Projected Green Functions for Scalar Fields with PT-Even Quaternionic Background Deformations

1. Introduction

Quantum field theory in curved spacetime starts from a physical geometry, a field operator, and a state prescription, and asks for the corresponding Green functions and local effective expansion [1,2,3,4]. Here the physical geometry is always a real Lorentzian $g_{\mu \nu }$. Quaternionic data enter only through an auxiliary principal-symbol coefficient on a complex carrier. This restricted ontology avoids assuming a quaternionic inverse metric, Levi–Civita connection, volume form, or dynamical background theory.

The paper’s central distinction is visible already at the operator level. For a carrier 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)

Strict projection and dynamical reduction are therefore not the same operation. In the quaternionic convention used below, a background-induced odd channel 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 heavy complementary gap this plus-sign identity produces a negative local EFT insertion because ${\mathcal{D}}_{--}^{-1}\simeq -M^{-2}$.

The auditable chain established in this paper is

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

(3)

The first two braces are the current 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]; PT-symmetric and pseudo-Hermitian frameworks are likewise established subjects [6,7,8,9,10,11]. Recent pseudoreal QFT work addresses non-Hermitian multiplets and interactions [12]. We do not claim priority over those subjects. Nor are projectors, Schur/Feshbach reduction [13], resolvent identities, functional traces, tadpoles, dimensional regularization, or heavy-field EFT expansions new.

The manuscript-specific contribution is their controlled assembly for one genuine quaternionic channel: a faithful carrier with $\rho \left(\mathbb{H}\right)\cap \mathrm{Herm}(2,\mathbb{C})=\mathbb{R}I$; a family of grading-odd pseudo-Hermitian quaternionic mixings; a clean separation of microscopic anti-linear $\mathcal{PT}$ from the linear grading $\mathrm{\Theta }$; 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.

Section 2 fixes the principal-symbol and grading data. Section 3 states the Green-function distinction. Section 4 derives the metric-channel 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 $\mathrm{\Theta }$-grading details. Earlier PTQ papers provide motivation but no companion result is an assumption of the derivation below [14,15,16,17].

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 dynamically reduced retained Green functions	Quaternionic exclusivity of Schur/Feshbach reduction
Quaternionic $W=\rho \left(e_2\right)$ channel and local-symbol metric audit	Universal Hilbert-space metric, Born rule, or interacting measure
$\delta {\mathcal{G}}^{\mu \nu }\to \mathcal{B}W\to -M^{-2}{\mathcal{B}}^2+\dots$	Observable response, backreaction, or phenomenology
Matched retained scalar propagator and one-loop insertions	Novel tadpole, self-energy, $Trlog$, or all-order renormalization theorem

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 dynamically reduced retained Green functions	Quaternionic exclusivity of Schur/Feshbach reduction
Quaternionic $W=\rho \left(e_2\right)$ channel and local-symbol metric audit	Universal Hilbert-space metric, Born rule, or interacting measure
$\delta {\mathcal{G}}^{\mu \nu }\to \mathcal{B}W\to -M^{-2}{\mathcal{B}}^2+\dots$	Observable response, backreaction, or phenomenology
Matched retained scalar propagator and one-loop insertions	Novel tadpole, self-energy, $Trlog$, 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.

The notation in (4) is definitional: ${\mathcal{G}}^{\mu \nu }$ is an auxiliary quaternionic-valued principal-symbol coefficient. It is not assumed to be the inverse of a quaternionic spacetime metric, a physical spacetime metric, a quaternionic Levi–Civita structure, or a source of a quaternionic volume form. In particular, no series that inverts a quaternionic covariant tensor is used. The associated carrier operator is specified at principal-symbol order by

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

(7)

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 channel 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) close the principal-symbol ontology: 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 2.1

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

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

(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 genuine quaternionic odd channel by

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

(12)

They satisfy

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

(13)

Thus W is grading-odd and $\mathrm{\Theta }$-pseudo-Hermitian, not Hermitian.

 Proposition 2.2

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

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

(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:

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

(15)

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, $\mathrm{\Theta }$ 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 $\mathrm{\Theta }$.

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 PT-even condition concerns the complete microscopic background channel, not W or ${\epsilon }_2$ in isolation. At principal-symbol level it is realized when

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

(17)

including the tensor pullback. Then $b_{\mathrm{metric}}W$ is microscopic-$\mathcal{PT}$ even. 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

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

(18)

is a kinematical carrier for $\rho \left(\mathbb{H}\right)$, $\mathrm{\Theta }$, 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 $P_{\mathrm{\Theta }}$ first selects a one-dimensional complex carrier line. The physical scalar theory then uses its chosen real structure, which we denote by

$${\mathcal{H}}_{+}^{\mathbb{R}}:=\{{\Psi }_{+}\in P_{\mathrm{\Theta }}{\mathcal{H}}_{\mathrm{kin}}:{\Psi }_{+}\mathrm{satisfies}\mathrm{the}\mathrm{declared}\mathrm{scalar}\mathrm{reality}\mathrm{condition}\}.$$

(19)

The symbol ${\varphi }_{+}$ is the real coordinate on this slice, obtained either by strict truncation or after a declared Schur reduction. Thus $P_{\mathrm{\Theta }}$ 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 $\mathrm{\Theta }$-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 $Q_{\mathrm{\Theta }}=I-P_{\mathrm{\Theta }}$ and ${\mathcal{H}}_{\mathrm{kin}}={\mathcal{H}}_{+}\oplus {\mathcal{H}}_{-}$, with ${\mathcal{H}}_{+}=P_{\mathrm{\Theta }}{\mathcal{H}}_{\mathrm{kin}}$. The auxiliary carrier operator has blocks

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

(20)

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.

