Projected Scalar Green Functions and Schur Reduction of PT-Symmetric Quaternionic Principal-Symbol Deformations

Abstract

We study the Green function of a real scalar field ${\varphi }_{+}$ on a fixed real Lorentzian spacetime $(M,g)$. Quaternionic data enter only through a fixed auxiliary two-component representation of the scalar principal-symbol deformation ${\mathcal{G}}^{\mu \nu }=g^{\mu \nu }I+\delta {\mathcal{G}}^{\mu \nu }$, which is not a spacetime metric. The grading $\Theta$ selects the retained component. Strict projection discards the complementary component, whereas exact Schur reduction retains the complementary block contribution before elimination. For $\delta {\mathcal{G}}_H^{\mu \nu }=E^{\mu \nu }{e}_2$, the faithful representation of $e_2$ produces off-diagonal mixing $\mathcal{B}W$ and the exact retained operator ${\mathcal{D}}_{\mathrm{eff},+}={\mathcal{D}}_{++}+\mathcal{B}{\mathcal{D}}_{--}^{-1}\mathcal{B}$. When the complementary block has a local separation scale M, the effective field theory (EFT) correction begins with $\delta {\mathcal{D}}_{\mathrm{eff},+}=-M^{-2}{\mathcal{B}}^2$, whose frozen principal symbol is $-{\left[E^{\mu \nu }{k}_{\mu }{k}_{\nu }\right]}^2/M^2$. Under the stated local microscopic $\mathcal{PT}$ assumptions, this retained correction is $\mathcal{PT}$-symmetric. It enters the standard scalar propagator, resolvent, and one-loop expansion. No quaternionic spacetime geometry, full interacting quaternionic quantum field theory, all-order renormalization theorem, or observational result is derived.

1. Introduction

The physical problem is a real scalar field ${\varphi }_{+}$ on a fixed real Lorentzian spacetime $(M,g)$. Its retained quadratic action, stated fully in Section 5, is $S_2=-{\displaystyle \frac{1}{2}}{\int }_M\sqrt{-g}{\varphi }_{+}{\mathcal{D}}_{+}{\varphi }_{+}$, and its Feynman Green function is the prescribed inverse of ${\mathcal{D}}_{+}$. Under the stated formal-symmetry and boundary assumptions, the corresponding free equation of motion is ${\mathcal{D}}_{+}{\varphi }_{+}=0$. This is the ordinary curved-spacetime setting of a physical geometry, field operator, and state prescription [1,2,3,4].

The only additional structure is a fixed auxiliary two-component representation used to organize a quaternionic deformation of the scalar principal symbol. Here, the principal symbol is the highest-derivative, local $k_{\mu }{k}_{\nu }$ part of the scalar kinetic operator, and the auxiliary carrier space is the two-dimensional complex bookkeeping representation used to realize its quaternionic coefficient. Its two components are not two independently quantized scalar fields. The linear grading $\Theta$ separates a retained + sector, whose chosen real slice is coordinatized by the physical field ${\varphi }_{+}$, from a complementary-sector used only in the reduction. The coefficient ${\mathcal{G}}^{\mu \nu }$ acting on this space is not a physical metric, an inverse quaternionic metric, or a new dynamical field. All connections, curvature, index operations, and volume factors remain those of $g_{\mu \nu }$.

The paper compares two operations on this representation. Strict projection sets the complementary component aside and retains only ${\mathcal{D}}_{++}$. Schur/Feshbach block reduction instead allows the complementary block to contribute through the off-diagonal coupling before elimination. The two operations therefore need not yield the same retained operator. For the decomposition ${\mathcal{H}}_{\mathrm{kin}}={\mathcal{H}}_{+}\oplus {\mathcal{H}}_{-}$,

$$\mathcal{D}=\left(\begin{array}{cc}{\mathcal{D}}_{++}& {\mathcal{D}}_{+-}\\ {\mathcal{D}}_{-+}& {\mathcal{D}}_{--}\end{array}\right),\begin{array}{cc}\hfill {\mathcal{D}}_{\mathrm{strict}}& ={\mathcal{D}}_{++},\hfill \\ \hfill {\mathcal{D}}_{\mathrm{eff},+}& ={\mathcal{D}}_{++}-{\mathcal{D}}_{+-}{\mathcal{D}}_{--}^{-1}{\mathcal{D}}_{-+}.\hfill \end{array}$$

(1)

In the quaternionic convention used below, the background-induced $\Theta$-odd mixing has ${\mathcal{D}}_{+-}=+\mathcal{B}$ and ${\mathcal{D}}_{-+}=-\mathcal{B}$, so the exact reduced operator is

$$\begin{array}{|c|}\hline {\mathcal{D}}_{\mathrm{eff},+}={\mathcal{D}}_{++}+\mathcal{B}{\mathcal{D}}_{--}^{-1}\mathcal{B}.\\ \hline\end{array}$$

(2)

With a gapped complementary block, this plus-sign identity produces a negative local effective field theory (EFT) insertion because ${\mathcal{D}}_{--}^{-1}\simeq -M^{-2}$. Here, M is the complementary-block separation scale controlling this expansion, as defined in Section 4.5; it is supplied reduction data, not an assumed physical particle mass.

The sequence from the supplied deformation to the retained insertion is

$$\underset{\mathrm{background}\mathrm{to}\mathrm{mixing}}{\underbrace{\delta {\mathcal{G}}_H^{\mu \nu }=E^{\mu \nu }{e}_2⟶\mathcal{B}W}}⟶\underset{\mathrm{mixing}\mathrm{to}\mathrm{retained}\mathrm{EFT}}{\underbrace{{\mathcal{D}}_{\mathrm{eff},+}⟶\delta {\mathcal{D}}_{\mathrm{eff},+}}}⟶G_F^{+}\mathrm{and}\mathrm{retained}\mathrm{loop}\mathrm{insertions}.$$

(3)

The first two braces are the manuscript-specific result. The final arrow only places the matched operator into standard scalar QFT. Dispersion or spectral response, renormalized $\langle T_{\mu \nu }\rangle$, background response kernels, backreaction, vacuum polarization, and phenomenology form a separate observable layer and are not calculated here.

The quaternionic and pseudo-Hermitian ingredients are deliberately bounded. Quaternionic quantum mechanics and field theory have a long history [5]; parity–time ($\mathcal{PT}$)-symmetric and pseudo-Hermitian frameworks are likewise established subjects [6,7,8,9,10,11,12]. Standard operator terminology calls A$\mathcal{PT}$-symmetric when $\mathcal{PT}A{\left(\mathcal{PT}\right)}^{-1}=A$ and anti-$\mathcal{PT}$-symmetric when the result is $-A$; related extensions and anti-pseudo-Hermitian distinctions are discussed in Refs. [13,14]. Recent pseudoreal QFT work addresses non-Hermitian multiplets and interactions [15]. We use established Schur/Feshbach reduction [16] and scalar QFT methods.

The manuscript-specific contribution is their controlled assembly for one quaternionically structured deformation component: a faithful auxiliary carrier with $\rho \left(\mathbb{H}\right)\cap \mathrm{Herm}(2,\mathbb{C})=\mathbb{R}I$; a family of $\Theta$-odd, $\Theta$-pseudo-Hermitian quaternionic mixings; a clean separation of microscopic anti-linear $\mathcal{PT}$ from the linear grading $\Theta$; a local-symbol real-spectrum and positive-metric audit; the explicit bridge $\delta {\mathcal{G}}^{\mu \nu }\to \mathcal{B}W$; the exact operator Schur sign; and the retained higher-derivative EFT operator that follows from it. A lower-derivative benchmark is retained only to display a transparent pole-mass shift and match the loop insertion coefficients.

