The Probability Sector of PT-Symmetric Quaternionic Spacetime

1. Introduction

The probabilistic structure of quantum theory remains conceptually subtle despite its empirical success. The Born rule provides an accurate prescription for measurement outcomes [3,4], yet in standard formulations it is introduced as an additional postulate rather than derived from deeper structural principles. This has motivated extensive efforts to reinterpret or constrain quantum probability through geometric, measure-theoretic, or symmetry-based arguments, including Gleason-type theorems [5,9], decision-theoretic approaches [6,8], and environment-assisted invariance [7,21,22].

Parallel developments in non-Hermitian and $\mathcal{PT}$-symmetric quantum mechanics have demonstrated that consistent quantum dynamics need not rely on naive Hermiticity. Instead, pseudo-Hermitian structures equipped with a positive metric operator G can support real spectra and norm-preserving evolution [10,11,12,13,14,15]. This suggests that the probabilistic sector of quantum theory is fundamentally linked to the metric structure of the state space rather than to a fixed kinematic norm.

A further extension arises in quaternionic quantum mechanics, where amplitudes are valued in a larger algebra [16,17,18,19,20]. Such extensions naturally encode additional geometric structure, but they also introduce a key ambiguity: the kinematic space exceeds the observable sector. Not all quaternionic components admit direct physical interpretation, raising the question of how probability should be defined in an extended kinematic framework.

The present work addresses this issue within the framework of PT-symmetric quaternionic spacetime (PTQ). In this setting, spacetime geometry is formulated in a metric-affine (Palatini) posture [23,27,29,30,31], with projective invariance and torsion playing a central role [24,25,26,27,28]. A defining feature of the PTQ framework is that the physical observable sector is not identified with the full kinematic space. Instead, it is obtained only after imposing both projective invariance and PT projection, restricting observables to real, PT-even quantities.

This point also fixes the literature posture of the present paper. Unlike Gleason-type, decision-theoretic, or envariance-based approaches [5,6,7,8,9,21,22], the aim here is not a universal derivation of quantum probability from abstract axioms, rationality requirements, or environment-assisted symmetry alone. And unlike standard pseudo-Hermitian or $\mathcal{PT}$-symmetric metric constructions [10,11,12,13,14,15], the present argument does not begin from a pre-given physical state space on which a metric is then selected. Its distinctive claim is PTQ-specific: probability is posed only after restriction to the PT-even, projectively invariant observable sector determined by the underlying geometric posture [2]. In that sense, the specifically PTQ-driven step is prior to pseudo-Hermitian probability assignment itself: projective invariance and PT projection first restrict the observable sector, and only on that restricted sector are metric admissibility and probability assignment subsequently posed.

This leads to a structural hierarchy: the full quaternionic-geometric sector is larger than the observable sector, and the probability theory must therefore be formulated only after projection. Consequently, probability is not defined on arbitrary quaternionic amplitudes, but only on the projected PT-even observable scalar sector.

The central question of this paper is therefore:

Given a PT-even projected observable sector in PTQ spacetime, what probability rule is structurally compatible with its geometry and dynamics?

We answer this question by showing that, once the observable sector is fixed, the admissible norm and the corresponding conserved probability current are constrained in a way that leads to the Born-rule form:

$$\begin{array}{c}\mathrm{projected}\mathrm{observability}⟹\mathrm{physical}G-\mathrm{inner}\mathrm{product}⟹\mathrm{conserved}\mathrm{probability}\mathrm{current}\\ ⟹P_i={\left|{\langle \psi \mid {\varphi }_i\rangle }_G\right|}^2.\end{array}$$

(1)

Within this route, the Born-rule form is no longer an independent postulate but appears as the probability assignment compatible with observability, metric consistency, and norm-preserving dynamics in the PT-even sector.

It is important to emphasize scope. The present work does not claim a universal derivation of the Born rule or a solution to the measurement problem. Rather, it establishes a structural result: once the observable sector and admissibility conditions are fixed within PTQ, the form of the probability rule is structurally constrained.

At the same time, the present manuscript should not be read as introducing the PTQ geometry or projected-operator posture ex nihilo. Several geometric and operator ingredients used here are continuous with earlier PTQ developments [1,2]. What is new in the present paper is the explicit integration of those ingredients into a single scope-limited chain, projection → metric admissibility → conserved current → Born-rule form, as a dedicated analysis of the probability sector. Where appendix-level overlap with earlier PTQ material is retained, it is included for auditability and logical continuity of the argument, not to enlarge the novelty claim.

The paper is organized as follows. Section 2 defines the geometric posture and observable sector. Section 3 derives the metric structure. Section 4 establishes the conserved probability current. Section 5 presents the PTQ route to the Born rule. Section 6 discusses interfaces with QFT and geometry. Section 7 discusses scope boundaries and open directions. Section 8 concludes.

2. Geometric Posture: PT Projection and the Observable Sector

This section fixes the geometric posture used throughout the paper and, in particular, makes precise what is meant by an observable quantity in the PT-symmetric quaternionic (PTQ) framework. Its role is foundational rather than phenomenological. Before one can discuss probability, inner products, or measurement amplitudes, one must first specify which sector of the quaternionic-geometric structure is admitted as physical. The central claim of the present section is that PTQ does not assign physical meaning to the full unprojected quaternionic algebra. Instead, observability is defined only after two restrictions are imposed:

1.

full projective invariance in the Route-A sense, implemented with a compensator one-form;

2.

restriction to a real, PT-even scalar sector via a PT projection operator.

These two requirements together define the observable sector that will later support the probability interpretation of the theory.

The importance of this point for the Born-rule question should be stated at the outset. If the physical sector were taken to be the full quaternionic kinematic space, then probability assignments would depend on components that are not retained by the PTQ observable map itself. Accordingly, the probability sector cannot be defined prior to projection. It must be constructed only after the physical scalar algebra has been restricted to the PT-even, projectively invariant sector.

2.1. Route A: Full Projective Equivalence and the Invariant Residue

We begin with the geometric posture in the affine sector. As in companion PTQ works [1,2], the relevant connection is treated in a Palatini or metric-affine spirit, so that the metric $g_{\mu \nu }$ and affine connection ${{\Gamma }^{\lambda }}_{\mu \nu }$ are regarded as independent geometric data at the level of kinematics. The key redundancy is the full projective transformation

$${{\Gamma }^{\lambda }}_{\mu \nu }⟶{{\Gamma }^{\lambda }}_{\mu \nu }+{\delta }_{\mu }^{\lambda }{\xi }_{\nu },$$

(2)

where ${\xi }_{\mu }$ is an arbitrary one-form. The trace part of torsion transforms nontrivially under this shift. Writing

$$T_{\mu }:={{\Gamma }^{\lambda }}_{\mu \nu },$$

(3)

one has, in four spacetime dimensions,

$$T_{\mu }⟶T_{\mu }+3{\xi }_{\mu }.$$

(4)

A scalar Stueckelberg completion compensates only the exact subset ${\xi }_{\mu }={\partial }_{\mu }\alpha$ and therefore does not realize the full one-form orbit. For this reason, the posture adopted here is Route A: projective invariance is completed by a compensator one-form $A_{\mu }$ transforming as

$$A_{\mu }⟶A_{\mu }+3{\xi }_{\mu }.$$

(5)

The projectively invariant residue is then defined by

$${\tilde{T}}_{\mu }:=T_{\mu }-A_{\mu }.$$

(6)

By construction, ${\tilde{T}}_{\mu }$ is invariant under the full one-form projective orbit. Equation (6) is therefore not a matter of notation only; it fixes the correct geometric variable for the physical trace sector.

The interpretive point is important. In the present posture, $A_{\mu }$ is treated spurionically rather than dynamically. Its role is to bookkeep the projective fiber choice so that the trace sector can be described directly on the quotient defined by the projective orbit. The physically meaningful information is thus carried not by the representative $T_{\mu }$ alone, but by the invariant residue ${\tilde{T}}_{\mu }$. In later sections, whenever we refer to the scalar-channel residue or to the projected probability sector, it is always this projectively invariant object that is intended.

2.2. PT Projection and the Observable Scalar Algebra

Projective invariance by itself does not yet specify which local quantities are observable. The second restriction is therefore a PT projection acting on scalar densities or, more generally, on quaternionic-valued scalar quantities that may appear in the effective description. We denote the corresponding projector by ${\Pi }_{\mathrm{PT}}$ and define it formally by

$${\Pi }_{\mathrm{PT}}\left[\mathcal{O}\right]:={\displaystyle \frac{1}{2}}\left(\mathcal{O}+{\mathcal{O}}^{\mathrm{PT}}\right),{\Pi }_{\mathrm{PT}}^2={\Pi }_{\mathrm{PT}},$$

(7)

where ${\mathcal{O}}^{\mathrm{PT}}$ denotes the image of $\mathcal{O}$ under the combined parity–time operation. Operationally, ${\Pi }_{\mathrm{PT}}$ removes PT-odd scalar contributions and restricts the observable sector to the real PT-even scalar algebra.

