Em01-F-v3 Constant-Weight Einstein–Scalar Substitution and Exact Positive-Scale Regrouping An explicitly normalized marked response insertion with the characteristic limit left open

We study a new gauge-fixed Einstein–scalar application with constant gravitational and coordinate weights. Its action, field normalization, free Hessians, contraction kernels, quadratic gravitational source, observation root, and density map are specified consistently. An explicit marked response insertion produces seven distributional rows.

The revision separates these derived equalities from convention agreements and constant-weight specializations of supplied weighted formulas. A comparison panel displays the missing coordinate factors and density-weight derivative terms without adding an unproved weighted model to the main theorem. The cubic source is checked against an explicit Einstein–Hilbert density, and the action–stress signs, coupling powers, polarization factors, and symmetric-component conversion are recorded.

We prove row-by-row substitution, exact transposition to cone measures, and a 42-entry positive-scale audit identity. On the strict time-ordering patch, an exact crossed-line reduction and trace-reversal identity compress this ledger into a combined scalar-profile formula. The full eight-dimensional convolution is preserved, including for nonfactorizing profiles. All cone tips and source incidences remain in the internal distributional construction.

A conservative order-five estimate with scale power twenty-one is proved. This is not a leading asymptotic exponent. Neither the complete characteristic limit nor its failure is established. The application is not identified with an inherited weighted relational observable, and no quantum admission or gauge-invariant completion is asserted.

and scope{ This paper proves an exact input-substitution and positive-scale reduction theorem for the marked insertion defined here. The gravitational and coordinate weights are identically one; the matter-interaction switch remains a separate function.

The comparison with supplied manuscripts is documentary and formula-level. It is not an independent verification of their parent sources and does not identify their observables with this one. The characteristic raw-limit question remains open.}

Figure fig:scope-Em01-F records that separation. Figure fig:split-Em01-F records one view of the same separation. Figure fig:chain-Em01-F records one view of the same separation. Figure fig:pair-Em01-F records one view of the same separation. Figure fig:gate-Em01-F records one view of the same separation. The drawings add no hypothesis.

The object of this paper is a new application, not a reconstruction of incompletely supplied predecessor data. The latter distinction is necessary because the supplied passages separate internal kernel construction from the still missing joint characteristic operation Packet.

The symbols below have different roles:

The proved application has

Neither $f$ nor $g$ is a gravitational ramp.

Fix the action, contractions, observations, and marked insertion specified in Sections sec:model–sec:insertion. Let $U⋐M^4$ be the separated time-ordering patch of Section sec:geometry. Let $\rho \in {C_c}^{\infty }({\mathbb{R}}^8;\mathbb{R})$ satisfy $\int \rho =1$, and let $\chi \in {C_c}^{\infty }({\mathbb{R}}^8)$ equal one near $\mathfrak{f}(\overline{U})$.

Every input in

is determined by the displayed formulas of this paper.

The seven kernels are exactly the coefficient kernels of the defined marked insertion. Their arbitrary compact joint-test actions have the cone-seed representation of Theorem thm:seed. The selected family has the exact 42-entry audit expansion of Theorem thm:ledger.

For sufficiently small $\delta >0$, the same complete family has the combined scalar-profile representation (eq:combined-seeds); this representation agrees with the audit expansion on every compact joint test.

For every $L⋐U$, there is $C_L>0$, independent of $\delta ,\lambda ,\hslash$, such that

The remaining fixed data may enter $C_L$. No raw-limit conclusion or leading scale exponent is asserted.

The overall factor $i$ in (eq:apptarget) is part of the selected-family convention. There is no $1/\hslash$ prefactor. Section sec:normalization explains its coupling and contraction multiplicity completely.

The proof of Theorem thm:main is completed in Section sec:estimate. The distinction between ordinary contractions, time ordering, and their domains is retained throughout; these are separate operations in perturbative algebraic quantum field theory FR (Sections 5–7).

Let

All spacetime raising and lowering use $\eta$. Distributional pairings are complex bilinear, with flat coordinate densities. Complex polarization is bilinear, not sesquilinear.

Let $S={\mathrm{Sym}}^2(({\mathbb{R}}^4{)}^{\ast })$. For $I=(i,j)$, $0\le i\le j\le 3$, define

Thus

An off-diagonal basis tensor has two entries equal to one; its algebraic dual has two entries equal to $1/2$. The algebraic dual is not simply the raised basis tensor.

Trace reversal is

Put

The subscripts in ${\mathsf{A}}^{\mu \nu }{}_L$ and ${\mathsf{b}}_L$ mean evaluation on $e_L$; they do not prescribe an extra fiber-metric contraction.