Schur/Feshbach reduction itself is standard; the question here is not how to eliminate a block in general. We ask whether a quaternionically structured auxiliary principal-symbol deformation can generate a controlled retained scalar EFT insertion once strict projection and block reduction are distinguished. Quaternionic algebra supplies a structured realization of the grading and mixing relations, which could otherwise be postulated independently in a complex two-component model.

Proposition 2 makes this explicit: for unit imaginary $u,v\in \mathbb{H}$ with $\mathrm{Re}\left(u\overline{v}\right)=0$, ${\Theta }_u=-i\rho \left(u\right)$ and $\rho \left(v\right)$ obey $\{{\Theta }_u,\rho \left(v\right)\}=0$ together with the pseudo-Hermitian and projected-block relations in (14). The result here is the explicit quaternionic realization and its bridge from the supplied principal-symbol deformation to the retained EFT operator.

Section 2 fixes the principal-symbol and grading data. Section 3 states the Green-function distinction. Section 4 derives the principal-symbol EFT operator. Section 5 and Section 6 place it in the curved-background propagator and retained loop expansion. Section 7 records the completed and deferred layers. The appendices retain only technical adiabatic and $\Theta$-grading details. Earlier PTQ papers provide motivation but no companion result is an assumption of the derivation below [17,18,19,20].

The principal claims and scope boundaries are summarized in

Table 1. Claims and scope boundaries.

Established Here	Not Established Here
Auxiliary quaternionic principal-symbol deformation on real $g_{\mu \nu }$	Quaternionic spacetime metric or quantum-gravity completion
Strict versus Schur-reduced retained Green functions	Quaternionic exclusivity of Schur/Feshbach reduction
Quaternionic mixing matrix $W=\rho \left(e_2\right)$ and local-symbol metric audit	Universal Hilbert-space metric, Born rule, or interacting measure
$\delta {\mathcal{G}}^{\mu \nu }\to \mathcal{B}W\to -M^{-2}{\mathcal{B}}^2+\cdots$	Observable response, backreaction, or phenomenology
Matched retained scalar propagator and one-loop insertions	Novel tadpole, self-energy, $\mathrm{Tr}\log$, or all-order renormalization theorem

.

2. Quaternionic Principal-Symbol Background and Grading Data

2.1. Physical Geometry and Auxiliary Principal Symbol

Let $(M,g)$ be a smooth globally hyperbolic spacetime, or a locally hyperbolic patch sufficient for the adiabatic construction. The physical metric $g_{\mu \nu }$ is real and Lorentzian, $g^{\mu \nu }$ is its ordinary real inverse, ∇ is its Levi–Civita derivative, $R=R\left[g\right]$, and $\sqrt{-g}{d}^{4}x$ is the baseline volume element. We introduce the contravariant coefficient

$${\mathcal{G}}^{\mu \nu }\left(x\right):=g^{\mu \nu }\left(x\right)I+\delta {\mathcal{G}}^{\mu \nu }\left(x\right),\delta {\mathcal{G}}^{\mu \nu }\left(x\right)={\displaystyle \sum _{a=1}^3}{\epsilon }_a\left(x\right)e_a{X}_{\left(a\right)}^{\mu \nu }\left(x\right),$$

(4)

where $X_{\left(a\right)}^{\mu \nu }$ are real symmetric tensors and

$$e_i{e}_j=-{\delta }_{ij}+{ϵ}_{ijk}{e}_k,\overline{e_i}=-e_i$$

(5)

defines the quaternion algebra $\mathbb{H}$. Small deformation and adiabatic conditions are imposed locally, schematically

$$\parallel \delta \mathcal{G}\parallel \ll \parallel g^{-1}\parallel ,{\displaystyle \frac{|\nabla {\epsilon }_a|}{\left|k\right||{\epsilon }_a|+{\mu }_{\mathrm{IR}}}}\ll 1,$$

(6)

for momenta and background scales in the EFT window; ${\mu }_{\mathrm{IR}}$ is a reference infrared scale that keeps the displayed ratio nonsingular.

Equation (4) defines an auxiliary quaternionic-valued principal-symbol coefficient ${\mathcal{G}}^{\mu \nu }$; no series that inverts a quaternionic covariant tensor is used. The associated second-order principal operator is specified by

$${\mathcal{K}}_{\mathrm{PTQ}}^{\left(\mathrm{pr}\right)}:={\mathcal{G}}^{\mu \nu }{\nabla }_{\mu }{\nabla }_{\nu }.$$

(7)

The symbol ${\mathcal{K}}_{\mathrm{PTQ}}^{\left(\mathrm{pr}\right)}$ denotes only this d’Alembert-type second-order principal part. Lower-derivative terms required by a chosen ordering relative to the baseline measure are separate EFT/adiabatic data; they do not define a new geometry.

For the displayed quaternionic deformation component set

$$E^{\mu \nu }\left(x\right):={\epsilon }_2\left(x\right)X_{\left(2\right)}^{\mu \nu }\left(x\right),\delta {\mathcal{G}}_H^{\mu \nu }=E^{\mu \nu }{e}_2.$$

(8)

The subscript H labels the explicitly quaternionic component, not a second metric. Equations (4)–(8) specify the principal-symbol data: all index movement outside the declared $X_{\left(a\right)}^{\mu \nu }$ data uses only the real $g_{\mu \nu }$ and $g^{\mu \nu }$.

2.2. Faithful Quaternionic Carrier

Use the faithful complex representation

$$\rho :\mathbb{H}⟶M_2\left(\mathbb{C}\right),\rho (a+b{e}_1+c{e}_2+d{e}_3)=\left(\begin{array}{cc}a+ib& c+id\\ -c+id& a-ib\end{array}\right),$$

(9)

where $a,b,c,d\in \mathbb{R}$. Thus,

$$\rho \left(e_1\right)=\left(\begin{array}{cc}i& 0\\ 0& -i\end{array}\right),\rho \left(e_2\right)=\left(\begin{array}{cc}0& 1\\ -1& 0\end{array}\right),\rho \left(e_3\right)=\left(\begin{array}{cc}0& i\\ i& 0\end{array}\right),$$

(10)

and $\rho \left(e_i\right)\rho \left(e_j\right)=-{\delta }_{ij}I+{ϵ}_{ijk}\rho \left(e_k\right)$.

Proposition 1

(Quaternionic image and Hermiticity). For the representation (9),

$$\rho \left(\mathbb{H}\right)\cap \mathrm{Herm}(2,\mathbb{C})=\mathbb{R}I.$$

(11)

In particular, $i\rho \left(e_2\right)\notin \rho \left(\mathbb{H}\right)$, although it is Hermitian.

Proof. 

If $A=\rho (a+b{e}_1+c{e}_2+d{e}_3)$, then $A^{†}=\rho (a-b{e}_1-c{e}_2-d{e}_3)$. Hence $A=A^{†}$ forces $b=c=d=0$. The off-diagonal entries of $i\rho \left(e_2\right)$ also have the opposite phase pattern from those allowed by (9). □

Define the complex-linear internal grading and the quaternionically realized $\Theta$-odd deformation matrix by

$$\Theta :=-i\rho \left(e_1\right)=\left(\begin{array}{cc}1& 0\\ 0& -1\end{array}\right),P_{\Theta }:={\displaystyle \frac{I+\Theta }{2}},W:=\rho \left(e_2\right)=\left(\begin{array}{cc}0& 1\\ -1& 0\end{array}\right).$$

(12)

They satisfy

$${\Theta }^2=I,[\Theta ,i]=0,W^{†}=-W,\{\Theta ,W\}=0,W^{†}=\Theta W\Theta ,P_{\Theta }W{P}_{\Theta }=0.$$

(13)

Thus, W is $\Theta$-odd, meaning $\Theta W\Theta =-W$, and is $\Theta$-pseudo-Hermitian, meaning $W^{†}=\Theta W\Theta$; it is not Hermitian. These are linear-grading statements and are distinct from the anti-linear $\mathcal{PT}$ terminology defined next.