The role of ${\Pi }_{\mathrm{PT}}$ should be understood carefully. It is not introduced merely as a cosmetic reality filter, nor as a post hoc device to discard inconvenient quaternionic components. Rather, it is part of the definition of the physical sector. In the PTQ framework, the full quaternionic-geometric structure is larger than the observable scalar sector. The latter is obtained only after projection. This is precisely why the present paper insists on a project-first and projection-explicit posture: the physical scalar algebra is not assumed to coincide with the full kinematic algebra.

The observable scalar sector is therefore defined by the condition

$$\mathcal{O}\in \mathrm{Obs}⟺{\Pi }_{\mathrm{PT}}\left[\mathcal{O}\right]=\mathcal{O},\mathcal{O}\mathrm{real}\mathrm{and}\mathrm{PT}-\mathrm{even}.$$

(8)

In particular, the combination of Equations (6) and (8) implies that observable scalar quantities must be built from projectively invariant data and must survive the PT projection as real PT-even objects.

The point is conceptually decisive for everything that follows. The PTQ framework does not say that every quaternionic quantity is measurable and that one may later decide how to interpret it. It says, instead, that the physical sector is selected before interpretation. Only projected, real, PT-even, projectively invariant quantities are admitted as physical inputs to the probability sector.

2.3. Scalar-Channel Focus and the Probability-Sector Domain

The present paper concerns the probability sector, not the full tensorial or bundle-valued structure of PTQ geometry. Accordingly, we work in the scalar channel appropriate to observables. This requires a further restriction of scope. Although the invariant residue ${\tilde{T}}_{\mu }$ is fundamentally a one-form, the probability sector discussed later will be formulated on domains where its scalar-channel representative is well defined. Concretely, we restrict attention to admissible domains on which

$${\overline{\nabla }}_{[\mu }{\tilde{T}}_{\nu ]}=0,$$

(9)

so that ${\tilde{T}}_{\mu }$ admits a longitudinal representative,

$${\tilde{T}}_{\mu }=\kappa {\partial }_{\mu }ϵ\mathrm{on}\mathrm{admissible}\mathrm{scalar}-\mathrm{channel}\mathrm{domains},$$

(10)

for some scalar representative $ϵ$ and proportionality factor $\kappa$ appropriate to the chosen background posture.

The logical order matters. The scalar $ϵ$ in Equation (10) is not the compensator that completes the projective orbit. That role is already played by $A_{\mu }$ in Route A. Rather, $ϵ$ is only a convenient representative of the already invariant residue ${\tilde{T}}_{\mu }$ on the scalar-channel domains relevant for the observable sector. This distinction prevents a common confusion: the probability sector is not built from a scalar replacing the full projective structure; it is built from the projected scalar channel of an invariant one-form sector.

This scalar-channel restriction is also what later permits a clean probability interpretation. The Born-rule question is a scalar observable question: it concerns norms, overlaps, and conserved probability densities. These require a well-defined projected scalar sector. For this reason, the present paper does not attempt to construct probabilities directly on the full unprojected quaternionic or higher-spin sector. Its claim is narrower and more controlled: once the PT-even scalar observable sector has been identified, one may ask what probability rule is naturally compatible with it.

2.4. The Observable Sector as a Proposition-Level Statement

The preceding discussion may be summarized in two proposition-level statements that will be used repeatedly in the rest of the paper.

Proposition 1  

(Observable-sector criterion). A scalar quantity is admissible as a physical observable in the PTQ probability sector only if it satisfies both of the following conditions:

1.

it is constructed from projectively invariant data, in particular from the residue ${\tilde{T}}_{\mu }$ rather than from a projectively non-invariant representative;

2.

it lies in the PT-even projected sector, i.e. it is real and satisfies Equation (8).

This proposition is not an independent theorem added on top of the framework. It is simply the operational meaning of combining Route A with the PT projection posture. Its relevance is that it excludes, from the outset, any probability assignment built on unprojected or projectively non-invariant scalar data.

Proposition 2  

(Projection precedes probability). Within PTQ, the probability sector must be defined only after restriction to the observable sector of Proposition 1. Consequently, any admissible norm, overlap, or probability rule must be formulated on the projected PT-even scalar sector rather than on the full unprojected quaternionic kinematic space.

Proposition 2 is the first pillar of the present paper’s route to the Born rule. It does not yet specify the physical inner product or the probability current. Those issues are deferred to Secs. 3 and 4. But it establishes the indispensable first step: PTQ probabilities are not defined on arbitrary quaternionic amplitudes; they are defined only on the projected observable sector.

2.5. Why This Posture Matters for the Born-Rule Question

We may now state the conceptual consequence that motivates the remainder of the paper. In ordinary presentations of quantum mechanics, the Born rule is often introduced directly as a probability prescription on amplitudes. In the PTQ setting, such a direct move would be premature. The amplitudes live in a geometric framework whose kinematic space is larger than its physical observable sector. Hence the prior question is unavoidable: what sector of the theory is actually physical?

The answer supplied by the present section is clear. Physical quantities must be real, PT-even, and projectively invariant. Therefore, the probability sector must inherit these same restrictions. This is why the route to the Born rule in PTQ begins not with a norm, but with a geometric selection principle. Only after the observable sector has been fixed can one ask which inner product is physically admissible, and only after that can one ask which conserved probability current and which measurement rule are natural. That sequence is not merely pedagogical. It is the structural content of the PTQ program.

For later reference, we summarize the logical outcome of this section in the schematic form

$$\begin{array}{c}\mathrm{full}\mathrm{quaternionic}-\mathrm{geometric}\sec \mathrm{tor}⟹\mathrm{Route}-\mathrm{A}\mathrm{invariant}\mathrm{trace}\sec \mathrm{tor}\\ ⟹{\Pi }_{\mathrm{PT}}-\mathrm{projected}\mathrm{observable}\mathrm{scalar}\sec \mathrm{tor}.\end{array}$$

(11)

The rest of the paper develops the consequences of Equation (11). Section 3 shows that consistent projected evolution selects a physical G-metric inner product. Section 4 shows that the same metric induces a conserved probability current. Only then, in Section 5, does the Born-rule form arise as the admissible probability assignment in the projected pseudo-Hermitian sector.

3. Metric Structure and Pseudo-Hermitian Admissibility

Section 2 established the first pillar of the present paper: the probability sector of PTQ must be defined only after restriction to the PT-even observable scalar sector, as summarized in Propositions 1 and 2. The next question is then unavoidable: once the physical sector has been selected, what inner product is admissible on that sector? The purpose of the present section is to answer this question.

The main claim is that the PTQ probability sector cannot consistently be built from the naive inner product inherited from an unprojected complex or quaternionic kinematic space. Rather, once one works in the projected PT-even sector, consistency of time evolution, spectral reality in the unbroken regime, and compatibility with the projected operator structure require a metric-compatible pseudo-Hermitian completion. This completion is encoded by a positive metric operator G, and the resulting G-inner product is the physical inner product relevant to the probability sector. In this sense, the metric is not freely specifiable once the observable sector and admissibility conditions are fixed; it is selected by the same structural constraints that define the observable sector.

3.1. PTQ Operators and Pseudo-Hermiticity

We begin with the operator-level problem. In the PTQ setting, quantum propagation is formulated on a background whose effective geometric data are generally quaternionic-valued prior to projection. Accordingly, the natural Dirac- or Klein–Gordon-type operators need not be Hermitian with respect to the naive kinematic inner product. This is not an accidental complication; it is a direct consequence of working in a geometric framework whose kinematic sector is larger than its physical observable sector.

Let H denote the effective Hamiltonian (or, more generally, the generator of evolution) restricted to the scalar probability sector. At the level of the unprojected kinematic space, one should not expect

$$H^{†}=H$$

(12)

to hold in general. The correct structural condition is instead pseudo-Hermiticity: there exists an invertible positive operator G such that

$$H^{†}G=GH.$$

(13)

Equation (13) is the appropriate compatibility condition between the projected PTQ dynamics and the physical norm. It ensures that the evolution generated by H is self-consistent with respect to the metric defined by G, even when H is not Hermitian in the naive sense.

This structure is already anticipated by the projected-operator language used in the PTQ QFT layer. Once the PT-even projection operator ${\Pi }_{\mathrm{PT}}$ has been imposed, the physically relevant quadratic operator takes the schematic form

$$D_{\mathrm{PTQ}}={\Pi }_{\mathrm{PT}}\left(D_{\mathrm{bare}}\right){\Pi }_{\mathrm{PT}},$$

(14)

where $D_{\mathrm{bare}}$ denotes the underlying kinematic operator prior to projection. The crucial point is that Equation (14) acts within the PT-even sector identified in Section 2, not on the full unprojected algebra. Therefore, the physically relevant notion of adjointness must also be defined on this restricted sector. Pseudo-Hermiticity, in the sense of Equation (13), is precisely the condition that stabilizes this projected dynamics.

