Landmark XI-B The free de Donder multiplet for the ramp Einstein–scalar sheet: causal propagators, residual gauge, and the layer dressing

Starting from the signed quadratic operator supplied by Landmark XI-A, we construct a free Einstein–scalar gauge-fixed multiplet on a finite Minkowski slab. One gauge fermion reproduces exactly the frozen de Donder term, fixes the ghost kinetic normalization, and introduces an independent Nakanishi–Lautrup field. The auxiliary field is retained for off-shell nilpotence.

We derive retarded and advanced Green operators for the graviton, scalar, ghost, and auxiliary blocks. The graviton inverse is minus trace reversal times the zero-data wave inverse; its coefficient is fixed by the handed-off action, with the Newtonian sign providing an independent check. A state-independent graded polynomial field algebra is constructed from the resulting causal kernels. The free differential is shown to preserve its equations and commutation relations.

The plateau algebra has trivial intersection with residual de Donder gauge. For classical de Donder fields, the ramp coordinate coefficient is the retarded response to a layer source, not a field supported only inside the layer. We give an explicit nonzero example and retain the fixed regulator width. The empty fixed-input Einstein jet of XI-A is not repaired: its gauge-fixed retarded response exists, but has nonzero divergence. No Hadamard state, interacting differential, tree, mixed loop, or relational BRST class is constructed.

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

Figure fig:scope-XI-B records that separation. The drawing adds no hypothesis.

: the free gate is constructed; the Einstein jet is not repaired{ The free graded multiplet, one gauge fermion, normalized causal Green operators, a nilpotent free differential, the trivial plateau residual algebra, and the classical layer formula are constructed.

The operator inverted here is the gauge-fixed operator. The empty fixed-input Einstein jet remains empty.

No interaction is contracted with the dressing. No claim is made that the plateau algebra acts by nontrivial residual transformations on the free gauge-fixed graviton, or that the ramp scalar defines an interacting BRST class.}

This manuscript begins where XI-A stopped: with a signed quadratic operator rather than a graviton two-point function. The construction is performed at the free background $(\eta ,0)$. A prescribed nonzero free scalar used in a diagnostic is an input to a coupling expansion, not a replacement background for the free Hessian.

There are three different objects:

They must not be identified merely because they involve derivatives of vector fields.

The polynomial algebra constructed below uses compactly smeared linear fields and their finite products. It has causal commutation relations, but no chosen Hadamard covariance or state. In particular, it is not the Wick algebra used in VII-C. This restricted algebraic construction suffices for the free multiplet and does not presume admission of a nonlocal relational composite.

The standard distinction between free gauge fixing and interacting BV quantization is respected BFR. We construct the former, including its free differential, without supplying an interacting master equation.

For the XI-A quadratic normalization and the conventions of Section sec:handoff, the following are established.

1. One gauge fermion yields the frozen term $S_{\mathrm{gf}}=\int D_{\nu }{D}^{\nu }$ after auxiliary-field elimination.

2. The full free operator has unique retarded and advanced Green operators on compact interior sources, with the stated causal support.

3. The free differential is nilpotent off shell with the auxiliary field retained, and descends to the causal polynomial field algebra.

4. The plateau algebra has zero intersection with residual de Donder gauge.

5. The regulated first coordinate coefficient of a classical de Donder field is a propagated layer functional.

These statements do not construct an interacting or relational BRST cohomology class.

The proofs are given in the successive sections. No tree calculation is part of the theorem.

Retain

Fields are smooth on a neighbourhood of the closed slab. Indices are raised and lowered with $\eta$; parentheses contain the factor $1/2$.

The switch satisfies

The fixed ramp is

where

Choose $U$ open and relatively compact with

Neither $g$ nor ${\sigma }_{\epsilon }$ occurs in the free kinetic operator.

For a symmetric tensor $\gamma$ define

Let $\mathcal{P}$ denote trace reversal, so $\bar{\gamma }=\mathcal{P}\gamma$. In four dimensions,

XI-A gives

This is not the normalization $-\frac{1}{2}\square \mathcal{P}$.

The inherited interaction is

with

It is not inserted into the free action or into the free differential.

The invariant XI-A action has a fixed-background Noether identity; its gauge-fixed action has an additional breaking term. The switching contribution includes $({\partial }_{\mu }g){T_0}^{\mu \nu }$. None of these facts is replaced by free gauge-fixed solvability XIA.