Proposition 2

(A quaternionic family of grading-odd mixings). Let $u,v\in \mathbb{H}$ be unit imaginary quaternions with $\mathrm{Re}\left(u\overline{v}\right)=0$, and set ${\Theta }_u=-i\rho \left(u\right)$ and $P_{u,+}=(I+{\Theta }_u)/2$. Then,

$$\begin{array}{cccccc}\hfill {\Theta }_u^{†}& ={\Theta }_u,\hfill & \hfill {\Theta }_u^2& =I,\hfill & \hfill \rho {\left(v\right)}^{†}& =-\rho \left(v\right),\hfill \\ \hfill \{{\Theta }_u,\rho \left(v\right)\}& =0,\hfill & \hfill {\Theta }_u\rho \left(v\right){\Theta }_u& =\rho {\left(v\right)}^{†},\hfill & \hfill P_{u,+}\rho \left(v\right)P_{u,+}& =0.\hfill \end{array}$$

(14)

This proposition supplies a quaternionic realization/family; no converse or exhaustive converse theorem is asserted.

2.3. Microscopic $\mathcal{PT}$ and the Linear Grading

Notation is fixed as follows:

$$\begin{array}{cc}\hfill \mathcal{PT}& =\mathrm{microscopic}\mathrm{anti-linear}\mathrm{transformation},\hfill \\ \hfill \theta & =\mathrm{spacetime}\mathrm{involution},\hfill \\ \hfill \Theta & =\mathrm{complex-linear}\mathrm{internal}\mathrm{grading}.\hfill \end{array}$$

(15)

We call an operator A$\mathcal{PT}$-symmetric under the microscopic anti-linear $\mathcal{PT}$ transformation if $\mathcal{PT}A{\left(\mathcal{PT}\right)}^{-1}=A$, and anti-$\mathcal{PT}$-symmetric if $\mathcal{PT}A{\left(\mathcal{PT}\right)}^{-1}=-A$. The qualifier “microscopic” identifies which $\mathcal{PT}$ transformation is meant: it includes the spacetime involution, coefficient pullback, and conjugation, and is not the internal grading $\Theta$. Pseudo-Hermiticity instead denotes an adjoint relation of the form $A^{†}=\eta A{\eta }^{-1}$ for an invertible Hermitian metric operator $\eta$ [8,9]. In the displayed internal relations below $\eta =\Theta ={\Theta }^{-1}$; “$\Theta$-pseudo-Hermitian” is therefore the precise term. In a local patch, microscopic $\mathcal{PT}$ acts with tensor pullback under $\theta$, complex conjugation, and quaternionic conjugation. In particular,

$$\mathcal{PT}i{\left(\mathcal{PT}\right)}^{-1}=-i,\mathcal{PT}W{\left(\mathcal{PT}\right)}^{-1}=-W.$$

(16)

By contrast, $\Theta$ does not act on the spacetime argument and commutes with explicit factors of i. Internal diagrammatic ${\mathbb{Z}}_2$ rules below are therefore derived only from $\Theta$.

Where this local transformation is invoked, we assume that the real baseline geometry is compatible with the involution on the patch, ${\theta }^{*}g=g$. The Levi–Civita derivative and the baseline scalar operator then transform covariantly. This is a local assumption, not a claim that an arbitrary curved spacetime admits a global $\mathcal{PT}$ isometry.

The title’s $\mathcal{PT}$-symmetric condition concerns the complete background deformation under the microscopic transformation, not W or ${\epsilon }_2$ in isolation. To express this condition, we introduce $b_{\mathrm{metric}}$ as the momentum-dependent mixing supplied by $E^{\mu \nu }$. At principal-symbol level the condition is realized when

$$k^{\theta }:={\left(d{\theta }_x^{-1}\right)}^{*}k,b_{\mathrm{metric}}(\theta x,k^{\theta })=-b_{\mathrm{metric}}(x,k),b_{\mathrm{metric}}(x,k):=E^{\mu \nu }\left(x\right)k_{\mu }{k}_{\nu },$$

(17)

where ${\theta }^{*}$ denotes pullback on spacetime tensors and $k^{\theta }$ is the separately defined induced cotangent momentum. Then, $b_{\mathrm{metric}}W$ is $\mathcal{PT}$-symmetric under the microscopic transformation. Equation (17) may arise from the combined profile $E^{\mu \nu }={\epsilon }_2{X}_{\left(2\right)}^{\mu \nu }$; it does not require ${\epsilon }_2$ alone to be odd.

2.4. Carrier and Retained-Sector Assumptions

The auxiliary doublet

$$\Psi \in \Gamma (E_{\mathrm{scalar}}\otimes {\mathbb{C}}^2),{\Psi }_{+}:=P_{\Theta }\Psi ,P_{\Theta }{\Psi }_{+}={\Psi }_{+}$$

(18)

is a kinematical carrier for $\rho \left(\mathbb{H}\right)$, $\Theta$, W, block operators, and their symbols. It is not quantized here as two independent complex fields, and no full interacting doublet functional integral is defined. The projector $P_{\Theta }$ first selects a one-dimensional complex carrier line. Choose a fixed basis vector $e_{+}$ spanning that line, with $P_{\Theta }{e}_{+}=e_{+}$. The physical scalar theory then uses its chosen real structure, on which

$${\Psi }_{+}\left(x\right)={\varphi }_{+}\left(x\right)e_{+},{\varphi }_{+}\left(x\right)\in \mathbb{R}.$$

(19)

Equivalently, in this chosen basis, the retained real slice is

$${\mathcal{H}}_{+}^{\mathbb{R}}:=\{{\Psi }_{+}={\varphi }_{+}{e}_{+}\in P_{\Theta }{\mathcal{H}}_{\mathrm{kin}}:{\varphi }_{+}\in \mathbb{R}\}.$$

(20)

Accordingly, $\Psi$ is the auxiliary doublet, ${\Psi }_{+}$ is its retained carrier vector, and ${\varphi }_{+}$ is only the physical real scalar coordinate on the retained slice, obtained either by strict truncation or after a declared Schur reduction. Thus, $P_{\Theta }$ alone is not asserted to turn an arbitrary complex component into a real number.

We assume a state and Feynman prescription for the retained operator, a $\Theta$-invariant state and regulator when grading rules are invoked, and a local soft/gapped regime when the heavy expansion is used. A compatible retained one-dimensional metric may be chosen as $G_{+}=1$. These assumptions are sufficient for the operator and Green-function construction; they are not a universal probability or quantization theorem.

3. Strict Projection Versus Schur Reduction

Let $Q_{\Theta }=I-P_{\Theta }$ and ${\mathcal{H}}_{\mathrm{kin}}={\mathcal{H}}_{+}\oplus {\mathcal{H}}_{-}$, with ${\mathcal{H}}_{+}=P_{\Theta }{\mathcal{H}}_{\mathrm{kin}}$. The auxiliary carrier operator has blocks

$$\begin{array}{cccc}\hfill {\mathcal{D}}_{++}& =P_{\Theta }\mathcal{D}{P}_{\Theta }{|}_{{\mathcal{H}}_{+}},\hfill & \hfill {\mathcal{D}}_{+-}& =P_{\Theta }\mathcal{D}{Q}_{\Theta },\hfill \\ \hfill {\mathcal{D}}_{-+}& =Q_{\Theta }\mathcal{D}{P}_{\Theta },\hfill & \hfill {\mathcal{D}}_{--}& =Q_{\Theta }\mathcal{D}{Q}_{\Theta }{|}_{{\mathcal{H}}_{-}}.\hfill \end{array}$$

(21)

Outer projectors make ${\mathcal{D}}_{++}$ an operator on ${\mathcal{H}}_{+}$ tautologically. They do not prove sector preservation; the latter requires ${\mathcal{D}}_{+-}={\mathcal{D}}_{-+}=0$ to the order retained.