The same point may be expressed in spectral language. In the unbroken PT regime, one demands that the projected dynamics admit a real spectrum and a consistent mode expansion within the physical sector. The naive Hermitian condition of the unprojected kinematic space is too strong and, in the present context, generally inappropriate. Pseudo-Hermiticity is the weaker but physically relevant condition: it preserves the reality of the physical sector while remaining compatible with the projected structure.

The role of PT symmetry is therefore not merely decorative. It is what makes the pseudo-Hermitian completion meaningful. The PT projection of Section 2 removes PT-odd scalar channels from the observable algebra, while Equation (13) organizes the remaining sector into a norm-preserving dynamical system. Taken together, they define the operator-level consistency conditions of the PTQ probability sector.

3.2. The G-Inner Product as the Physical Inner Product

Once the existence of a positive metric operator G has been established, the physical inner product is no longer ambiguous. It is defined by

$${\langle \psi \mid \chi \rangle }_G:=\langle \psi \mid G\chi \rangle ,$$

(15)

for any two states $\psi$ and $\chi$ in the projected PT-even sector. Equation (15) is the correct norm structure for the probability sector. The point is not merely that this inner product can be written down, but that it is the one under which the projected dynamics become norm-compatible.

To see this, consider the evolution equation

$$i{\partial }_t\psi =H\psi .$$

(16)

Taking the time derivative of the G-norm and using Equation (13), one obtains

$$\begin{array}{cc}\hfill {\displaystyle \frac{d}{dt}}{\langle \psi \mid \psi \rangle }_G& ={\displaystyle \frac{d}{dt}}\langle \psi \mid G\psi \rangle \hfill \\ \hfill & =\langle {\partial }_t\psi \mid G\psi \rangle +\langle \psi \mid G{\partial }_t\psi \rangle \hfill \\ \hfill & =-i\langle \psi \mid H^{†}G\psi \rangle +i\langle \psi \mid GH\psi \rangle \hfill \\ \hfill & =0.\hfill \end{array}$$

(17)

Thus, the G-inner product is precisely the one preserved by the PTQ evolution. This is why it is physically distinguished.

Equation (17) should be read together with the observable-sector restriction of Section 2. The PTQ framework does not first posit an arbitrary projected sector and then independently choose a convenient metric on it. The logic is tighter. The projected sector identifies the admissible physical states; pseudo-Hermiticity then selects the metric with respect to which their evolution is norm preserving. In that sense, the physical inner product is not an auxiliary structure pasted onto the theory after the fact; it is part of the consistent completion of the projected dynamics.

This is also why the present paper does not adopt the naive kinematic norm. A naive norm on the full quaternionic kinematic space would fail to respect the sequence already established in Section 2:

$$\mathrm{projection}⟹\mathrm{physical}\sec \mathrm{tor}⟹\mathrm{admissible}\mathrm{inner}\mathrm{product}.$$

(18)

The direction of implication matters. One first restricts to the physical sector and only then asks what norm is preserved by the physical evolution on that sector.

For later use, it is convenient to record the following statement.

Proposition 3  

(G-metric physicality criterion). Let H be the generator of evolution on the PT-even projected sector selected by Section 2. If there exists a positive invertible operator G satisfying Equation (13), then the physically admissible inner product on that sector is the G-inner product of Equation (15). In particular, the corresponding norm is preserved by the evolution generated by H.

This proposition is not intended as a new theorem beyond the standard pseudo-Hermitian logic. Its role in the present paper is to make explicit which inner product is relevant to the PTQ probability sector and why the naive norm is not.

3.3. Why the Metric Is Not Freely Specifiable

At this point one may still ask whether the choice of G is merely conventional. The answer, within the scope of the present paper, is no. The metric is constrained by three simultaneous requirements:

1.

compatibility with the projected PT-even sector of Section 2;

2.

pseudo-Hermitian consistency, Equation (13);

3.

positivity and stability of the induced norm.

These conditions do not determine an arbitrary family of equally physical metrics. They identify the class of metrics admissible for the PTQ probability sector, and within the present posture they exclude the naive kinematic norm as physically inappropriate.

The admissibility claim made here should be read carefully. We do not claim a theorem that no other generalized norm could ever be imagined in a broader non-Hermitian or non-associative framework. Such a claim would be too strong for the present paper. The actual claim is more precise and more useful: once the observable sector, the PT-even restriction, and pseudo-Hermitian evolution are fixed, the admissible probability-compatible metric is no longer an independent degree of freedom. It is selected by consistency of the very structure that defines the physical sector.

One may summarize this by saying that G is canonical within the admissible equivalence class relevant to the PTQ probability sector. What matters for the present work is not a metaphysical uniqueness claim, but the fact that probability interpretation is no longer freely specifiable once projected PTQ dynamics have been fixed. There is a distinguished norm structure within the present PTQ posture, and that structure is the one induced by G.

This observation can also be reformulated in operator language. Suppose one attempts to equip the projected sector with a different positive metric $G^{\prime }$ unrelated to the pseudo-Hermitian structure of H. Then, generically,

$$H^{†}{G}^{\prime }\ne G^{\prime }H,$$

(19)

so the induced norm fails to be preserved by the physical evolution. Such a metric may exist algebraically, but it is not admissible as a physical probability metric in the sense required here. Hence the role of G is not merely to provide one possible inner product among many; it is to provide the inner product compatible with projected PTQ evolution.

The point is strong enough to record explicitly.

Proposition 4  

(Metric non-arbitrariness in the PTQ probability sector). Within the PTQ probability sector defined by Proposition 2, the physical metric operator is not freely specifiable after the projected dynamics are chosen. Rather, admissibility requires compatibility with the pseudo-Hermitian condition Equation (13) and preservation of the induced norm. Consequently, the probability-compatible metric is fixed up to the admissible equivalence class of the projected PT-even sector.

Proposition 4 supplies the second pillar of the paper’s route to the Born rule. Section 2 established that probability must be defined only after projection. The present section now adds that, once projection is fixed, the physical inner product is not freely specifiable but must be defined with respect to the metric operator G. This is the precise sense in which PTQ selects a metric-compatible norm.

3.4. From Projected Observability to Metric-Compatible Probability

We may now summarize the structural content of this section. Section 2 reduced the full quaternionic-geometric sector to the PT-even observable scalar sector, cf. Equation (11). The present section shows that a second reduction is then forced: within that projected observable sector, only a G-metric inner product is compatible with the physical evolution. The combined logic may be written schematically as

$$\begin{array}{c}{\Pi }_{\mathrm{PT}}-\mathrm{projected}\mathrm{observability}⟹\mathrm{pseudo}-\mathrm{Hermitian}\mathrm{consistency}\\ ⟹\mathrm{physical}G-\mathrm{inner}\mathrm{product}.\end{array}$$

(20)

Equation (20) provides the missing bridge between the geometric posture of Section 2 and the probability-current analysis of Section 4. Once the physical inner product has been fixed, one may define the corresponding density and ask whether it is conserved under the same evolution. That is the subject of the next section.

The role of the present section in the paper is therefore exact: it does not yet produce the Born rule, but it removes the arbitrariness in the choice of norm on the physical sector. Only after this step does the question of probability density and probability current become well posed. For this reason, the metric structure is not a secondary technical issue. It is the second foundational pillar of the PTQ route to the Born rule.

4. Conserved Probability Current from the PTQ Probability Sector

Sections 2 and 3 established the first two pillars of the present paper. First, the probability sector of PTQ must be defined only after restriction to the projected PT-even observable scalar sector; see Propositions 1 and 2. Second, once that sector is fixed, the physically admissible norm is not the naive kinematic one, but the G-inner product selected by pseudo-Hermitian consistency; see Propositions 3 and 4. The purpose of the present section is to establish the third pillar: the same structure that selects the physical metric also induces a distinguished conserved probability current.

This point is essential for the logic of the paper. A candidate probability rule is not physically meaningful unless the quantity it assigns is preserved by the corresponding time evolution. In the PTQ setting, this requirement is especially nontrivial because the physical sector is projected and the norm is metric-dependent. The main claim of this section is therefore the following: once the PT-even projected sector and the pseudo-Hermitian G-metric structure are fixed, the admissible probability density is

$$j^0:={\psi }^{†}G\psi ,$$

(21)

and the associated total probability is conserved under the same projected evolution. This conserved current is not chosen ad hoc. It is the current naturally induced by the very structure that defines the PTQ probability sector.

4.1. From the G-Inner Product to a Probability Density

The starting point is the G-inner product introduced in Equation (15),

$${\langle \psi \mid \chi \rangle }_G:=\langle \psi \mid G\chi \rangle .$$

(22)

Because this is the physically admissible norm on the projected PT-even sector, the local density associated with it is obtained in the obvious way:

$$j^0\left(x\right):={\psi }^{†}\left(x\right)G\psi \left(x\right).$$

(23)

Equation (23) is the PTQ analogue of the standard probability density. Its meaning is straightforward: it is the local integrand of the global G-norm,

$${\langle \psi \mid \psi \rangle }_G=\int d^{3}x{j}^0\left(x\right).$$

(24)

Thus the notion of probability density is not introduced independently of the norm. It is the local form of the physically selected inner product.

