by scalar $P_m$-images and graviton ${\mathsf{P}}_{\mathrm{gr}}$-images, with closure on the admissible compact microlocal kernel stages. The bisolution property makes these dynamical relations compatible with ${\star }_H$.

Let $X_{\xi }u={\xi }^{\mu }{\partial }_{\mu }u$ and $Z_{\xi }=X_{\zeta }+\frac{1}{2}{X_{\xi }}^2$. The classical background transformation is

For fixed smooth backgrounds,