Two Retained Inverses

Strict projection defines

$${\mathcal{D}}_{\mathrm{strict}}:={\mathcal{D}}_{++},G_F^{\mathrm{strict}}:={\left({\mathcal{D}}_{++}\right)}_F^{-1}.$$

(22)

Here and below the subscript F means that the inverse uses a specified state, boundary condition, and $i0$ prescription. Whenever ${\left({\mathcal{D}}_{--}\right)}_F^{-1}$ exists with the same prescription, it defines

$${\mathcal{D}}_{\mathrm{eff},+}:={\mathcal{D}}_{++}-{\mathcal{D}}_{+-}{\left({\mathcal{D}}_{--}\right)}_F^{-1}{\mathcal{D}}_{-+},G_F^{\mathrm{red}}:={\left({\mathcal{D}}_{\mathrm{eff},+}\right)}_F^{-1}.$$

(23)

These are different physical approximations unless the Schur correction vanishes or is beyond the declared accuracy.

Proposition 3

(Projected full inverse). Under the common inverse prescription just stated,

$$P_{\Theta }{\mathcal{D}}_F^{-1}{P}_{\Theta }{|}_{{\mathcal{H}}_{+}}={\left({\mathcal{D}}_{++}-{\mathcal{D}}_{+-}{\left({\mathcal{D}}_{--}\right)}_F^{-1}{\mathcal{D}}_{-+}\right)}_F^{-1}.$$

(24)

Thus, ${\left(P_{\Theta }\mathcal{D}{P}_{\Theta }\right)}^{-1}$ on ${\mathcal{H}}_{+}$ cannot be replaced unconditionally by $P_{\Theta }{\mathcal{D}}^{-1}{P}_{\Theta }$.

Proof. 

Block Gaussian elimination gives (24). The operator domains and inverse prescription must be common to all blocks; the statement is not an identity between unspecified formal reciprocals. □

At local-symbol level, with blocks $Q_{AB}(x,k)$, the same distinction is

$$\begin{array}{cccc}\hfill Q_{\mathrm{strict}}& =Q_{++},\hfill & \hfill G_F^{\mathrm{strict}}\left(k\right)& ={\displaystyle \frac{i}{Q_{\mathrm{strict}}+i0}},\hfill \\ \hfill Q_{\mathrm{eff},+}& =Q_{++}-Q_{+-}{Q}_{--}^{-1}{Q}_{-+},\hfill & \hfill G_F^{\mathrm{red}}\left(k\right)& ={\displaystyle \frac{i}{Q_{\mathrm{eff},+}+i0}}.\hfill \end{array}$$

(25)

The local symbol is an audit and EFT tool; it does not erase derivative ordering when coefficients vary. Section 4 now supplies the quaternionically realized off-block operator and verifies that its operator and symbol signs agree.

4. Quaternionic Pseudo-Hermitian Core and Principal-Symbol EFT

4.1. Local Doublet Symbol and Positive Metric

For real local-symbol data $a,d,b$, define

$$Q=a{P}_{+}+d{P}_{-}+bW=\left(\begin{array}{cc}a& b\\ -b& d\end{array}\right),P_{+}=P_{\Theta },$$

(26)

so that

$$Q_{+-}=+b,Q_{-+}=-b,Q^{†}=\Theta Q\Theta .$$

(27)

Its eigenvalues are

$${\lambda }_{\pm }={\displaystyle \frac{a+d}{2}}\pm \sqrt{{\displaystyle \frac{{(a-d)}^2}{4}}-b^2},$$

(28)

and the nondegenerate real-spectrum domain is

$$|a-d|>2\left|b\right|.$$

(29)

To exhibit a positive metric for the local doublet symbol on this domain, define

$$G_{\mathrm{full}}=\left(\begin{array}{cc}1& r\\ r& 1\end{array}\right),r={\displaystyle \frac{2b}{a-d}},$$

(30)

which obeys

$$Q^{†}{G}_{\mathrm{full}}=G_{\mathrm{full}}Q,G_{\mathrm{full}}>0⟺\left|r\right|<1⟺|a-d|>2\left|b\right|.$$

(31)

For $b\ne 0$, $[G_{\mathrm{full}},\Theta ]\ne 0$. Consequently, $P_{\Theta }$ is an algebraic grading projector, not automatically the $G_{\mathrm{full}}$-orthogonal projector.

The scope of (30) is pointwise in the local symbol. Since r may depend on $(x,k)$, its momentum dependence may represent position-space nonlocality. If locally $G_{\mathrm{full}}=S^{†}S$, then $h=SQ{S}^{-1}$ is Hermitian pointwise in (29); this does not construct a global Hilbert-space metric, a functional measure, or an interacting quasi-Hermitian QFT. After reduction to a real one-dimensional retained symbol, one may take the bounded retained metric $G_{+}=1$.

4.2. Two Off-Block Mixings with Different Purposes

We introduce $b_{\mathrm{loc}}$ only as a lower-derivative pedagogical benchmark: a constant mixing provides a transparent pole-shift example and a simple loop-normalization and subtraction check. It is not generated by the principal-symbol deformation and is not the background-induced result. The quantity $b_{\mathrm{metric}}$ is the mixing derived from $E^{\mu \nu }$ and enters the manuscript-specific higher-derivative result. In $b_{\mathrm{loc}}$, $\mu$ is a reference mass scale used to give the mixing mass dimension two, and $\epsilon$ sets its dimensionless amplitude:

$$\begin{array}{cccc}\hfill b_{\mathrm{loc}}& :=\epsilon {\mu }^2,\hfill & \hfill & \mathrm{lower-derivative}\mathrm{pedagogical}\mathrm{benchmark},\hfill \\ \hfill b_{\mathrm{metric}}(x,k)& :=E^{\mu \nu }\left(x\right)k_{\mu }{k}_{\nu },\hfill & \hfill & \mathrm{background-induced}\mathrm{principal-symbol}\mathrm{mixing}.\hfill \end{array}$$

(32)

For the principal-symbol mixing example $a=k^2-m^2$, $d=k^2-M^2$, and $b=E^{\mu \nu }{k}_{\mu }{k}_{\nu }$, with the same $k^2$ convention in both diagonal symbols, the local real-spectrum condition (29) becomes

$$2\left|E^{\mu \nu }{k}_{\mu }{k}_{\nu }\right|<|M^2-m^2|.$$

(33)

Together with $|k^2|\ll M^2$ and the small/adiabatic assumptions (6), this defines a controlled overlap window for this principal-symbol example. It is neither necessary for every ordering nor sufficient for global stability.

4.3. From the Background to an Off-Block Differential Operator

Applying $\rho$ to (8) gives

$$\rho \left(\delta {\mathcal{G}}_H^{\mu \nu }\right)=E^{\mu \nu }W,\rho \left(\delta {\mathcal{G}}_H^{\mu \nu }\right)k_{\mu }{k}_{\nu }=b_{\mathrm{metric}}(x,k)W.$$

(34)

At differential-operator level, a general representative may contain lower-derivative terms with the same principal symbol. To fix the differential ordering, we choose the representative

$${\mathcal{B}}_{\mathrm{sym}}\varphi :={\nabla }_{\mu }\left(E^{\mu \nu }\left(x\right){\nabla }_{\nu }\varphi \right)=E^{\mu \nu }{\nabla }_{\mu }{\nabla }_{\nu }\varphi +\left({\nabla }_{\mu }{E}^{\mu \nu }\right){\nabla }_{\nu }\varphi ,$$

(35)

for real symmetric $E^{\mu \nu }$. Hereafter, $\mathcal{B}$ in the displayed operator construction denotes this ${\mathcal{B}}_{\mathrm{sym}}$ representative; other orderings with the same principal symbol require their own adjoint and lower-order audit. With

$${(f,h)}_g:={\int }_M{d}^{4}x\sqrt{-g}{f}^{*}h,$$

(36)

integration by parts for compact support, or boundary conditions that remove the boundary term, gives

$${(f,{\mathcal{B}}_{\mathrm{sym}}h)}_g=-{\int }_M{d}^{4}x\sqrt{-g}{\left({\nabla }_{\mu }f\right)}^{*}{E}^{\mu \nu }{\nabla }_{\nu }h={({\mathcal{B}}_{\mathrm{sym}}f,h)}_g.$$

(37)

Thus, ${\mathcal{B}}_{\mathrm{sym}}^{†}={\mathcal{B}}_{\mathrm{sym}}$ only in the formal adjoint sense under these assumptions; essential self-adjointness and a universal operator domain are not asserted.