VII-C's Hadamard graviton matrix is not an input. X-D is not used. The text of X-B-v3 is not supplied here; no externally numbered result from it is presumed. The scalar sector is constructed directly below, with its off-shell and free-shell presentations kept separate.

The independent fields and their gradings are:

The algebra is complexified to accommodate the factor $i$ in the antighost transformation. No positivity or ghost reality condition is needed for this release.

Define

Thus the requested parenthesis convention is retained. XI-A's variation $K\xi$ is recovered by setting $c=2\xi$. This fixes a ghost normalization; it introduces neither a new metric gauge condition nor a gauge-parameter family.

The free odd derivation is specified on independent fields by

It commutes with spacetime derivatives and raises ghost number by one. Its action on products is the graded Leibniz rule.

The derivation (eq:s) satisfies ${s_0}^2=0$ on the independent fields and on their polynomial differential algebra.

Proof. The second variation of $\gamma$ is $\mathcal{R}(s_{0}c)=0$. The second variation of $\bar{c}$ is $i{s}_{0}b=0$. The remaining fields already have zero first variation. The square of an odd derivation is an even derivation; hence vanishing on the generators proves vanishing on their polynomial algebra.

The quadratic Fierz–Pauli action is copied in its signed form:

The free scalar action is

Choose exactly one fermion:

The free auxiliary action

has

Its auxiliary equation is $b_{\mu }=D_{\mu }\gamma$. Eliminating $b$ gives precisely $S_{\mathrm{gf}}$ and the ghost term $i\int {\bar{c}}_{\mu }\square c^{\mu }$.

Proof. Equation (eq:basic) gives $s_0{D}^{\mu }\gamma =\frac{1}{2}\square c^{\mu }$. The minus sign from applying an odd derivation to the second factor of $\bar{c}(2D-b)$ yields (eq:sPsi). The auxiliary variation is $2(D^{\mu }\gamma -b^{\mu })$. Finally,

All action identities are bulk variational identities, modulo divergences and tested with compact interior variations. No BRST boundary charge or boundary variational principle is asserted.

The Fierz–Pauli part is invariant under its linear gauge map, and the scalar is unchanged by $s_0$. Consequently $s_0{S}_0=0$ as a free bulk variational functional, since ${s_0}^2\Psi =0$. This does not contradict the XI-A breaking: the bosonic gauge-fixing term alone is not gauge invariant. The independent auxiliary and ghost terms produce the free BRST-invariant completion.

After elimination, $s_0\bar{c}=iD\gamma$, and therefore

This vanishes on the free ghost shell, not on unrestricted ghost configurations. We retain $b$ in the off-shell multiplet.

For bookkeeping, name the antifields

An antifield has opposite parity and ghost number $-1-\mathrm{gh}(\text{field})$. These names denote external source slots only. They are not additional propagating generators here; no antifield differential, BV antibracket, or interacting master action is claimed.

For $\mu \ge 0$, let ${\Omega }_{\mu }=\sqrt{-\Delta +{\mu }^2}$. For a smooth source $F$ define

The quotient has its continuous zero-frequency value. For nondecaying smooth sources these formulas are interpreted locally by finite propagation. On compact interior sources they are the restrictions of the full Minkowski retarded and advanced operators.

The operators (eq:ER)–(eq:EA) invert $\square +{\mu }^2$ with, respectively, zero initial and zero final Cauchy data. For $F\in {C_c}^{\infty }({M_T}^{\circ })$,

and

These support bounds are compact in the closed slab. Moreover $({{\mathsf{E}}_{\mu }}^{\mathrm{R}}{)}^{\ast }={{\mathsf{E}}_{\mu }}^{\mathrm{A}}$ on interior test functions.

Proof. Differentiating the sine formulas twice verifies the equation and the prescribed endpoint data. Uniqueness follows from the homogeneous Cauchy problem. Applying this uniqueness to ${{\mathsf{E}}_{\mu }}^{\pm }(\square +{\mu }^2)F-F$ gives the left inverse on compact interior tests. Finite propagation gives the support bounds. The intersection of a causal future or past of a compact set with a bounded-time closed slab is compact. Green's identity proves transposition: the retarded solution has zero data at $0$, the advanced solution has zero data at $T$, and there is no spatial boundary contribution. These are the standard smooth Cauchy properties BGP (Chapter 3).

The endpoint prescriptions select the inverses. They do not require all free solutions or ghosts to have zero Cauchy data.

Under the flat Lorentz pairings, the graviton operator is

Its retarded and advanced inverses are

Thus the coefficient in $c\mathcal{P}{{\mathsf{E}}_0}^{\mathrm{R}}$ is $c=-1$.