In the basis (eq:basis), define

For the ordered labels $(00,01,02,03,11,12,13,22,23,33)$,

If ${\mathcal{E}}^{\mu \nu }$ is a contravariant Euler derivative, its coordinate Euler components and propagated covariant components are related by

For arbitrary symmetric $h,k$,

No additional diagonal/off-diagonal weights occur in a contraction of a coefficient dual to $h_I$ with $h_I$.

Proof. The basis tensors have disjoint component supports. A diagonal contraction has one term; an off-diagonal contraction has two terms and the corresponding metric sign. This gives the displayed diagonal matrix.

For $I=00$, $E_{00}={\mathcal{E}}^{00}$ and $J^{00}={\mathcal{E}}^{00}$. For $I=01$,

These agree with the corresponding entries of ${\mathsf{G}}^{-1}$.

The general statement follows from the completeness identity

It also proves the full-index pairing formula after substituting the component expansions. Therefore an extra off-diagonal factor in an already dualized coefficient pairing would count the same conversion a second time.

locked conventions{ Inverse: ${G_{\gamma }}^{\mathrm{R}}=-\mathsf{P}{E_0}^{\mathrm{R}}$. A coefficient-$-4$ inverse belongs to a different normalization.

Polarization: for a homogeneous quadratic map $J$, the half-polarization $J(G,h)=\frac{1}{2}(J[G+h]-J[G]-J[h])$ satisfies $DJ[G]h=2J(G,h)$. The source and density maps below use the mixed derivative.

Components: the algebraic dual already handles symmetric components. No additional $1,2$ diagonal/off-diagonal rule may be inserted without a derived change of convention.}

Write $w=(a,b,z,y)$. The label $J$ is dual to the marked tensor input of Section sec:insertion. For $U⋐M^4$, set

Strong distributional convergence is uniform convergence on bounded test sets. Such sets have common compact support and bounded derivative seminorms of every order.

Set

and choose a smooth orthonormal transverse frame $E_A=(0,{\mathbf{e}}_A)$, $A=1,2$, on a sphere subchart. Use

There are eight tangential coordinates $(a,\mathbf{n},\alpha ,\beta )$ and eight normal coordinates $(r,u,v,\xi )$.

Near $r=u=v=\xi =0$, $0<\alpha <\beta$, these are smooth coordinates, and

The flat density is

The map and inverse have bounded derivatives on compact subcharts with positive lower bounds for $\alpha ,\beta -\alpha ,\beta ,\beta -u$.

Proof. For $s=b-a$, the inverse is

For $d=y-a$,

The null-frame identities give the cone equations.

The absolute Jacobian from $(\beta ,u)$ to $(s^0,|\mathbf{s}|)$ is two; its spatial radial factor is $(\beta -u{)}^2$. At fixed $\mathbf{n}$, the map $(\alpha ,v,\xi )\mapsto d$ has absolute determinant two. The full map is block triangular after choosing $a,s$. This proves the density and the stated bounds.

Let

Their codimensions are eight and five. Along $N$,

The three differentials have rank two. Their crossed restrictions to $S_{\mathrm{resp}}$ have rank one along $N$. Also, throughout the chart,

Choose a small relatively compact patch about

We require throughout its closure

and all positive separation bounds above. In particular $b\ne a$ and $b\ne z$. Strict separation of $a,y,b$ does not remove $z=a$.

Write

Then the additional internal cone equations are exactly

The $K_{5,+}$ tip line has argument $-r$, and the $K_{5,-}$ tip line has argument $r$; their retarded time orientations must not be interchanged.

If $\beta -A_r>0$, the equation $(b-z{)}^2=0$ gives

This is a geometric constraint, not a scaling rule for $A_r$.

For actual smoothing displacements

one has

Thus

These equations retain the residual variable and its cone-tip transitions. The geometry does not assign a common rescaling weight to $r$.

Fix $m\ge 0$, $\lambda \in \mathbb{R}\setminus \{0\}$, and $\hslash >0$. Coefficient expansions are formal at $\lambda =0$. The metric variable and background are

Use

Let

The gauge-fixing functional is explicitly quadratic; no nonlinear gauge-fixing interaction is implicit.

With $g\in {C_c}^{\infty }(M;\mathbb{R})$, define

The action is evaluated on compact perturbations, or under compact local variations. Its linear gravitational term is a divergence.

The gravitational action is unswitched. The coordinate prescription below also has constant weight. The matter interaction is switched by $g$. The model is used off shell; no existence result for an invariant switched Einstein jet is assumed.

For a compact covariant metric variation,

The curved matter stress in the invariant matter density is

Its covariant metric variation satisfies

Indeed,

These identities give (eq:mattervariation). For (eq:EHvariation), vary the Ricci tensor by

Metric compatibility puts the derivative contribution into a divergence; the remaining inverse-metric and determinant variations give the displayed Einstein tensor and sign.