Our local inverse-propagator symbol convention is ${\sigma }_{\mathrm{inv}}\left({\nabla }_{\mu }{\nabla }_{\nu }\right)=k_{\mu }{k}_{\nu }$; Fourier-transform minus signs are absorbed into the definition of the quadratic operator $\mathcal{D}$. Consequently,

$$\sigma \left({\mathcal{B}}_{\mathrm{sym}}\right)(x,k)=E^{\mu \nu }\left(x\right)k_{\mu }{k}_{\nu }=b_{\mathrm{metric}}(x,k).$$

(38)

Thus, the carrier deformation is $\mathcal{B}W$. When $E^{\mu \nu }\left(x\right)$ varies, $\mathcal{B}$ is a differential operator with position-dependent coefficients—it is not multiplication by $b_{\mathrm{metric}}$.

The spacetime operator ${\mathcal{B}}_{\mathrm{sym}}$ and internal matrix W act on different factors. Using (37) and (13),

$${\left({\mathcal{B}}_{\mathrm{sym}}W\right)}^{†}=-{\mathcal{B}}_{\mathrm{sym}}W,\Theta \left({\mathcal{B}}_{\mathrm{sym}}W\right)\Theta =-{\mathcal{B}}_{\mathrm{sym}}W.$$

(39)

Hence, ${\left({\mathcal{B}}_{\mathrm{sym}}W\right)}^{†}=\Theta \left({\mathcal{B}}_{\mathrm{sym}}W\right)\Theta$. This extends the pseudo-Hermitian audit from the frozen principal symbol only to the declared formally symmetric off-block representative; it is not a theorem for all orderings or for a full interacting carrier theory. The diagonal blocks retain their separately declared assumptions.

Because W has the off-diagonal signs in (12), the complete carrier operator takes the form

$$\mathcal{D}=\left(\begin{array}{cc}{\mathcal{D}}_{++}& +\mathcal{B}\\ -\mathcal{B}& {\mathcal{D}}_{--}\end{array}\right),{\mathcal{D}}_{+-}=+\mathcal{B},{\mathcal{D}}_{-+}=-\mathcal{B}.$$

(40)

Any additional off-block term would require its own declared deformation component and is not included in the present single-component construction.

Proposition 4

($\mathcal{PT}$-symmetric Schur re-entry of a $\Theta$-odd quaternionic mixing). Assume that ${\theta }^{*}g=g$ on the local patch, that the boundary/state prescription respects the declared transformation there, and, including coefficient transformation and tensor pullback, that

$$\mathcal{PT}\mathcal{B}{\left(\mathcal{PT}\right)}^{-1}=-\mathcal{B},\mathcal{PT}W{\left(\mathcal{PT}\right)}^{-1}=-W,\mathcal{PT}{\mathcal{D}}_{--}{\left(\mathcal{PT}\right)}^{-1}={\mathcal{D}}_{--}.$$

(41)

Then, $\mathcal{B}W$ and $\mathcal{B}{\left({\mathcal{D}}_{--}\right)}_F^{-1}\mathcal{B}$ are $\mathcal{PT}$-symmetric under the declared microscopic transformation. Strict $P_{\Theta }$ projection removes the Θ-odd retained block at linear order, whereas Schur reduction regenerates a retained $\mathcal{PT}$-symmetric operator at second order.

Proof. 

The spacetime operator and internal matrix act on different factors, so the two minus signs give $\mathcal{PT}\left(\mathcal{B}W\right){\left(\mathcal{PT}\right)}^{-1}=\mathcal{B}W$. The common state/boundary prescription implies $\mathcal{PT}{\left({\mathcal{D}}_{--}\right)}_F^{-1}{\left(\mathcal{PT}\right)}^{-1}={\left({\mathcal{D}}_{--}\right)}_F^{-1}$ under the declared condition. The two factors of $\mathcal{B}$ then make the Schur term even. Finally, $P_{\Theta }W{P}_{\Theta }=0$ proves the strict linear statement. □

At principal-symbol level, the first condition in (41) reduces to (17). This proposition does not assert that all $\mathcal{PT}$-symmetric quaternionic backgrounds have this form, nor does it identify the anti-linear microscopic symmetry with the linear grading. For the chosen representative, covariance of ∇ under the local isometry and the odd pullback of $E^{\mu \nu }$ imply $\mathcal{PT}{\mathcal{B}}_{\mathrm{sym}}{\left(\mathcal{PT}\right)}^{-1}=-{\mathcal{B}}_{\mathrm{sym}}$. Thus, $\mathcal{B}$ and W are individually anti-$\mathcal{PT}$-symmetric, while $\mathcal{B}W$ and the quadratic Schur term are $\mathcal{PT}$-symmetric under the declared microscopic transformation [13,14]. These references support the standard terminology, not the manuscript-specific Schur result. The anti-linear classifications do not replace their separate $\Theta$-grading and $\Theta$-pseudo-Hermiticity properties.

4.4. Exact Operator Schur Sign

Substituting (40) into (23) gives

$$\begin{array}{cc}\hfill {\mathcal{D}}_{\mathrm{eff},+}& ={\mathcal{D}}_{++}-{\mathcal{D}}_{+-}{\left({\mathcal{D}}_{--}\right)}_F^{-1}{\mathcal{D}}_{-+}\hfill \\ \hfill & =\begin{array}{|c|}\hline {\mathcal{D}}_{++}+\mathcal{B}{\left({\mathcal{D}}_{--}\right)}_F^{-1}\mathcal{B}\\ \hline\end{array}.\hfill \end{array}$$

(42)

The plus sign is fixed by ${\mathcal{D}}_{+-}=+\mathcal{B}$ and ${\mathcal{D}}_{-+}=-\mathcal{B}$. Applying the local symbol map ${\mathcal{D}}_{++}\mapsto a$, ${\mathcal{D}}_{--}\mapsto d$, and $\mathcal{B}\mapsto b$ yields

$$Q_{\mathrm{eff},+}=a+{\displaystyle \frac{b^2}{d}},Q_{\mathrm{strict}}=a,$$

(43)

which agrees with the direct Schur complement of (26).