This is precisely where the PTQ route differs from a naive generalization of ordinary quantum mechanics. One does not begin with an arbitrary local density and then try to justify it later. Instead, one first identifies the projected observable sector, then the admissible G-metric, and only then the density implied by that metric. In this sense, the density in Equation (23) is not optional. It is the density naturally associated with the physical norm of the theory.

The dependence on the projected sector remains crucial here. Since the states $\psi$ are understood to lie in the PT-even observable sector, the density $j^0$ is not a quantity defined on the full unprojected quaternionic kinematic space. It is defined only for projected states and only with respect to the physical metric G. Accordingly, the probability density inherits both restrictions already established in the previous sections:

$$\mathrm{PT}-\mathrm{even}\mathrm{projected}\mathrm{state}⟹\mathrm{physical}G-\mathrm{norm}⟹\mathrm{physical}\mathrm{density}{j}^0.$$

(25)

4.2. Conservation from Pseudo-Hermitian Time Evolution

We now show why the density Equation (23) is conserved under the PTQ evolution. Let the projected dynamics be governed by

$$i{\partial }_t\psi =H\psi ,$$

(26)

where H acts on the projected PT-even sector and satisfies the pseudo-Hermitian compatibility condition

$$H^{†}G=GH.$$

(27)

Then the time derivative of the total G-norm is

$$\begin{array}{cc}\hfill {\displaystyle \frac{d}{dt}}\int d^{3}x{j}^0& ={\displaystyle \frac{d}{dt}}\int d^{3}x{\psi }^{†}G\psi \hfill \\ \hfill & =\int d^{3}x\left[\left({\partial }_t{\psi }^{†}\right)G\psi +{\psi }^{†}G\left({\partial }_t\psi \right)\right]\hfill \\ \hfill & =\int d^{3}x\left[i{\psi }^{†}{H}^{†}G\psi -i{\psi }^{†}GH\psi \right]\hfill \\ \hfill & =0,\hfill \end{array}$$

(28)

where the final equality follows directly from Equation (27). Equation (28) is the global conservation law for total probability in the PTQ probability sector.

The importance of Equation (28) should not be understated. It shows that probability conservation is not an extra postulate layered onto the theory after the metric has been chosen. Rather, it follows immediately from the same pseudo-Hermitian structure that selected the metric in the first place. The density $j^0$ is conserved because the G-norm is the norm preserved by the physical evolution.

This fact is precisely what makes the current physically distinguished. If one were to choose instead a different density not derived from the metric-compatible norm, there would be no reason to expect it to remain conserved under the same projected evolution. Indeed, such a density would generically fail to track the physical state space defined by the PT-even sector and its pseudo-Hermitian time evolution. The PTQ current is therefore not merely one possible current among many; it is the current singled out by the structure of the theory.

4.3. Local Current Structure and the Continuity Form

Equation (28) is the integrated statement of probability conservation. For the purposes of the present paper, this global statement is already sufficient to support the later Born-rule discussion. Nevertheless, it is useful to indicate the corresponding local structure.

Let $j^{\mu }$ denote the current whose time component is Equation (23). In the projected PT-even sector, the continuity equation takes the schematic form

$${\partial }_{\mu }{j}^{\mu }=0,$$

(29)

or, on a curved background,

$${\nabla }_{\mu }{j}^{\mu }=0,$$

(30)

with the understanding that the precise covariant derivative and spatial components depend on the form of the projected kinetic operator used in the effective PTQ description. The essential point is not the detailed form of $j^i$ for every model realization, but the fact that the time component is fixed by the G-inner product and that the corresponding continuity relation follows from projected pseudo-Hermitian evolution.

This is fully consistent with the PTQ QFT layer, where the projected kinetic operator $D_{\mathrm{PTQ}}$ acts within the PT-even sector and propagators are replaced by their projected versions. From that viewpoint, the conserved current is the local expression of the same selection rule that governs projected propagation and projected traces. The operator language and the current language are therefore two descriptions of the same underlying structure: the PT-even projected sector carries a metric-compatible evolution, and that evolution preserves a distinguished norm and its associated current.

For later use, we record the following proposition.

Proposition 5  

(Conserved probability density in the PTQ sector). Let ψ evolve in the PT-even projected sector under a generator H satisfying the pseudo-Hermitian condition Equation (13). Then the density

$$j^0={\psi }^{†}G\psi$$

(31)

defines the physical probability density, and its spatial integral is conserved under the evolution. Equivalently, the PTQ probability sector admits a distinguished conserved current whose time component is fixed by the G-metric structure.

Proposition 5 supplies the third pillar of the paper’s route to the Born rule. Section 2 fixed the observable sector. Section 3 fixed the physically admissible metric. The present section now shows that these data together determine a conserved probability density.

4.4. Why This Current Is Not Freely Specifiable

One may still ask whether the current Equation (23) is merely one convenient choice among many conserved quantities. Within the scope of the present paper, the answer is again no. The reason parallels the non-arbitrariness of the metric itself.

Suppose one attempts to define an alternative density

$${\tilde{j}}^0:={\psi }^{†}Q\psi ,$$

(32)

for some positive operator Q unrelated to the pseudo-Hermitian metric structure. Then conservation of $\int d^{3}x{\tilde{j}}^0$ would require

$$H^{†}Q=QH.$$

(33)

If Q fails to satisfy Equation (33), the corresponding density is not preserved by the physical evolution. In other words, a candidate density is physically admissible only if it is induced by a metric compatible with the same pseudo-Hermitian dynamics. But that is precisely the role already played by G.

Accordingly, the current of the PTQ probability sector is not freely specifiable for exactly the same reason that the metric is not freely specifiable: it is fixed by compatibility with projected evolution. The sequence of constraints is therefore now complete:

$$\mathrm{projected}\mathrm{observability}⟹\mathrm{physical}G-\mathrm{metric}⟹\mathrm{conserved}\mathrm{probability}\mathrm{current}.$$

(34)

Equation (34) is the third foundational step of the paper.

4.5. Relation to Improved Currents and Geometric Flow Language

The current identified above is the immediate probability current relevant to the PTQ probability sector. It is important, however, to place it in the broader conceptual context of the PTQ program. In companion operational and thermodynamic formulations, symmetry-protected or posture-protected structures are often expressed in terms of improved currents, boundary improvements, or coarse-grained flow statements. Those formulations are useful because they show that PTQ already admits a broader “symmetry ⇒ conserved/improved flow ⇒ auditable invariant” language.

The present paper does not claim that such thermodynamic or coarse-grained flow statements directly derive the Born rule. That would overstate the scope. The correct statement is more modest: the thermodynamic and improved-flow language provides supporting intuition for why a projected conserved current is natural in PTQ, but the immediate current used in the probability sector is the pseudo-Hermitian G-current of Equation (23). In other words, the geometric-flow language and the probability-current language are compatible, but they are not identical in logical role.

This distinction is important for rigor. The current that enters the Born-rule route must be the one preserved by the projected quantum evolution. That current is the G-current. Broader entropy-flow or improvement-current structures may illuminate why such a conserved quantity is natural in the PTQ program, but they do not replace the pseudo-Hermitian derivation of the physical probability density.

4.6. From Metric-Compatible Evolution to Probability Assignment

We may now summarize the outcome of the present section. Given the G-inner product fixed in Section 3, the present section identifies the corresponding conserved density and current:

$$\begin{array}{c}{\Pi }_{\mathrm{PT}}-\mathrm{projected}\mathrm{observability}⟹\mathrm{physical}G-\mathrm{inner}\mathrm{product}⟹j^0={\psi }^{†}G\psi \\ ⟹{\displaystyle \frac{d}{dt}}\int d^{3}x{j}^0=0.\end{array}$$

(35)

Equation (35) is exactly the statement needed for the next section. Once the physical observable sector, the admissible metric, and the conserved probability current have all been fixed, the probability rule is no longer freely specifiable independently. This is why the route to the Born rule in Section 5 can now be stated in a disciplined and auditable way.

The role of the present section is therefore limited but necessary. It does not yet write down the Born rule itself; it fixes the conserved current required for the probability interpretation used in the next section. Only after this step does the Born-rule form become admissible.

4.7. A Compact Illustrative Toy Model

Before turning to the general PTQ probability assignment, it is useful to record a minimal two-level example that makes the preceding logic concrete. The purpose of this toy model is illustrative only. It is not a derivation of PTQ from a $2\times 2$ Hamiltonian, nor a substitute for the geometric construction of Secs. 2–4. Its only role is to display, in the simplest reviewer-friendly setting, why the naive norm need not be the physical one and how a conserved G-norm leads to the corresponding Born-rule form. What it does illustrate is the metric/current/probability part of the argument once an admissible sector has already been fixed. What it does not illustrate is the PTQ-specific step by which that sector is first restricted through projective invariance and PT projection before any pseudo-Hermitian probability assignment is made.