3.1. Two Retained Inverses

Strict projection discards the complementary carrier channel and defines

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

(21)

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

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

(22)

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

 Proposition 3.1

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

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

(23)

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

 Proof. 

Block Gaussian elimination gives (23). 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

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

(24)

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

4. Quaternionic Pseudo-Hermitian Core and Metric-Channel EFT

4.1. Local Doublet Symbol and Positive-Metric Witness

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

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

(25)

so that

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

(26)

Its eigenvalues are

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

(27)

and the nondegenerate real-spectrum domain is

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

(28)

On this domain the matrix

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

(29)

obeys

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

(30)

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

The scope of (29) 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 (28); 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 Channels with Different Roles

We keep two nonconflated choices for b:

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

(31)

The first gives a clean pole-mass example. It is not derived from $\delta {\mathcal{G}}^{\mu \nu }$ without additional matching. The second is the actual bridge from the background coefficient and produces the higher-derivative retained operator derived below.

For the displayed metric benchmark $a=k^2-m^2$, $d=k^2-M^2$, and $b=E^{\mu \nu }{k}_{\mu }{k}_{\nu }$, with the same $k^2$ convention in both diagonal symbols, the local real-spectrum condition (28) becomes

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

(32)

Together with $|k^2|\ll M^2$ and the small/adiabatic assumptions (6), this defines a controlled overlap window for this benchmark. 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

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

(33)

At differential-operator level, a general representative may contain lower-derivative terms with the same principal symbol. For the central operator closure we fix the explicit representative

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

(34)

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

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

(35)

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

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

(36)

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,

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

(37)

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 (36) and (13),

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

(38)

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

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

(39)

Any additional off-block term would require its own declared channel and is not included in the present one-channel closure.

 Proposition 4.1

(PT-even Schur re-entry of a quaternionic odd channel). 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

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

(40)

Then $\mathcal{B}W$ and $\mathcal{B}{\left({\mathcal{D}}_{--}\right)}_F^{-1}\mathcal{B}$ are microscopic-$\mathcal{PT}$ even. Strict $P_{\mathrm{\Theta }}$ projection removes the grading-odd retained block at linear order, whereas Schur reduction regenerates a retained microscopic-$\mathcal{PT}$-even 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, $P_{\mathrm{\Theta }}W{P}_{\mathrm{\Theta }}=0$ proves the strict linear statement. □

At principal-symbol level the first condition in (40) reduces to (17). This proposition does not assert that all PT-even 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 $\mathcal{PT}{\mathcal{B}}_{\mathrm{sym}}{\left(\mathcal{PT}\right)}^{-1}=-{\mathcal{B}}_{\mathrm{sym}}$.

4.4. Exact Operator Schur Sign

Substituting (39) into (22) gives

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

(41)

The plus sign is fixed by ${\mathcal{D}}_{+-}=+\mathcal{B}$ and ${\mathcal{D}}_{-+}=-\mathcal{B}$; replacing it by a minus sign would contradict the quaternionic block convention. Indeed, applying the local symbol map ${\mathcal{D}}_{++}\mapsto a$, ${\mathcal{D}}_{--}\mapsto d$, and $\mathcal{B}\mapsto b$ yields

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

(42)

which agrees with the direct Schur complement of (25).

4.5. Heavy-Gap Operator Expansion

Write the complementary operator as

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

(43)

where ${\mathcal{K}}_{-}$ contains the declared soft covariant kinetic term and may include specified curvature and lower-derivative adiabatic structures. We do not require a global Lorentzian operator-norm theorem. Here M denotes the local spectral-gap scale of the complementary auxiliary carrier block in the declared EFT patch; it need not be the mass of an independently propagating physical particle, and no second physical scalar is introduced. The following is a local-symbol/EFT expansion in a patch where the complementary sector is gapped and $\parallel {\mathcal{K}}_{-}\parallel /M^2\ll 1$ in the stated soft sense:

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

(44)

Consequently,

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

(45)

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 the gap closes, the exact resolvent in (41), rather than the local series, is required.

4.6. Principal-Symbol Matching and Position-Space Ordering

Freeze $E^{\mu \nu }$ within a local adiabatic patch. From (37) and (45),

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

(46)

This sign is not inserted by hand: it follows from the exact plus sign in (41) 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 metric channel 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

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

(47)

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,

1.

a leading $EE{\nabla }^4$ structure;

2.

$E(\nabla E){\nabla }^3$ terms;

3.

$E\left({\nabla }^{2}E\right){\nabla }^2$ terms; and

4.