4.5. Complementary-Block Separation and Local Expansion

Write the complementary operator as

$${\mathcal{D}}_{--}={\mathcal{K}}_{-}-M^2,$$

(44)

where ${\mathcal{K}}_{-}$ contains the declared soft covariant kinetic term and may include specified curvature and lower-derivative adiabatic structures. Here, M is the complementary-block separation scale: it separates the invertible complementary block from the soft operator ${\mathcal{K}}_{-}$ in the declared local EFT patch. Both ${\mathcal{D}}_{--}$ and M are supplied reduction inputs; they are not determined by $\delta {\mathcal{G}}_H^{\mu \nu }$ alone. Thus, M controls block elimination and is not assumed to be the mass of an independently propagating particle. The expansion requires $\parallel {\mathcal{K}}_{-}\parallel /M^2\ll 1$ in the already-declared local-symbol/soft EFT sense, not a global Lorentzian operator-norm or spectral theorem:

$${\left({\mathcal{D}}_{--}\right)}_F^{-1}=-M^{-2}-M^{-4}{\mathcal{K}}_{-}+\mathcal{O}\left(M^{-6}{\mathcal{K}}_{-}^2\right).$$

(45)

Consequently,

$$\begin{array}{|c|}\hline {\mathcal{D}}_{\mathrm{eff},+}={\mathcal{D}}_{++}-M^{-2}{\mathcal{B}}^2-M^{-4}\mathcal{B}{\mathcal{K}}_{-}\mathcal{B}+\mathcal{O}\left(M^{-6}\mathcal{B}{\mathcal{K}}_{-}^2\mathcal{B}\right)\\ \hline\end{array}.$$

(46)

The dimensions are consistent: $\left[\mathcal{D}\right]=\left[\mathcal{B}\right]=\left[{\mathcal{K}}_{-}\right]=2$, so every displayed correction has mass dimension two. With the canonical four-dimensional scalar dimension, the corresponding quadratic action term ${\varphi }_{+}{\mathcal{B}}_{\mathrm{sym}}^2{\varphi }_{+}/M^2$ is a dimension-six EFT operator. No convergence claim is made beyond the local soft/gapped and adiabatic regime. If this separation fails, the local series is not justified; the exact resolvent in (42) remains applicable only where its prescribed inverse exists.

4.6. Principal-Symbol Matching and Position-Space Ordering

Freeze $E^{\mu \nu }$ within a local adiabatic patch. From (38) and (46),

$$\delta Q_{\mathrm{eff}}^{\mathrm{metric}}(x,k)=-{\displaystyle \frac{1}{M^2}}{\left[E^{\mu \nu }{k}_{\mu }{k}_{\nu }\right]}^2+\mathcal{O}\left(M^{-4}\right).$$

(47)

This sign is not inserted by hand: it follows from the exact plus sign in (42) and the gapped inverse $d^{-1}\simeq -M^{-2}$. It is identical to $Q_{\mathrm{eff},+}=a+b_{\mathrm{metric}}^2/d$ with $d\simeq -M^2$. The background-induced principal-symbol mixing therefore generates a retained four-derivative principal-symbol operator beginning at order $M^{-2}$.

At leading frozen-coefficient order its position-space form is schematically

$$\delta {\mathcal{D}}_{\mathrm{eff},+}^{\left(M^{-2}\right)}\simeq -{\displaystyle \frac{1}{M^2}}{\left(E^{\mu \nu }{\nabla }_{\mu }{\nabla }_{\nu }\right)}^2.$$

(48)

For varying $E^{\mu \nu }\left(x\right)$ the correct expression is the composition $-M^{-2}{\mathcal{B}}_{\mathrm{sym}}^2$; coefficients must not be commuted through derivatives. Expanding that composition organizes, without requiring their full coefficients here,

• A leading $EE{\nabla }^4$ structure;
• $E(\nabla E){\nabla }^3$ terms;
• $E\left({\nabla }^{2}E\right){\nabla }^2$ terms;
• Covariant-derivative reordering and curvature commutators.

• These terms are ordered by (6). Thus, the retained EFT remains locally controlled in the declared regime without pretending that $b_{\mathrm{metric}}(x,k)$ is a multiplication operator in position space.

4.7. Mass-like Benchmark and Matching Coefficients

For the separate lower-derivative benchmark

$$a=k^2-m^2,d=k^2-M^2,b=b_{\mathrm{loc}}=\epsilon {\mu }^2,\left|k^2\right|\ll M^2,$$

(49)

one has

$${\displaystyle \frac{1}{d}}=-{\displaystyle \frac{1}{M^2}}-{\displaystyle \frac{k^2}{M^4}}+\mathcal{O}\left(M^{-6}\right)$$

(50)

and therefore

$$Q_{\mathrm{eff},+}=k^2-m^2-{\displaystyle \frac{b_{\mathrm{loc}}^2}{M^2}}-{\displaystyle \frac{b_{\mathrm{loc}}^2{k}^2}{M^4}}+\mathcal{O}\left(M^{-6}\right).$$

(51)

The leading physical pole shift is

$$\Delta m_{\mathrm{pole}}^2=+{\displaystyle \frac{{\epsilon }^2{\mu }^4}{M^2}}.$$

(52)

To compare the lower-derivative insertion coefficients, introduce the generic parameterization below. Here, $M_{\epsilon }$ supplies the overall mass dimension, and $\Lambda$ is only a reference EFT scale used to express the momentum expansion coefficients. Unlike the complementary-block elimination scale M, $\Lambda$ is not a second complementary-sector gap; no equality between M and $\Lambda$ is assumed. The generic retained insertion is

$$\delta Q_{\epsilon }\left(k\right)={\epsilon }^2{M}_{\epsilon }^2\left(c_0+c_2{\displaystyle \frac{k^2}{{\Lambda }^2}}+\cdots \right),$$

(53)

and the mass-like benchmark fixes

$$M_{\epsilon }^2{c}_0=-{\displaystyle \frac{{\mu }^4}{M^2}},{\displaystyle \frac{M_{\epsilon }^2{c}_2}{{\Lambda }^2}}=-{\displaystyle \frac{{\mu }^4}{M^4}}.$$

(54)

Equations (47) and (54) have different roles: the former is the actual background-induced higher-derivative result, while the latter is a pedagogical lower-derivative match.

4.8. Retained Determinants and the Formal Carrier Identity

The physical retained one-loop quantities are

$${\Gamma }_{\mathrm{strict}}^{\left(1\right)}={\displaystyle \frac{i}{2}}{\mathrm{Tr}}_{+}\log {\mathcal{D}}_{++},{\Gamma }_{\mathrm{red},+}^{\left(1\right)}={\displaystyle \frac{i}{2}}{\mathrm{Tr}}_{+}\log {\mathcal{D}}_{\mathrm{eff},+},$$

(55)

under the manuscript’s retained state and regulator assumptions. Separately, if the indicated inverses exist, a finite-mode or block-compatible Gaussian regularization is used, and all blocks share a boundary/state prescription, block elimination gives the algebraic identity

$$det\mathcal{D}=det{\mathcal{D}}_{--}det{\mathcal{D}}_{\mathrm{eff},+}$$

(56)

To avoid implying a fully defined interacting auxiliary functional integral, we denote the associated block-elimination expression only formally as

$${\Gamma }_{\mathrm{aux},\mathrm{formal}}^{\left(1\right)}:={\displaystyle \frac{i}{2}}{\mathrm{Tr}}_{-}\log {\mathcal{D}}_{--}+{\displaystyle \frac{i}{2}}{\mathrm{Tr}}_{+}\log {\mathcal{D}}_{\mathrm{eff},+}.$$

(57)

No regulator-independent multiplicative-determinant theorem is asserted, and no full pseudo-Hermitian interacting functional measure is constructed. The formal carrier identity does not promote $\Psi$ to a physical doublet QFT.

5. Projected Green Functions on a Curved Background

Choose explicitly either the strict operator ${\mathcal{D}}_{+}={\mathcal{D}}_{++}$ or the reduced operator ${\mathcal{D}}_{+}={\mathcal{D}}_{\mathrm{eff},+}$ at the declared EFT order. In the reduced case, the local physical action uses the EFT truncation, for example (46); the exact state-prescribed Schur/Feynman kernel (42) remains the parent resolvent expression when the local expansion is not invoked. The retained Green function is then expanded perturbatively in this local EFT insertion in Section 5.2. The physical quadratic action is only for the retained real scalar,

$$S_2\left[{\varphi }_{+}\right]=-{\displaystyle \frac{1}{2}}{\int }_M{d}^{4}x\sqrt{-g}{\varphi }_{+}{\mathcal{D}}_{+}{\varphi }_{+}.$$

(58)

Any lower-derivative ordering terms needed to make a chosen second-order baseline formally symmetric with respect to $\sqrt{-g}{d}^{4}x$ are included in ${\mathcal{D}}_{+}$. Under this formal-symmetry assumption and the already-declared boundary conditions, variation gives the retained free equation of motion

$${\displaystyle \frac{\delta S_2}{\delta {\varphi }_{+}}}=0⟹{\mathcal{D}}_{+}{\varphi }_{+}=0.$$

(59)

No quaternionic volume element is used. The bi-distribution $G_F^{+}$ defined below is the Feynman inverse associated with this retained equation under the declared state and $i0$ prescription.