Consider the standard unbroken-$\mathcal{PT}$ toy Hamiltonian

$$H_{\mathrm{toy}}=\left(\begin{array}{cc}i\gamma & r\\ r& -i\gamma \end{array}\right),r,\gamma \in \mathbb{R},r>\left|\gamma \right|.$$

(36)

This operator is non-Hermitian whenever $\gamma \ne 0$, since $H_{\mathrm{toy}}^{†}\ne H_{\mathrm{toy}}$. Accordingly, the naive norm $\langle \psi \mid \psi \rangle ={\psi }^{†}\psi$ is not, in general, the relevant conserved quantity. Indeed, for $i{\partial }_t\psi =H_{\mathrm{toy}}\psi$ one finds

$${\displaystyle \frac{d}{dt}}\left({\psi }^{†}\psi \right)=i{\psi }^{†}\left(H_{\mathrm{toy}}^{†}-H_{\mathrm{toy}}\right)\psi =2\gamma \left(|{\psi }_1{|}^2-{\left|{\psi }_2\right|}^2\right),$$

(37)

which is not identically zero.

A positive metric operator can nevertheless be chosen so that the same dynamics become pseudo-Hermitian. One convenient choice is

$$G_{\mathrm{toy}}=\left(\begin{array}{cc}1& -i\gamma /r\\ i\gamma /r& 1\end{array}\right),$$

(38)

for which

$$H_{\mathrm{toy}}^{†}{G}_{\mathrm{toy}}=G_{\mathrm{toy}}{H}_{\mathrm{toy}}.$$

(39)

Because $r>\left|\gamma \right|$, the eigenvalues of $G_{\mathrm{toy}}$ are $1\pm \gamma /r$, so $G_{\mathrm{toy}}$ is positive in the unbroken regime. The corresponding physical norm is therefore

$${\langle \psi \mid \psi \rangle }_{G_{\mathrm{toy}}}:={\psi }^{†}{G}_{\mathrm{toy}}\psi ,$$

(40)

and, by the same computation as in Section 3,

$${\displaystyle \frac{d}{dt}}{\langle \psi \mid \psi \rangle }_{G_{\mathrm{toy}}}=0.$$

(41)

Equivalently, the admissible probability density in this toy model is the $G_{\mathrm{toy}}$-density ${\psi }^{†}{G}_{\mathrm{toy}}\psi$, not the naive Euclidean one.

If $\left\{{\varphi }_i\right\}$ is a $G_{\mathrm{toy}}$-orthonormal eigenbasis of $H_{\mathrm{toy}}$, then the associated probability assignment takes the expected G-Born form,

$$P_i=|{\langle \psi \mid {\varphi }_i\rangle }_{G_{\mathrm{toy}}}{|}^2.$$

(42)

This compact example does not prove the PTQ result. It only illustrates, in finite dimension, the same scope-limited logic used in the main text: once the admissible sector and the compatible metric are fixed, the conserved norm and the corresponding Born-rule form follow together.

5. The PTQ Route to the Born Rule

We are now in a position to state the central claim of this paper in its proper scope. The purpose of the present section is not to assert that the Born rule has been derived from an unrestricted starting point, nor to claim that the measurement problem is solved in all of its interpretational aspects. Rather, the claim is narrower and more structural: within the PT-symmetric quaternionic (PTQ) posture established in the previous sections, the Born-rule form appears as the probability assignment compatible with the restricted sector already fixed by the earlier analysis:

$$\begin{array}{cc}\hfill {\Pi }_{\mathrm{PT}}-\mathrm{projected}\mathrm{observability}& ⟹\mathrm{physical}G-\mathrm{inner}\mathrm{product}\hfill \\ \hfill & ⟹\mathrm{conserved}\mathrm{probability}\mathrm{current}\hfill \\ \hfill & ⟹P_i=|{\langle \psi \mid {\varphi }_i\rangle }_G{|}^2.\hfill \end{array}$$

(43)

The significance of this result is not that it removes every foundational ambiguity of quantum theory, but that it identifies a constrained and geometrically motivated probability sector in PTQ spacetime.

5.1. Minimal Derivation Logic

The logical ingredients have already been established in the preceding sections. Section 2 fixed the observable sector by proving that probability-relevant quantities must be defined only after restriction to the PT-even, projectively invariant scalar sector; see in particular Propositions 1 and 2. Section 3 then showed that, once the physical sector is fixed, the admissible norm is not the naive kinematic norm but the G-metric norm selected by pseudo-Hermitian consistency; see Equation (20) and Propositions 3 and 4. Here and below, “admissible” means compatible simultaneously with the projected observable sector, pseudo-Hermitian metric compatibility, and norm-preserving evolution. Section 4 finally identified the corresponding conserved probability current, whose time component is fixed by the same G-metric structure; see Proposition 5 and Equation (35).

The present section simply closes that chain. The route may therefore be stated in four conceptual steps.

Step 1: Probabilities are defined only on the projected observable sector.

The first step is negative but indispensable: PTQ does not assign direct physical meaning to the full quaternionic kinematic space. As established in Section 2, physical observables must be real, PT-even, and projectively invariant. Hence the probability sector cannot be specified prior to projection. It must be constructed only after the restriction to the PT-even observable scalar sector. This is exactly the content of Proposition 2.

Accordingly, PTQ already departs from a naive transplantation of ordinary complex quantum mechanics into a larger quaternionic arena. The relevant question is not how to define probabilities on arbitrary quaternionic amplitudes and then discard unphysical pieces afterwards. The correct question is: once the physical sector has been selected by the observable map of Section 2, what norm and what probability assignment remain compatible with that projected dynamics? This prior observable-sector restriction is the PTQ-specific step of the present argument; the later pseudo-Hermitian probability assignment is read only within that already restricted domain.

Step 2: The projected dynamics select the physical inner product.

The second step is supplied by Section 3. Within the PT-even sector, the relevant generator of evolution is not required to be Hermitian in the naive sense. Instead, consistency is encoded by the pseudo-Hermitian condition (13), and the corresponding physical norm is the G-inner product of Equation (15). This is the precise content of Proposition 3.

The conceptual consequence is decisive: once the projected dynamics are fixed, the probability-compatible metric is no longer an independent choice. By Proposition 4, the admissible inner product is fixed up to the equivalence class relevant to the projected PT-even sector. Thus the probability sector is not free to choose an arbitrary norm after the theory has been specified. The physical norm is structurally selected.

Step 3: The selected norm induces a conserved probability current.

The third step was established in Section 4. Because the physical norm is the G-norm, the corresponding local density is $j^0={\psi }^{†}G\psi$, and its spatial integral is conserved under the same projected pseudo-Hermitian evolution; see Equation (23), Equation (28), and Proposition 5. This means that the PTQ probability sector does not merely possess an admissible norm; it also possesses the associated conserved probability current.

This point closes the gap between kinematics and probability interpretation. A candidate probability rule is physically meaningful only if the quantity it assigns is preserved by the corresponding evolution. In PTQ, that conserved quantity is already fixed by the G-metric structure. Therefore, once Equation (35) has been established, the remaining question is no longer whether there exists a distinguished probability assignment, but only what explicit form that assignment should take.

Step 4: The Born-rule form is the admissible probability assignment.

Let $\left\{{\varphi }_i\right\}$ denote a complete set of G-orthonormal eigenstates in the unbroken PT-even sector. Then the probability associated with finding the system in the state ${\varphi }_i$ is

$$P_i=|{\langle \psi \mid {\varphi }_i\rangle }_G{|}^2.$$

(44)

Equation (44) is precisely the Born-rule form, now written with the physical G-inner product already fixed by the earlier sections.

Its justification is no longer merely analogical. It follows from the conjunction of the three structural facts already established:

1.

only the PT-even projected sector is observable (Proposition 2);

2.

within that sector, pseudo-Hermitian consistency selects the physical metric and hence the physical norm (Propositions 3 and 4);

3.

the same metric induces the conserved probability current (Proposition 5).

Therefore, the Born-rule expression (44) is not inserted as an isolated postulate. It is the probability rule adapted to the PTQ probability sector.

This is also the sense in which the present paper differs from more standard pseudo-Hermitian or $\mathcal{PT}$-symmetric metric-based probability constructions. Those frameworks typically begin with a specified operator sector and then identify a compatible positive metric. Here, by contrast, the argument is PTQ-specific and logically earlier: the probability sector is admissible only after restriction to the PT-even, projectively invariant observable sector fixed in Section 2, and the metric and Born-rule form are then read within that already restricted domain.

This does not yet amount to a universal derivation in the strongest foundational sense. What it does establish is a nontrivial reduction of arbitrariness: once the observable sector, the admissible metric, and the conserved current are fixed, the probability rule is no longer freely specifiable independently. Within the present PTQ posture, the Born-rule form is admissible in the projected PT-even sector.

5.2. What Is Derived, and What Is Not

It is important to state with precision what has and has not been achieved.

What is established.