Proof. Trace reversal commutes with the wave operator and satisfies ${\mathcal{P}}^2=\mathrm{id}$. Hence

and the left inverse follows on compact tests. Any coefficient $c$ instead gives $-c\mathrm{id}$, forcing $c=-1$. The support statement follows from Lemma lem:Green.

This fixes the normalization from XI-A's Hessian. The same sign reproduces its Newtonian comparison: for $H=q-\eta$ and a static positive source,

A coefficient $-2$ would invert a different Hessian. The Newtonian check is not permission to change the quadratic normalization.

The scalar equation operator is $P_m$, with inverse ${{\mathsf{E}}_{\mathsf{m}}}^{\pm }$. Its action Hessian is $-P_m$, with inverse $-{{\mathsf{E}}_{\mathsf{m}}}^{\pm }$. This distinction determines the scalar causal bracket below.

Let $D^{\ast }$ be the formal adjoint of $D$. With flat index identifications,

Write $A=2G_{\mathrm{lin}}$. The even metric–auxiliary operator is

Its source equation is

The unique retarded and advanced inverses are

G^{\mathrm{aux}}^\pm=
 \begin{pmatrix}
 G^{\mathrm{gr}}^\pm&G^{\mathrm{gr}}^\pm D^*\\
 DG^{\mathrm{gr}}^\pm&DG^{\mathrm{gr}}^\pm D^*-\frac12\operatorname{id}
 \end{pmatrix}
 =
 \begin{pmatrix}
 -\mathcal P\mathsf E_0^\pm&\mathcal R\mathsf E_0^\pm\\
 D(-\mathcal P\mathsf E_0^\pm)&0
 \end{pmatrix}.
 \label{eq:Gaux}

The identities in the second matrix hold on compact interior sources.

Proof. The auxiliary equation gives $b=D\gamma -j/2$. Substitution reduces the metric equation to

The handed-off identity is $A+2D^{\ast }D=-\square \mathcal{P}$. Equation (eq:Ggr) therefore gives the first matrix and uniqueness.

For the simplification, direct calculation gives

Consequently ${G^{\mathrm{gr}}}^{\pm }{D}^{\ast }=\mathcal{R}{{\mathsf{E}}_0}^{\pm }$ and DG^{\mathrm{gr}}^\pm D^*=\frac12\operatorname{id}. The last block cancels the algebraic contact term. The source elimination proves both Green identities; derivatives do not enlarge causal support.

The vanishing $bb$ block does not make $b$ a discarded field: the $\gamma b$ and $b\gamma$ blocks are nonzero. Nor is an independent wave inverse for $b$ postulated.

Use left Grassmann variations. The ghost kinetic term yields

In the order $(c,\bar{c})$, with dual vector indices understood,

Matrix multiplication proves the two Green identities. The scalar, auxiliary, and ghost blocks together give the full free multiplet Green operator. No Feynman inverse is selected.

Let ${\mathcal{P}}_{\mathrm{off}}$ be the complex graded commutative polynomial algebra generated by compact interior smearings of the five independent fields. Products are finite. They have smooth finite-rank multivariable kernels, so no coincident-field composite is needed to define them.

This is an off-shell algebra: no field equation is used in Lemma lem:nilpotence. A second, free-shell algebra is obtained by imposing the linear equations and causal commutation relations. Making that second presentation does not retroactively impose free equations on classical relational functionals.

Put

The full causal matrix is the retarded-minus-advanced inverse of the action Hessian. Its nonzero blocks are

The appropriate vector Kronecker symbols accompany the ghost blocks. These are causal commutator kernels, not Hadamard two-point functions.

The algebra ${\mathcal{A}}_0$ is generated over $\mathbb{C}[[\hslash ]]$ by the compact interior smeared fields, subject to:

1. the auxiliary free equations

   $$A\gamma +2D^{\ast }b=0,b=D\gamma ,P_m\varphi =0,\square c=\square \bar{c}=0;$$

2. the graded commutation relations

   $$[{\Phi }_A(f),{\Phi }_B(k){]}_{\mathrm{gr}}=i\hslash ⟨f,{\Delta }_{AB}k⟩1.$$

Only finite polynomial degree is used at each formal $\hslash$ coefficient.

Definition def:algebra gives a nonzero associative graded polynomial algebra. All pairings in (eq:CCR) exist.

Proof. Smooth compact tests can be paired with the displayed causal distributions. Transposition of the Green operators implies graded antisymmetry: the even blocks are antisymmetric, while the odd–odd block is symmetric under simultaneous exchange of its slots.