5.1. Feynman Inverse and State Prescription

For a specified Hadamard state, or an adiabatic state of sufficient order where applicable, and a common $i0$ prescription, the retained Feynman bi-distribution is defined by

$$G_F^{+}(x,x^{\prime }):={}_G\langle 0|T\left\{{\varphi }_{+}\left(x\right){\varphi }_{+}\left(x^{\prime }\right)\right\}|0\rangle ,{\mathcal{D}}_{+}^{\left(x\right)}{G}_F^{+}(x,x^{\prime })=-{\delta }_g(x,x^{\prime }),$$

(60)

where ${\delta }_g(x,x^{\prime })={\delta }^{\left(4\right)}(x-x^{\prime })/\sqrt{-g\left(x\right)}$ and, in the one-dimensional retained sector, $G_{+}=1$ is sufficient. Equation (60), rather than a reciprocal of a quaternionic scalar, is the primary definition.

In a normal neighborhood, the leading local-momentum representation is

$$G_F^{+}(x,x^{\prime })\simeq \int {\displaystyle \frac{d^{4}k}{{\left(2\pi \right)}^4}}\mathcal{A}(x,x^{\prime };k)e^{i{k}_{\mu }{\sigma }^{\mu }(x,x^{\prime })}{\displaystyle \frac{i}{Q_{+}(x,k)+i0}}+\mathcal{O}(\nabla R,\nabla E),$$

(61)

where ${\sigma }^{\mu }$ is the tangent separation constructed from the real metric, $\mathcal{A}$ contains the usual transport/measure factors, and

$$Q_{+}(x,k)=g^{\mu \nu }{k}_{\mu }{k}_{\nu }-m^2-\xi R+\delta Q_{+}(x,k).$$

(62)

For a reduced calculation, $\delta Q_{+}$ includes (47) and the declared adiabatic corrections. Curvature and derivative terms can be organized by covariant symbol or heat-kernel methods [21,22]; only the amount needed to define the local EFT expansion is used here.

The state and $i0$ prescription are part of the inverse data. On a general curved spacetime, no global microscopic-$\mathcal{PT}$ isometry is assumed; the symmetry audit is local to a patch admitting the involution $\theta$ and the transformation law (41). In time-dependent patches, the adiabatic state is selected by a slowly varying positive-frequency branch. Strong particle production or a gap closure lies outside the approximation, because either invalidates the local derivative expansion.

5.2. Retained EFT Insertion

Write ${\mathcal{D}}_{+}={\mathcal{D}}_{0,+}+\delta {\mathcal{D}}_{+}$ with both inverses defined by the same state prescription. The standard resolvent identity gives

$$G_F^{+}=G_{0,F}^{+}-G_{0,F}^{+}\delta {\mathcal{D}}_{+}{G}_{0,F}^{+}+\mathcal{O}\left({\left(\delta {\mathcal{D}}_{+}\right)}^2\right),G_{0,F}^{+}:={\left({\mathcal{D}}_{0,+}\right)}_F^{-1}.$$

(63)

For the principal-symbol deformation at leading gapped order,

$$\delta {\mathcal{D}}_{+}=-M^{-2}{\mathcal{B}}^2+\mathcal{O}\left(M^{-4}\right),$$

(64)

so (63) is the final interface from the auxiliary background to retained propagation. The induced higher-derivative term is treated perturbatively within the EFT window; it is not resummed to infer extra high-energy poles outside that window.

At local-symbol level,

$${\displaystyle \frac{i}{Q_0+\delta Q_{+}+i0}}={\displaystyle \frac{i}{Q_0+i0}}-{\displaystyle \frac{i\delta Q_{+}}{{(Q_0+i0)}^2}}+\mathcal{O}\left(\delta Q_{+}^2\right).$$

(65)

All background coefficients remain slowly varying until the local calculation is assembled; restoring x dependence does not authorize commuting them through derivatives.

5.3. Linear Grading Selection Rule

The diagrammatic ${\mathbb{Z}}_2$ statement is conventional once the independent linear grading is declared. If

$$\Theta {\Phi }_A\left(x\right){\Theta }^{-1}={\eta }_A{\Phi }_A\left(x\right),{\eta }_A=\pm 1,$$

(66)

and the state, interaction, and regulator are $\Theta$-invariant, a correlator with odd net external $\Theta$-grade vanishes. Retained external legs and traces are restricted by $P_{\Theta }$; in a strict calculation the propagator is the sector inverse (22), whereas in a reduced calculation the complementary block contribution has already re-entered through (64). The proof and compact graph bookkeeping appear in Appendix B. Neither spacetime parity signs, Levi–Civita pseudotensor signs, anti-linear conjugation, nor explicit factors of i alter this internal $\Theta$ grade.

6. Matched Retained Scalar Loop Insertion

This section places the retained operator from Section 4 in ordinary scalar perturbation theory.

Consider the retained field $\varphi ={\varphi }_{+}$ with

$$S\left[\varphi \right]=S_2\left[\varphi \right]-\int d^{4}x\sqrt{-g}\left({\displaystyle \frac{g_0}{3!}}{\varphi }^3+{\displaystyle \frac{{\lambda }_0}{4!}}{\varphi }^4\right).$$

(67)

In a frozen local patch, write $Q_{+}=Q_0+\delta Q_{+}$, $Q_0=k^2-m^2$. The two-point insertion convention consistent with (65) is

$$G_0\left(k\right)\left[i\delta Q_{+}\left(k\right)\right]G_0\left(k\right)=-{\displaystyle \frac{i\delta Q_{+}\left(k\right)}{{(Q_0\left(k\right)+i0)}^2}},G_0\left(k\right)={\displaystyle \frac{i}{Q_0\left(k\right)+i0}}.$$

(68)

For the actual principal-symbol deformation,

$$\delta Q_{+}^{\mathrm{metric}}\left(k\right)=-{\displaystyle \frac{{\left[E^{\mu \nu }{k}_{\mu }{k}_{\nu }\right]}^2}{M^2}}+\mathcal{O}\left(M^{-4}\right),$$

(69)

whereas the lower-derivative benchmark uses (53) with the coefficients (54).

6.1. Tadpole and Cubic Self-Energy

The quartic tadpole is

$${\Pi }_{\mathrm{tad}}^{\left(1\right)}\left(x\right)={\displaystyle \frac{{\lambda }_0}{2}}{G}_F^{+}(x,x),$$

(70)

or, in local momentum organization through first insertion order,

$${\Pi }_{\mathrm{tad}}^{\left(1\right)}={\displaystyle \frac{{\lambda }_0}{2}}\int {\displaystyle \frac{d^{4}p}{{\left(2\pi \right)}^4}}\left[{\displaystyle \frac{i}{Q_0\left(p\right)+i0}}-{\displaystyle \frac{i\delta Q_{+}\left(p\right)}{{(Q_0\left(p\right)+i0)}^2}}\right]+\mathcal{O}\left(\delta Q_{+}^2\right).$$

(71)

The retained internal trace is trivial for the one-dimensional sector.