The present analysis identifies a structured PTQ route to the Born-rule form. More explicitly:

1.

the probability sector is anchored in the PT-even observable map rather than in the full quaternionic kinematic space;

2.

the physical inner product is selected by pseudo-Hermitian consistency through the metric operator G;

3.

the corresponding G-density defines a conserved probability current;

4.

the probability of measurement outcomes is therefore represented by the G-Born rule (44).

In this precise sense, PTQ makes the Born-rule form structurally admissible rather than purely ad hoc.

What is not established.

At the same time, the present paper does not claim the following:

1.

a derivation of the Born rule from a completely unrestricted starting point;

2.

a solution of the measurement problem, including state reduction, branch selection, or the ontology of outcomes;

3.

a proof that the PTQ route is the uniquely possible route to quantum probability in all generalized quantum theories;

4.

a treatment of broken-PT or strongly non-adiabatic sectors, in which the present projected pseudo-Hermitian construction may require modification.

These limitations are not weaknesses concealed by the formalism; they are the explicit scope boundaries of the present contribution.

Interpretive status.

The correct conclusion is therefore intermediate. PTQ does not yet claim to have solved the full philosophical problem of quantum probability. What it does provide is a disciplined and auditable framework in which the Born-rule form is supported simultaneously by geometric projection, metric compatibility, and conserved flow. That is already a substantial sharpening over a framework in which the probability rule is merely appended by analogy.

We may summarize the result as follows: within the projected pseudo-Hermitian sector of PTQ spacetime, the Born-rule form is the probability rule compatible with observability, norm preservation, and conserved current structure. This is the precise sense in which PTQ offers a route to the Born rule.

6. Interfaces: QFT, Geometry, and Future Tests

The preceding sections have isolated the probability sector of PTQ and shown how its internal logic closes: projected observability selects the physical sector, pseudo-Hermitian consistency selects the physical metric, and the same metric induces the conserved probability current from which the Born-rule form becomes natural; see Equations (11), (20), (35), and (43). The purpose of the present section is to explain how this result interfaces with the broader PTQ program [1,2].

This paper is not intended as an isolated note on generalized quantum probability. Its role is more specific. It identifies a scope-limited connecting layer between the already existing geometric, operator, and effective-field-theoretic ingredients of PTQ to a disciplined probability interpretation. For that reason, the probability sector developed here should be read not as a detached appendix to the PTQ literature, but as an interface between several previously separate lines of development. Its physical significance within the PTQ program is straightforward. If projected observables and projected operator sectors are taken seriously as defining the admissible physical domain, then the probability rule should not be appended independently of that same observable map. Rather, the probability sector should be formulated on the same projected PT-even sector that already governs what counts as observable and what counts as an admissible mode of evolution.

6.1. QFT Interface: Projected Propagators and PT-Even Subsectors

The first interface is to the projected QFT layer. In companion PTQ constructions [1], the quantum field-theoretic description is not formulated on the full unprojected quaternionic kinematic space, but on a PT-even projected subsector. At the level of quadratic operators, this is already encoded in the schematic projected operator (14). Accordingly, propagators, Green’s functions, and loop traces are understood to be evaluated within a restricted PT-even sector rather than in the unrestricted quaternionic algebra.

The relevance of the present paper to that QFT layer is immediate. A projected propagator formalism requires more than a kinematic projector. It also requires a precise statement of what norm and what probability interpretation are physically admissible on the projected subsector. Without such a statement, the projected operator formalism remains incomplete at the level of physical interpretation. The current paper supplies exactly that missing layer: the PT-even subsector is not only the sector on which projected propagators are defined, but also the sector on which the physical G-inner product and the conserved probability current exist.

The connection may be summarized schematically as

$$\begin{array}{c}\mathrm{projected}\mathrm{operator}\sec \mathrm{tor}⟹\mathrm{pseudo}-\mathrm{Hermitian}\mathrm{metric}\sec \mathrm{tor}⟹\\ \mathrm{probability}-\mathrm{compatible}\mathrm{projected}\mathrm{QFT}\sec \mathrm{tor}.\end{array}$$

(45)

In this sense, the present paper clarifies what the projected QFT layer of PTQ means at the level of physical state space. It does not merely say which modes propagate; it says how those modes are to be endowed with a probability interpretation once the projected operator structure has been fixed.

This also explains why the present work is naturally upstream of a fuller PTQ diagrammatic or loop-level analysis. Projected propagators and PT-restricted traces are mathematically meaningful before a probability rule is discussed, but their physical interpretation is incomplete until one knows which norm is preserved and which current is conserved. The probability sector derived here therefore provides the conceptual closure needed by the QFT interface, without itself attempting to reproduce the full diagrammatic machinery.

6.2. Geometric Interface: Projective Residue and Torsion Structure

The second interface is geometric. Section 2 already established that the observable sector is defined only after imposing both Route-A projective invariance and PT projection. In particular, the projectively invariant residue ${\tilde{T}}_{\mu }$ of Equation (6) is the physically relevant trace-sector object, and the observable scalar algebra is defined only after PT projection via Equation (8). The probability sector constructed in the present paper inherits this entire geometric posture.

This point matters because it prevents a misleading reading of the results. The probability sector of PTQ is not based on an abstract pseudo-Hermitian structure that could have been introduced independently of geometry. Rather, the admissible norm and current were shown to depend on a prior geometric selection principle: only projected, PT-even, projectively invariant scalar data are physically admissible. Thus the geometric residue structure and the probability sector are not separate modules. They are sequentially ordered layers of the same framework.

The geometric interface may therefore be expressed as

$$\begin{array}{c}\mathrm{Route}-\mathrm{A}\mathrm{projective}\mathrm{quotient}⟹\mathrm{invariant}\mathrm{residue}{\tilde{T}}_{\mu }⟹{\Pi }_{\mathrm{PT}}-\mathrm{projected}\mathrm{scalar}\sec \mathrm{tor}\\ ⟹\mathrm{probability}\sec \mathrm{tor}.\end{array}$$

(46)

The significance of Equation (46) is that it places the Born-rule question on the correct side of the geometric hierarchy. Probability is not assigned to the full bundle-valued or torsionful kinematic data. It is assigned only after the scalar observable channel has been isolated from the projective and PT constraints.

This is also where the present paper interfaces with the more structural PTQ works [1,2] on pure-trace torsion alignment, route-specific residue necessity, and the impossibility of removing the relevant trace sector by trivial redefinitions. Those works identify which geometric data survive as physically meaningful. The present paper adds that, once those data are further restricted to the observable scalar sector, the admissible probability rule is no longer freely specifiable. In this way, geometry determines not only what remains observable, but also what form the probability sector may consistently take.

6.3. Thermodynamic and Flow Interface

The third interface is to the thermodynamic and operational flow language of the broader PTQ program. In companion work [1], PTQ often organizes physically meaningful statements in the form

$$\mathrm{symmetry}/\mathrm{posture}⟹\mathrm{conserved}\mathrm{or}\mathrm{improved}\mathrm{flow}⟹\mathrm{auditable}\mathrm{invariant}.$$

(47)

This language appears naturally in entropy-inspired formulations, in projected flow descriptions, and in operational frameworks where one seeks observable diagnostics tied to symmetry-protected structures.

The probability current constructed in Section 4 fits naturally into this broader pattern. The crucial distinction, however, is that the present paper does not identify probability current with a generic thermodynamic or coarse-grained flow. The current relevant here is the pseudo-Hermitian G-current, whose time component is $j^0={\psi }^{†}G\psi$ and whose conservation follows from projected pseudo-Hermitian evolution; see Equation (28). The thermodynamic and improved-flow language does not replace this derivation. Rather, it provides a wider conceptual setting in which a projected conserved current is no longer surprising.

For that reason, the relation between the present paper and the thermodynamic interface is best described as one of compatibility rather than identity. The probability current used here is the microscopic or operator-level current relevant to the probability sector. Entropy flow, improved currents, and coarse-grained monotonicity statements may provide supporting intuition or effective descriptions at other levels of organization, but they do not by themselves determine the Born-rule sector. That determination still relies on the projected pseudo-Hermitian structure developed in Secs. 2–4.

This distinction is important both conceptually and strategically. Conceptually, it preserves the rigor of the present argument by preventing a category error between probability current and thermodynamic flow. Strategically, it shows how the probability sector developed here can serve as the microscopic anchor for broader PTQ flow-based formulations without collapsing into them.

6.4. Future Tests and Programmatic Consequences

Although the present paper is primarily foundational, it does suggest several directions for future testing and extension.

First, on the QFT side, one should examine in detail how the G-metric probability sector interacts with projected propagators, PT-even mode expansions, and loop-level effective actions. The present paper implies that any such construction should respect not only the projector structure but also the metric-compatible norm and current identified here. Concretely, one should ask whether projected propagators, completeness relations for PT-even modes, and the normalization of intermediate states can all be formulated within the same admissible G-metric sector, and whether failure of any one of these conditions signals an inconsistency in the proposed projected description. In particular, a projected construction that preserves propagator-level projection but fails at mode completeness or normalization within the same G-sector would not realize the present closure in a consistent way. This provides a concrete consistency criterion for future PTQ QFT developments.