The equations annihilate the causal matrix on both sides, so the pairing descends to the test space modulo equation tests. It is a constant even graded antisymmetric form on that quotient. The associated central graded Lie algebra satisfies the Jacobi identity because every bracket is central. Its enveloping algebra, with the central generator set to the unit and scaled by $i\hslash$, supplies the required algebra. Equivalently, symmetrization gives the constant graded Moyal product on polynomials. This establishes associativity and the polynomial vector-space presentation without choosing a state FR.

No extension to all local or microcausal polynomial functionals is asserted. In particular, (eq:vertex) is not being multiplied in ${\mathcal{A}}_0$.

The derivation $s_0$ preserves the free equation ideal.

Proof. Since $A\mathcal{R}=0$ and $s_{0}b=0$,

Also,

The ghost equation for $c$ is unchanged, while

Taking the divergence of the metric–auxiliary equation gives $\square b=0$: the divergence of $A\gamma$ is zero and the divergence of $2D^{\ast }b$ is $-\square b$. The scalar equation is unchanged. Thus every transformed equation belongs to the same linear equation ideal.

The eliminated relations then give ${s_0}^2\bar{c}=\frac{i}{2}\square c=0$ in the free-shell algebra, while independent $b$ already provided off-shell nilpotence.

The field rule (eq:s) induces a ghost-number-one nilpotent graded derivation of ${\mathcal{A}}_0$.

Proof. Equation stability is Lemma lem:equations. It remains to check the commutation relations, not merely to postulate a Leibniz rule.

The only nontrivial mixed check is the initially zero relation between $\gamma$ and $\bar{c}$. The ghost block gives

The auxiliary block gives

with its tensor indices understood. Therefore

These are distributional identities tested with compact smearings; no pointwise field multiplication is involved.

All other transformed relations contain only zero cross-blocks, or $cc$ or $bb$ brackets, which vanish. The nonzero $\gamma \gamma$, $\varphi \varphi$, and $c\bar{c}$ brackets have constant right-hand sides and zero transformed left-hand sides. Hence $s_0$ preserves the relation ideal. Nilpotence descends from Lemma lem:nilpotence.

Thus the free algebra is a differential graded algebra. No cohomology classification or interacting quantum differential ${\hat{s}}_{\mathrm{int}}$ follows from this statement. The construction contains no stress-tensor vertex.

A classical de Donder field has $D\gamma =0$. Under $\gamma \mapsto \gamma +K\xi$,

Its residual de Donder parameters therefore satisfy $\square \xi =0$.

Proof. A vector field in ${\mathfrak{g}}_U$ vanishes on a neighbourhood of ${\Sigma }_0$. Every component has zero initial Cauchy data. If it also solves the homogeneous wave equation, Cauchy uniqueness forces it to vanish everywhere on the slab.

This does not say that all residual parameters on Minkowski space vanish. Smooth noncompact wave solutions exist. It says that the frozen plateau support class contains none except zero.

Consequently ${\mathfrak{g}}_U$ is not a nontrivial residual action on classical de Donder free solutions. More generally, a compact shift $K\xi$ preserving the gauge-fixed free equation would require $K\square \xi =0$; the compactly supported vector $\square \xi$ would then be a Killing vector and hence zero, again giving $\xi =0$.

The admitted plateau algebra remains an algebra acting on the classical background prescription. It is not thereby promoted to an interacting action on the free graviton algebra. The graded transformation involving the ghost is a different construction. In particular, ghosts are not required to be compactly supported plateau wave solutions.

For XI-A's source-response coefficient put

The gauge-fixed retarded solution is

Taking its divergence on the free scalar shell gives

For the named input

one has ${T_0}^{00}=1/2$ and ${T_0}^{0i}=0$. Thus

A nonzero compact interior switch has ${\partial }_{t}g\not\equiv 0$, so this divergence is nonzero.

Both (eq:response) and its divergence exist as smooth retarded responses. The failed requirement is $D{\gamma }_1=0$, not existence of the divergence. The ungauge-fixed Einstein equation would require conservation of $g{T}_0$ by its Bianchi identity. XI-A's empty ${\mathcal{J}}_g({\varphi }_m)$ is therefore unchanged. This diagnostic is not a failure of the Green construction and uses no contraction with the interaction.

The density coefficient inherited from X-A is

Its divergence gives exactly the de Donder combination:

The regulated first coordinate coefficient, with zero initial data, is therefore

This is the divergence-form ramp prescription, not a multiplied-equation replacement.