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 benchmark

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

(48)

one has

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

(49)

and therefore

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

(50)

The leading physical pole shift is

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

(51)

For the generic retained insertion

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

(52)

the mass-like benchmark fixes

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

(53)

Equations (46) and (53) have different roles: the former is the actual background-induced higher-derivative channel, 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

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

(54)

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

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

(55)

and the conservative notation

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

(56)

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. The physical quadratic action is then only for the retained real scalar,

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

(57)

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 ${\mathcal{D}}_{+}$. No quaternionic volume element is used.

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

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

(58)

where ${\delta }_g(x,x^{\prime })={\delta }^{\left(4\right)}(x-x^{\prime })/\sqrt{-g\left(x\right)}$ and, in the one-dimensional retained channel, $G_{+}=1$ is sufficient. Equation (58), rather than a reciprocal of a quaternionic scalar, is the primary definition.

In a normal neighborhood the leading local-momentum representation is

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

(59)

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

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

(60)

For a reduced calculation, $\delta Q_{+}$ includes (46) and the declared adiabatic corrections. Curvature and derivative terms can be organized by covariant symbol or heat-kernel methods [18,19]; 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 (40). 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

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

(61)

For the metric channel at leading gapped order,

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

(62)

so (61) 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,

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

(63)

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

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

(64)

and the state, interaction, and regulator are $\mathrm{\Theta }$-invariant, a correlator with odd net external $\mathrm{\Theta }$-grade vanishes. Retained external legs and traces are restricted by $P_{\mathrm{\Theta }}$; in a strict calculation the propagator is the sector inverse (21), whereas in a reduced calculation odd-channel virtual effects have already re-entered through (62). 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 $\mathrm{\Theta }$ grade.

6. Matched Retained Scalar Loop Insertion

This section shows only how the retained operator produced in Section 4 enters ordinary scalar perturbation theory. The integrals, combinatorics, and regularization are standard and are not claimed as novel.

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

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

(65)

In a frozen local patch write $Q_{+}=Q_0+\delta Q_{+}$, $Q_0=k^2-m^2$. The two-point insertion convention consistent with (63) is

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

(66)

For the actual metric channel,

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

(67)

whereas the lower-derivative benchmark uses (52) with the coefficients (53). The latter is useful for the clean constant insertion and pole example; it is not a derivation of $b_{\mathrm{loc}}$ from the background.

6.1. Tadpole and Cubic Self-Energy

The quartic tadpole is

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

(68)

or, in local momentum organization through first insertion order,

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

(69)

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

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

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

(70)

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

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

(71)

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 (70); alternative conventions that define the 1PI amplitude as $i\mathrm{\Sigma }$ redistribute the overall factor of i but not the insertion rule (66). These formulas apply to (67) 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

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

(72)

This is ${\mathrm{\Gamma }}_{\mathrm{strict}}^{\left(1\right)}$ or ${\mathrm{\Gamma }}_{\mathrm{red},+}^{\left(1\right)}$ according to the operator chosen; it is not the formal auxiliary quantity (56).

As a local subtraction check, take the constant part $\delta Q_{+}={\epsilon }^2{c}_0{M}_{\epsilon }^2$. After Wick rotation define, for $d=4-2{ϵ}_{\mathrm{DR}}$,

$$I_E\left(m^2\right)={\mu }_{\mathrm{DR}}^{4-d}\int {\displaystyle \frac{d^d{p}_E}{{\left(2\pi \right)}^d}}{\displaystyle \frac{1}{p_E^2+m^2}},{\displaystyle \frac{1}{{\overline{ϵ}}_{\mathrm{DR}}}}:={\displaystyle \frac{1}{{ϵ}_{\mathrm{DR}}}}-{\gamma }_E+log4\pi .$$

(73)

Dimensional regularization gives

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

(74)

and the divergent Euclidean local density is

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

(75)

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 metric-channel 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 genuine quaternionic off-block operator $\mathcal{B}W$. This formulation needs no quaternionic inverse metric and keeps all curvature, volume, and covariant-derivative data tied to the real baseline geometry.

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

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

(76)

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 $Trlog$ expansion.

The conceptual mechanism is equally explicit. Microscopic $\mathcal{PT}$ is anti-linear and independent of the complex-linear grading $\mathrm{\Theta }$. Under the compensating transformation law for $\mathcal{B}$, both $\mathcal{B}$ and W are microscopic-$\mathcal{PT}$ odd while their product is even. Strict $P_{\mathrm{\Theta }}$ projection removes the linear grading-odd block, but Schur reduction returns a retained even contribution at second order. This does not make Schur/Feshbach reduction quaternionic-exclusive; quaternionic algebra supplies the structured signs and pseudo-Hermitian odd channel.

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 witness $G_{\mathrm{full}}$, but that witness 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 gap M, the grading $\mathrm{\Theta }$, state choice, regulator, and slow-background regime remain inputs. No all-order renormalization theorem or full pseudo-Hermitian functional measure 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 closed, auditable chain from an auxiliary quaternionic principal-symbol deformation to a retained scalar EFT insertion and its Green-function interface.