Second, on the geometric side, it would be desirable to study more explicitly how the probability sector changes when one leaves the scalar-channel domains characterized by Equation (9), or when nontrivial bundle-valued sectors are retained. The current paper deliberately restricts attention to the scalar observable channel. Whether and how the probability sector extends beyond that channel remains an open question. In particular, one should check whether positivity of the admissible metric, conservation of the corresponding norm, and the projected observable interpretation remain compatible near PT-broken regimes or under genuinely non-scalar-channel extensions. Loss of that joint compatibility would be the clearest indication that the present scope-limited closure has ceased to apply and must be replaced rather than extended by continuity.

Third, on the phenomenological and operational side, one may ask whether the probability sector identified here leaves indirect signatures in effective observables. The present work does not claim any immediate empirical discriminator of the Born-rule route itself. However, the fact that projected observability, metric compatibility, and conserved current are all structurally linked suggests that future PTQ observables may have to satisfy nontrivial consistency relations between geometric, dynamical, and statistical sectors. At a minimum, one may ask whether candidate effective descriptions that share the same projected observable map also admit mutually compatible metric, current, and probability assignments, or whether some apparently acceptable constructions fail precisely at that joint consistency level.

These directions show that the present paper is not an endpoint. Its contribution is to place the probability question in the correct structural location within the PTQ program. Once that location is fixed, future QFT, geometric, and operational extensions can be asked in a more disciplined way.

7. Discussion and Scope Boundary

The purpose of the present section is to state as precisely as possible the regime of validity, interpretive boundary, and non-claims of the construction developed in Secs. 2–6. The main body of the paper has already established a constrained route from projected observability to the Born-rule form. The role of the present section is therefore not to restate that route as a conclusion, but to clarify the exact domain in which it should be read.

7.1. Regime of Validity

The construction developed in this paper is formulated in the projected PT-even sector of PTQ spacetime. More specifically, it assumes:

1.

the observable scalar sector is selected by the combined Route-A projective posture and PT projection, as developed in Section 2;

2.

the projected dynamics admit a pseudo-Hermitian metric completion in the sense of Equation (13);

3.

the associated probability density and current are defined with respect to the corresponding G-metric structure, as in Secs. 3 and 4.

Accordingly, the present analysis should be read as a result about the unbroken PT-even projected sector, not about the full quaternionic kinematic space in all regimes.

This restriction is not a defect added after the fact. It is built into the logic of the paper. The central chain

$$\begin{array}{c}\mathrm{projected}\mathrm{observability}⟹\mathrm{physical}G-\mathrm{inner}\mathrm{product}⟹\mathrm{conserved}\mathrm{probability}\mathrm{current}\\ ⟹\mathrm{Born}-\mathrm{rule}\mathrm{form}\end{array}$$

(48)

is meaningful only when each arrow is well defined. If one leaves the PT-even observable sector, abandons the projected scalar-channel description, or enters a regime in which pseudo-Hermitian consistency fails, then the present construction may no longer apply without modification.

7.2. What Is Not Claimed

To avoid overstatement, it is useful to list explicitly what the present paper does not claim.

1.

It does not claim a universal derivation of the Born rule from first principles valid for all conceivable quantum frameworks.

2.

It does not claim a solution of the measurement problem. In particular, collapse, branch selection, and the ontology of outcomes remain outside the scope of the present work.

3.

It does not claim that every non-Hermitian, quaternionic, or PT-symmetric theory must reduce to the specific probability sector described here.

4.

It does not treat broken-PT regimes, strongly non-adiabatic sectors, or generic bundle-valued sectors beyond the scalar observable channel.

These non-claims are important because the contribution of the paper is structural rather than universal. What has been shown is that within the PTQ posture fixed in this paper, the probability sector is no longer freely specifiable once the observable sector and admissible dynamics have been specified. That result is significant, but it should not be inflated into a claim stronger than the paper actually supports.

7.3. Internal Versus Universal Admissibility

A related clarification concerns the meaning of admissibility. Throughout the paper, we have emphasized that the admissible metric and the associated probability current are not freely specifiable. This statement should be read in an internal sense.

More precisely, once the projected observable sector has been fixed (Proposition 2) and the pseudo-Hermitian compatibility condition has been imposed (Propositions 3 and 4), the physically admissible norm and the corresponding probability current are no longer independently freely specifiable. In that sense, the Born-rule form identified in Section 5 is structurally selected.

This is not the same as claiming a universal metaphysical uniqueness theorem for quantum probability. The present paper does not exclude the possibility that other generalized frameworks may realize probability differently. Its claim is narrower: within the projected pseudo-Hermitian sector of PTQ, the admissible probability rule is structurally constrained.

7.4. Relation to Broader PTQ Layers

The probability sector isolated here should also be distinguished from, but not separated from, the broader PTQ program.

On the geometric side, the present construction depends on the Route-A projective quotient, the invariant residue ${\tilde{T}}_{\mu }$, and the PT projection of the scalar sector. On the operator side, it depends on the pseudo-Hermitian metric-compatible structure of the projected dynamics. On the QFT side, it provides the norm and current structure needed to interpret projected propagators and PT-even subsectors physically. On the flow/thermodynamic side, it is compatible with the broader PTQ language in which symmetry or posture gives rise to conserved or improved quantities, but it is not identical to those coarse-grained or effective descriptions.

The practical lesson is that the present paper should not be read either too narrowly or too broadly. It is not merely a formal appendix about one possible inner product, but neither is it a complete theory of quantum measurement. It occupies an intermediate but important layer: the layer at which PTQ geometry and PTQ dynamics first become a disciplined probability sector.

7.5. Open Directions Beyond the Present Scope

Several natural directions remain open.

First, one would like to know how the present probability-sector construction extends beyond the scalar observable channel, especially in settings where the projected representation of ${\tilde{T}}_{\mu }$ becomes insufficient.

Second, one should study more carefully the fate of the construction in broken-PT regimes, where the existence of a stable pseudo-Hermitian metric sector may become nontrivial.

Third, it remains to be explored whether the probability-sector structure identified here induces indirect constraints on PTQ phenomenology or on projected QFT observables.

These questions are not pursued in the present paper. Their existence does not weaken the current result; rather, it indicates that the present analysis identifies the structural layer on which such future extensions should be built.

8. Conclusion

In this work we have isolated and analyzed the probability sector of PT-symmetric quaternionic spacetime from a structural point of view. The central contribution of the paper is not the claim of a universal derivation of the Born rule from an unrestricted starting point, but the identification of a coherent route by which the Born-rule form becomes admissible within the projected pseudo-Hermitian sector of PTQ.

That route proceeds through three steps. First, the probability sector must be defined only after restriction to the real, PT-even, projectively invariant observable scalar sector. Second, consistency of the projected dynamics selects a physically distinguished G-inner product rather than a naive kinematic norm. Third, the same metric structure induces a conserved probability current, $j^0={\psi }^{†}G\psi$, whose spatial integral is preserved under the projected evolution. Once these elements are fixed, the probability assignment for G-orthonormal states takes the Born-rule form

$$P_i=|{\langle \psi \mid {\varphi }_i\rangle }_G{|}^2.$$

(49)

The significance of the result lies in the reduction of arbitrariness. Within the PTQ framework studied here, probability is not appended independently after the geometry and dynamics have been chosen. In that precise sense, the Born-rule form is not merely imported by analogy, but appears as the admissible probability rule within the projected sector.

This is also why the result matters physically inside the PTQ program even in the absence of any direct experimental confirmation at the present stage. If one accepts projected observables and projected operator sectors as part of the physical posture of PTQ, then one should also require the probability rule to be compatible with that same projection rather than appended independently afterwards. The present paper supplies exactly that compatibility statement in a scope-limited form.

This also clarifies the role of the present paper within the broader PTQ program. It is neither a replacement for the geometric works nor a substitute for future QFT or phenomenological developments. Rather, it supplies a scope-limited connecting layer between geometric selection, pseudo-Hermitian dynamics, and physical probability. That connection is what allows the PTQ program to move from a collection of formal ingredients toward a disciplined probabilistic framework.

Future work may extend this construction in several directions, including broken-PT sectors, non-scalar observable channels, and the detailed interaction between the G-metric probability structure and projected QFT observables. More concretely, future work should test whether projected propagators, PT-even mode expansions, and the associated normalization rules can all be maintained within a single admissible metric sector, and how that structure changes near PT-broken regimes or beyond the scalar channel. It should also test whether candidate PTQ effective descriptions satisfy the linked consistency requirements of geometric projection, dynamical admissibility, and statistical interpretation at the same time. Those developments lie beyond the present scope. What the current paper establishes is the more basic point: once the observable sector and admissible dynamics are fixed within PTQ, the probability rule is no longer freely specifiable independently.