For every smooth classical tensor satisfying $D\gamma =0$,

Its source vanishes outside $0<t<\epsilon$. If that source has spatially compact support, then

The propagated coefficient need not vanish for $t>\epsilon$. It is nonzero precisely when the source vector in (eq:layer) is nonzero.

Proof. Set $D\gamma =0$ in (eq:Xgeneral). The derivative ${\sigma }_{\epsilon }^{\prime }$ is smooth, vanishes off the ramp, and is flat at both endpoints. Finite propagation proves the support inclusion. Since $\square {\mathsf{X}}_1=-{\sigma }_{\epsilon }^{\prime }{a_1}^{0\cdot }$, zero coefficient implies zero source. The converse follows from zero-data uniqueness.

Strictly, the closed support of ${\sigma }_{\epsilon }^{\prime }$ can meet both $0$ and $\epsilon$, even though the function vanishes at those endpoints. The support statement is about a smooth layer source, not a hypersurface distribution.

A nonzero tensor in the layer can have ${a_1}^{0\mu }=0$ and hence zero dressing. For example, a constant purely spatial trace-free tensor has this property. Thus “nonzero layer data” must refer to the projected source in (eq:layer), not to arbitrary nonzero metric components.

Let $\lambda \ne 0$ and ${\gamma }_{\mu \nu }=\lambda {\eta }_{\mu \nu }$. It solves the homogeneous gauge-fixed free equation and $D\gamma =0$. Its coordinate coefficient is

In particular, for $t\ge \epsilon$,

Proof. The constant tensor has zero derivatives, and ${a_1}^{\mu \nu }=\lambda {\eta }^{\mu \nu }$. For its spatially constant source, the zero-frequency wave formula gives

The ramp satisfies $\chi (1-s)=1-\chi (s)$, so ${{\int }_0}^{\epsilon }{\sigma }_{\epsilon }(s)\mathrm{d}s=\epsilon /2$. This yields (eq:layerexample).

No spatial decay was imposed on classical configurations. This example is legitimate under the local smooth Cauchy prescription and shows explicitly why the output cannot be labelled layer-supported.

For a compact interior covector test $a_{\mu }$, transposition gives the de Donder-layer identity

The resulting linear kernel is smooth at fixed $\epsilon$. Its support is contained in the closed layer intersected with the causal past of $\mathrm{supp}a$.

Equation (eq:layerkernel) is a classical restriction to de Donder tensors. The full auxiliary multiplet is not restricted by $D\gamma =0$: its equation is $b=D\gamma$. Imposing $D=0$ as an additional algebraic quotient would require a separate analysis and is not done here.

A boundary-supported trace is suggested by formally shrinking the layer. No convergence theorem, sharp limit, or identification with a sharp Cauchy kernel is asserted. In particular, a smooth closed-slab kernel need not be a compactly supported interior test. This classical kernel fact is not an admission theorem for a quantum relational composite.

Neither ${\mathsf{X}}_1$ nor its scalar dressing is declared $s_0$-closed. The plateau covariance of X-A does not prove such a claim for unrestricted ghost Cauchy data.

VII-C concerns a sharp zero-Cauchy insertion, a fixed Hadamard Wick matrix in the opposite signature, and a specified normal-topology extension. Those are not the data of the causal polynomial algebra constructed here VIIC.

For precision, its four extension requirements are recorded below. “Unmet” means not established or not instantiated by this manuscript, not disproved for every possible successor construction.

Formal coefficient extraction is available algebraically. That does not establish E4. Likewise the tensor $\mathcal{P}$ derived from the XI-A operator is not an imported Hadamard block. A causal inverse does not select a positive-frequency bisolution.

We do not form the word $B_f\star W_g\star B_{f^{\prime }}$, do not evaluate a reflected light-ray residue, and do not subtract one. There is no conclusion here that the ramp resolves VII-C's obstruction, nor that VII-C obstructs the present restricted free algebra.

The ending is Constructed for the free gate. The existence of its causal Green operators is not evidence against XI-A's on-shell obstruction. Different equations are being solved.

XI-C receives:

1. the signed operator $-\square \mathcal{P}$ and its causal inverses;

2. the single gauge fermion, auxiliary equations, ghost normalization, and full free causal matrix;

3. the free graded polynomial algebra and its differential;

4. the residual-gauge result (eq:residual);

5. the classical layer formula, with its support qualifications;

6. the inherited vertex (eq:vertex), still uncontracted;

7. the distinction between the classical plateau action and a residual action on the gauge-fixed free graviton.