For the cubic interaction, let ${\Sigma }_{+}$ enter $G^{-1}=G_0^{-1}-{\Sigma }_{+}$ on ${\mathcal{H}}_{+}$. Then,

$${\Sigma }_{+}^{\left(1\right)}\left(k\right)={\displaystyle \frac{g_0^2}{2}}\int {\displaystyle \frac{d^{4}p}{{\left(2\pi \right)}^4}}{G}_F^{+}\left(p\right)G_F^{+}(k-p).$$

(72)

Writing $A\left(p\right)=Q_0\left(p\right)+i0$, its part linear in the matched insertion is

$$\delta {\Sigma }_{+}^{\left(1\right)}\left(k\right)={\displaystyle \frac{g_0^2}{2}}\int {\displaystyle \frac{d^{4}p}{{\left(2\pi \right)}^4}}\left[{\displaystyle \frac{\delta Q_{+}\left(p\right)}{A{\left(p\right)}^{2}A(k-p)}}+{\displaystyle \frac{\delta Q_{+}(k-p)}{A\left(p\right)A{(k-p)}^2}}\right].$$

(73)

Here, the displayed sign follows directly from $(-i\delta Q/A^2)(i/A^{\prime })=+\delta Q/\left(A^2{A}^{\prime }\right)$ under the definition (72); alternative conventions that define the 1PI amplitude as $i\Sigma$ redistribute the overall factor of i but not the insertion rule (68). These formulas apply to (69) as a derivative EFT insertion or to the mass-like benchmark, with curvature/gradient corrections restored by the adiabatic organization.

6.2. Retained Functional Trace and Local Counterterm Check

For either declared retained operator, the regulated one-loop action is

$${\Gamma }_{+}^{\left(1\right)}={\displaystyle \frac{i}{2}}{\mathrm{Tr}}_{+}\log {\mathcal{D}}_{+},\delta {\Gamma }_{+}^{\left(1\right)}={\displaystyle \frac{i}{2}}{\mathrm{Tr}}_{+}\left({\mathcal{D}}_{0,+}^{-1}\delta {\mathcal{D}}_{+}\right)+\mathcal{O}\left(\delta {\mathcal{D}}_{+}^2\right).$$

(74)

This is ${\Gamma }_{\mathrm{strict}}^{\left(1\right)}$ or ${\Gamma }_{\mathrm{red},+}^{\left(1\right)}$ according to the operator chosen; it is not the formal auxiliary quantity (57).

As a local subtraction check for the lower-derivative benchmark, take the constant part $\delta Q_{+}={\epsilon }^2{c}_0{M}_{\epsilon }^2$. After Wick rotation, ${\mu }_{\mathrm{DR}}$ denotes the dimensional-regularization scale, distinct from the benchmark scale $\mu$. Define, for $d=4-2{ϵ}_{\mathrm{DR}}$,

$$I_E\left(m^2\right)={\mu }_{\mathrm{DR}}^{4-d}\int {\displaystyle \frac{d^d{p}_E}{{\left(2\pi \right)}^d}}{\displaystyle \frac{1}{p_E^2+m^2}},{\displaystyle \frac{1}{{\overline{ϵ}}_{\mathrm{DR}}}}:={\displaystyle \frac{1}{{ϵ}_{\mathrm{DR}}}}-{\gamma }_E+\log 4\pi .$$

(75)

Dimensional regularization gives

$$I_E\left(m^2\right)=-{\displaystyle \frac{m^2}{16{\pi }^2}}\left[{\displaystyle \frac{1}{{\overline{ϵ}}_{\mathrm{DR}}}}+1-\log \left({\displaystyle \frac{m^2}{{\mu }_{\mathrm{DR}}^2}}\right)\right]+\mathcal{O}\left({ϵ}_{\mathrm{DR}}\right),$$

(76)

and the divergent Euclidean local density is

$${\left.{\displaystyle \frac{\delta {\Gamma }_{+,E}^{\left(1\right)}}{V}}\right|}_{\mathrm{div}}=-{\displaystyle \frac{m^2}{32{\pi }^2{\overline{ϵ}}_{\mathrm{DR}}}}\delta Q_{+}.$$

(77)

It is cancelled, in this convention, by the opposite local counterterm. This check demonstrates the ordinary local loop interface for a supplied retained insertion. It is not an all-order counterterm-closure theorem, and no observable is extracted from it. The explicitly evaluated dimensional-regularization divergence is therefore only the lower-derivative benchmark/subtraction check. The principal-symbol insertion requires the corresponding local higher-derivative counterterm basis; its complete curved-space renormalization is not part of this interface calculation.

7. Discussion and Conclusions

The manuscript closes two layers of the QFT-on-background construction. In Layer A, the real physical geometry $g_{\mu \nu }$ is supplemented only by the auxiliary coefficient $\delta {\mathcal{G}}_H^{\mu \nu }=E^{\mu \nu }{e}_2$, whose faithful carrier image produces the quaternionically realized off-diagonal operator $\mathcal{B}W$.

In Layer B, the quaternionic signs fix ${\mathcal{D}}_{+-}=+\mathcal{B}$ and ${\mathcal{D}}_{-+}=-\mathcal{B}$, hence

$$\mathcal{B}W⟶{\mathcal{D}}_{\mathrm{eff},+}={\mathcal{D}}_{++}+\mathcal{B}{\mathcal{D}}_{--}^{-1}\mathcal{B}⟶\delta {\mathcal{D}}_{\mathrm{eff},+}=-M^{-2}{\mathcal{B}}^2-M^{-4}\mathcal{B}{\mathcal{K}}_{-}\mathcal{B}+\cdots .$$

(78)

The frozen symbol gives $-{\left[E^{\mu \nu }{k}_{\mu }{k}_{\nu }\right]}^2/M^2$ and the varying-coefficient result retains the exact composition $-M^{-2}{\mathcal{B}}^2$. This is the principal manuscript-specific result. The same retained operator then enters the ordinary resolvent, tadpole, self-energy, and $\mathrm{Tr}\log$ expansion.

The conceptual mechanism is equally explicit. The microscopic $\mathcal{PT}$ transformation is anti-linear and independent of the complex-linear grading $\Theta$. Under the declared local transformation law, the second-order Schur term is $\mathcal{PT}$-symmetric. The reduction remains non-exclusive to quaternionic systems. Quaternionic algebra supplies the particular faithful $\rho \left(\mathbb{H}\right)$ realization: the $\Theta /W$ relation, its block signs, and the orthogonal $u,v$ family of Proposition 2. The manuscript-specific result is the explicit bridge (78) from $E^{\mu \nu }{e}_2$ through $\mathcal{B}W$ and the Schur-reduced operator to $-{\mathcal{B}}^2/M^2+\cdots$, $G_F^{+}$, and the retained loop insertion. No exclusivity theorem is claimed.

The supporting audits remain deliberately local. The faithful image excludes $i\rho \left(e_2\right)$ as a quaternionic generator. The doublet symbol has the real-spectrum domain $|a-d|>2\left|b\right|$ and the positive local-symbol metric $G_{\mathrm{full}}$, but that matrix is not promoted to a universal QFT metric. The carrier $\Psi$ is not a physical interacting doublet, and the determinant factorization is only the declared formal block identity. The complementary-block separation scale M, the grading $\Theta$, state choice, regulator, and slow-background regime remain inputs. No all-order renormalization theorem or full pseudo-Hermitian functional measure or full quaternionic QFT is claimed.

The local amplitudes ${\epsilon }_a\left(x\right)$ are parameters of this principal-symbol deformation. They are not identified with phenomenological weak-field matching parameters or force ratios used in separate constructions unless an independent matching derivation is supplied.

Layer C is reserved for later work: dispersion or spectral response, renormalized stress tensors, background response kernels, vacuum polarization, backreaction, and phenomenology require their own background model, state, renormalization, and matching analysis. No such observable calculation is a result of the present paper. The achieved output is instead the explicit sequence from an auxiliary quaternionic principal-symbol deformation to a retained scalar EFT insertion and its Green-function interface.