Where $g=1$, the invariant part of the model therefore has metric equation

This equation concerns the invariant part, not the additional gauge-fixing Euler equation.

The comparison dictionary, when adopting the Newtonian normalization reported in the supplied XI/XII passages, is

It is a convention agreement with those passages Packet, not an identification with every use of $\kappa$ elsewhere.

The covariant tensor pairing is

At the zero-field background,

The mixed free block vanishes. For the equation-normalized Green operators of $P_m={\square }_{\eta }+m^2$, the action-Hessian inverses are

Proof. The linearized Einstein tensor is

The gravitational Euler derivative in the $\gamma$-variable has linear covariant output $2{\mathcal{G}}_1(\gamma )$, by (eq:EHvariation) and $\delta \mathbf{g}=\lambda \delta \gamma$. The lowered gauge-fixing Euler derivative is

It cancels the divergence terms, leaving $-{\square }_{\eta }\mathsf{P}\gamma$.

The free scalar Euler derivative is $-P_m\varphi$. Every matter interaction correction contains at least one metric field and two scalar fields, so there is no free mixed block at $(\eta ,0)$. Homogeneity gives the quadratic action.

Finally $\mathsf{P}$ commutes with ${\square }_{\eta }$ and squares to the identity, so direct composition proves (eq:inverse).

In particular,

The coefficient is forced to be $c=-1$. The reported X-track coefficient $-4$ cannot be used with this Hessian Packet.

The three counts are distinct:

For different species, the mixed term in $\frac{1}{2}({\Gamma }_{\gamma }+{\Gamma }_{\varphi }{)}^2$ is ${\Gamma }_{\gamma }{\Gamma }_{\varphi }$. The two species assignments cancel the factor $1/2$. The mixed matter derivative already includes the choice of either scalar field in its quadratic stress. No further factor of two is added.

Thus a later definition with an additional $1/\hslash$ response normalization would have a different quantum degree. It would not be the family (eq:apptarget). The distinction between contraction expansion and interacting response normalization is also explicit in the standard functional framework FR (Sections 5–6).

At flat background use

Then

The mass sign therefore agrees with $P_m$.

Its polarization is

Define

The expansions are

with

The mixed derivative is

For the defined insertion,

The full mixed metric-field variation carries ${\lambda }^2{D}_{\mathrm{dens}}(G,h)$.

Proof. Expand the inverse matrix and square-root determinant:

Multiplication gives $a_1,a_2,b_1$. Differentiation of $a_2[G+th]$ at $t=0$ gives (eq:Ddensity). The insertion uses this derivative, not its half-polarization.

For example,

These follow both from (eq:Ddensity) and from directly inverting the corresponding one-parameter metric matrices. For arbitrary $h,G$, bilinearity and basis completeness give

Thus the full-index and ten-component density calculations agree.

The first matter-interaction coefficient is $\lambda V_g$, where

Its mixed derivative is

Proof. The first density coefficient is

The scalar derivative doubles the symmetric polarization.

For a covariant symmetric field $h$, let

For $n=1,2,3$, set

With

the declared cubic density is

It is the coefficient of $\lambda$ in the unintegrated Einstein–Hilbert density of (eq:action). The source is calculated from this density with unit weight. No density-weighted integration-by-parts equivalence is assumed.

At zero scalar field, write the lowered metric Euler equation as

The source is homogeneous quadratic. The label ${J_3}^{\mathrm{app}}$ may equivalently denote ${\mathcal{J}}_2$, emphasizing that it comes from the cubic action coefficient, not from a cubic field-degree source.

Put

Then $C={\Gamma }_1$, $C_2={\Gamma }_2$. Define

For compact $h,k$,

where

Proof. Expand (eq:EHvariation) in the field variable $\mathbf{g}=\eta +\lambda h$. Its lowered Euler output is

The coefficient of ${\lambda }^2$ inside the last density is

Multiplication by $2/\lambda$ gives (eq:J2evaluated) at order $\lambda$. The coefficient of the action at that order is precisely (eq:ell3). Formal coefficient extraction commutes with compact variation, proving (eq:cubicvariation). The quadratic gauge-fixing term has no contribution at this order.

For an independent density-level check, define

The coordinate Ricci formula and metric compatibility give

where

Hence ${\ell }_3$, modulo a displayed divergence, equals the first-jet cubic density

Here the superscript $(1)$ refers to first-jet dependence, not coupling order.

For a variation $k$,

Thus its entire directional variation is

The exact divergence identity proves that its compactly integrated variation is also (eq:cubicvariation). This supplies a second density representation for checking the source. It is valid here because the weight is constant. Multiplication by a nonconstant gravitational weight would generate additional terms from the divergence and is not authorized by this check.

The insertion uses

The dynamical field $h$ occupies the $a$-slot; the marked field $G$ occupies the $z$-slot.