It does not receive a Hadamard state, Feynman prescription, tree, mixed loop, renormalized relational product, or interacting BRST class. The first future contraction must name its domain and its covariance; neither is supplied by the word “propagator.”

The free construction closes:

At the same time,

The former is a residual-gauge statement, the latter a coupled source obstruction. Neither is a failure of the constructed gauge-fixed Green operator.

The reduced graviton Hessian is $-\square \mathcal{P}$, so its inverse is $-\mathcal{P}{{\mathsf{E}}_0}^{\pm }$.

The scalar action Hessian is $-(\square +m^2)$, so its inverse is $-{{\mathsf{E}}_{\mathsf{m}}}^{\pm }$, even though the equation inverse is $+{{\mathsf{E}}_{\mathsf{m}}}^{\pm }$.

The retarded-minus-advanced scalar Hessian kernel is therefore $-{\delta }_m$. With the convention (eq:CCR),

Since ${\partial }_{x^0}{\delta }_m$ at equal times is the spatial delta, this gives $[\varphi ,{\partial }_t\varphi ]=i\hslash {\delta }_{{\mathbb{R}}^3}$ in the canonical ordering. Thus the action and commutator signs agree.

The half-normalized metric differential is $s_0\gamma =\mathcal{R}c$. The gauge fermion produces $i\bar{c}\square c$, not $2i\bar{c}\square c$.

Its causal anticommutator is

while $[\gamma ,b]=i\hslash \mathcal{R}{\delta }_0$. These exact coefficients produce the cancellation in Theorem thm:differential. Replacing one coefficient without rescaling the ghost would destroy that compatibility.

One could instead use the full ghost normalization $s_0\gamma =K\tilde{c}$, with $c=2\tilde{c}$. Then the corresponding ghost kinetic coefficient changes. That is an equivalent reparametrization, not a second gauge chosen in this paper.

For an unrestricted tensor,

Only the classical condition $D\gamma =0$ removes the first term. The gauge-fixed equation $-\square \bar{\gamma }=0$ does not itself impose $D\gamma =0$. The auxiliary field records that distinction.

The proofs use the explicit quadratic action, compact interior variation, ordinary smooth wave Cauchy theory, algebraic trace reversal, and graded central commutation relations. They do not use a graviton Hadamard matrix from another manuscript.

The supplied XI-A is used for its signed action and the Einstein-jet obstruction. The ramp coefficient is derived from the displayed divergence-form prescription. VII-C is used only to identify the separate hypotheses that are not constructed here.

level. The free algebra is a formal algebra of compactly smeared linear fields and finite products. No Hilbert-space representation or state is selected.

level. Initial and final data specify causal inverses. No independent boundary algebra, BRST boundary charge, or admission topology is constructed.

level. The switched stress-tensor vertex remains a named classical interaction. It is not used in any contraction with the ramp dressing and is not included in $s_0$.

. No successful pdflatex execution, proof-assistant verification, executed symbolic computation, independent referee review, or journal acceptance is asserted.

XIA

N. Cheung, Landmark XI-A: A switched Einstein–scalar action for the ramp relational scalar: sign conventions, Noether identities, and an empty fixed-input Einstein jet. Supplied companion manuscript. Used for the signed quadratic operator, the switching conventions, and the fixed-input Einstein-jet obstruction.

VIIC

Landmark VII-C: A mixed-Wick obstruction to a normal-topology boundary extension for sharp Cauchy harmonic insertions. Supplied comparison manuscript. Used only for the extension checklist; its free matrix and products are not inputs to the present construction.

BGP

C. Bär, N. Ginoux, and F. Pfäffle, Wave Equations on Lorentzian Manifolds and Quantization, ESI Lectures in Mathematics and Physics, European Mathematical Society, 2007. arXiv:0806.1036. Used for smooth Cauchy theory, causal Green operators, and finite propagation.

FR

K. Fredenhagen and K. Rejzner, Perturbative algebraic quantum field theory, arXiv:1208.1428v2, 2013. arXiv:1208.1428. Background for the state-independent polynomial deformation; all block normalizations and compatibility checks are derived in this manuscript.

BFR

R. Brunetti, K. Fredenhagen, and K. Rejzner, “Quantum gravity from the point of view of locally covariant quantum field theory,” Communications in Mathematical Physics 345 (2016), 741–779. arXiv:1306.1058. Comparison framework for the distinction between free gauge fixing and an interacting BV construction.