We therefore conclude with the following statement. PT-symmetric quaternionic spacetime does not yet provide a complete theory of quantum probability in the strongest possible sense, but it does provide a nontrivial structural framework in which the Born-rule form becomes admissible in the projected pseudo-Hermitian sector. This is the precise and limited sense in which PTQ offers a scope-limited structural result for the probability sector of quantum theory.

Appendix A.1. Guiding Principle of the Notation

The notation of the paper is organized according to the following structural sequence:

$$\begin{array}{cc}\hfill & \mathrm{kinematic}\mathrm{quaternionic}-\mathrm{geometric}\sec \mathrm{tor}⟹\mathrm{projectively}\mathrm{invariant}\sec \mathrm{tor}\hfill \\ \hfill & ⟹{\Pi }_{\mathrm{PT}}-\mathrm{projected}\mathrm{observable}\sec \mathrm{tor}\hfill \\ \hfill & ⟹\mathrm{pseudo}-\mathrm{Hermitian}G-\mathrm{metric}\sec \mathrm{tor}⟹\mathrm{probability}\sec \mathrm{tor}.\hfill \end{array}$$

(A50)

Every symbol appearing in the main text should be understood relative to one of these layers. The main source of potential confusion is that a quantity may exist at the kinematic level but fail to survive projection, or may survive projection but fail to define a physical probability quantity until the metric-compatible structure has been specified. Equation (A50) summarizes the intended reading order.

Appendix A.2. Geometric and Projective Symbols

We first collect the geometric symbols introduced in Section 2.

g _μν .

The spacetime metric. In the present paper it serves as part of the metric-affine / Palatini kinematic data and is not, by itself, sufficient to define the probability sector.

Γ ^λ _μν .

The affine connection. It transforms under the Route-A full projective transformation according to Equation (2).

ξ _μ .

The arbitrary one-form parameter of the full projective orbit in Equation (2).

T _μ .

The torsion trace,

$$T_{\mu }:={T^{\lambda }}_{\lambda \mu },$$

(A51)

introduced in Equation (3). Under the full projective transformation it shifts according to Equation (4). As emphasized in Section 2, $T_{\mu }$ by itself is not the invariant physical trace-sector quantity.

A _μ .

The compensator one-form of Route A, transforming as in Equation (5). Its role is spurionic rather than dynamical in the present paper: it completes the full projective one-form orbit and allows one to define an invariant trace-sector residue.

${\tilde{T}}_{\mu }$ .

The projectively invariant residue,

$${\tilde{T}}_{\mu }:=T_{\mu }-A_{\mu },$$

(A52)

defined in Equation (6). Whenever the main text refers to the physical trace-sector object relevant for the scalar probability channel, it is this residue that is intended. It is the correct variable on the projective quotient.

ϵ .

A scalar-channel representative defined only on admissible domains satisfying Equation (9), such that Equation (10) holds. Crucially, $ϵ$ is not the compensator of the projective orbit. It is only a scalar representative of the already invariant residue ${\tilde{T}}_{\mu }$ on scalar-channel domains. This distinction is essential to avoid conflating Route-A completion with scalar-channel reduction.

Route A.

The term “Route A” refers to the projective posture in which the full one-form projective orbit is completed by a compensator one-form $A_{\mu }$, rather than only by an exact scalar Stueckelberg completion. This posture is fixed in Section 2 and is the geometric background assumed throughout the rest of the paper.

Appendix A.3. Projection and Observable-Sector Symbols

We next collect the symbols that define the observable sector.

Π PT .

The PT projection operator introduced in Equation (7). It acts on scalar quantities and removes PT-odd contributions, projecting onto the real PT-even scalar sector. This projector is part of the definition of the physical sector, not a post hoc filter.

O .

A generic scalar quantity in the effective description. It becomes an observable only if it satisfies the condition of Equation (8).

Obs .

The observable scalar sector defined in Equation (8). By Proposition 1, a scalar quantity belongs to $\mathrm{Obs}$ only if it is both projectively invariant and PT-even after projection.

Observable sector.

Throughout the paper, the phrase “observable sector” means the projected PT-even, projectively invariant scalar sector. It never means the full quaternionic kinematic space. This convention is fixed by Proposition 2 and is indispensable for the later probability interpretation.

Appendix A.4. Operator and Metric Symbols

We next summarize the operator-level and metric-level notation introduced in Section 3.

H .

The effective generator of evolution (or Hamiltonian) restricted to the scalar probability sector. At the naive kinematic level it need not satisfy Equation (12). Its physical consistency is instead encoded by the pseudo-Hermitian condition Equation (13).

D_bare .

A schematic unprojected kinematic quadratic operator, appearing inside the projected operator construction.

D_PTQ .

The projected quadratic operator introduced schematically in Equation (14). It acts within the PT-even projected sector and represents the operator-level realization of the PTQ QFT posture relevant for the present paper.

G .

The positive metric operator that defines the physical inner product on the projected PT-even sector. Its defining compatibility condition is

$$H^{†}G=GH,$$

(A53)

namely Equation (13). The main text does not use G as an arbitrary auxiliary metric, but as the probability-compatible metric selected by pseudo-Hermitian consistency, as stated in Propositions 3 and 4.

〈ψ∣χ〉 G .

The physical inner product,

$${\langle \psi \mid \chi \rangle }_G:=\langle \psi \mid G\chi \rangle ,$$

(A54)

defined in Equation (15). Whenever the paper refers to the physical norm or the probability-compatible norm, this G-inner product is meant.

Pseudo-Hermiticity .

In the present paper, this term always refers to compatibility with Equation (13), not to naive Hermiticity on the full unprojected space. Pseudo-Hermiticity is the operator-level structure that makes the projected PT-even sector norm preserving.

Appendix A.5. Probability-Sector Symbols

We now list the symbols directly associated with the probability sector developed in Secs. 4 and 5.

ψ, χ, ϕ _i .

States in the PT-even projected sector. Unless explicitly stated otherwise, they are always understood as belonging to the physical sector selected by Section 2 and endowed with the G-metric structure of Section 3.

j ⁰ .

The physical probability density, defined locally by Equation (23),

$$j^0={\psi }^{†}G\psi .$$

(A55)

This is the time component of the distinguished conserved current identified in Proposition 5.

j ^μ .

The conserved probability current whose time component is $j^0$. Its continuity equation is written schematically in Equations (29) and (30). The paper does not require a universal closed form for every spatial component; what is essential is that the current is induced by the same G-metric structure that fixes the physical norm.

P _i .

The measurement probability associated with the G-orthonormal eigenstate ${\varphi }_i$, defined by the Born-rule form

$$P_i=|{\langle \psi \mid {\varphi }_i\rangle }_G{|}^2,$$

(A56)

namely Equation (44). This is the final probability assignment justified in Section 5.

Probability sector.

Throughout the paper, “probability sector” means the sector in which all of the following are simultaneously in force:

1.

projected observability in the sense of Proposition 2;

2.

pseudo-Hermitian metric compatibility in the sense of Propositions 3 and 4;

3.

conserved probability current in the sense of Proposition 5.

It is therefore a narrower notion than either the full geometric sector or the full projected kinematic sector.

Appendix A.6. Logical Map of the Paper

For convenience, we summarize the logical function of the main equations and propositions.

Sec. 2: geometric posture.

• Equation (2): full projective shift of the connection;
• Equation (6): definition of the invariant residue ${\tilde{T}}_{\mu }$;
• Equation (7): definition of the PT projector;
• Equation (8): definition of the observable sector;
• Proposition 1: observable-sector criterion;
• Proposition 2: projection precedes probability;
• Equation (11): summary of the geometric reduction.

Sec. 3: metric structure.

• Equation (13): pseudo-Hermitian compatibility;
• Equation (15): definition of the physical G-inner product;
• Proposition 3: physicality of the G-metric;
• Proposition 4: non-arbitrariness of the metric;
• Equation (20): summary of the metric selection.

Sec. 4: conserved current.

• Equation (23): local probability density;
• Equation (28): global probability conservation;
• Proposition 5: conserved probability density;
• Equation (35): summary of the current structure.

Sec. 5: Born-rule route.

• Equation (43): the full PTQ route to the Born rule;
• Equation (44): the Born-rule form in the G-metric sector.

Appendix A.7. Compact Symbol Table

For ease of reference, we conclude with a compact table of the most frequently used symbols. The table is purely mnemonic and introduces no new content.

Symbol	Meaning in this paper
$g_{\mu \nu }$	spacetime metric (kinematic geometric datum)
${T^{\lambda }}_{\lambda \mu }$	affine connection
${\xi }_{\mu }$	one-form projective shift parameter
$T_{\mu }$	torsion trace before projective completion
$A_{\mu }$	Route-A compensator one-form
${\tilde{T}}_{\mu }$	projectively invariant residue, Eq. (6)
$ϵ$	scalar-channel representative of ${\tilde{T}}_{\mu }$ on admissible domains
${\Pi }_{\mathrm{PT}}$	PT projection operator, Eq. (7)
$\mathrm{Obs}$	projected PT-even observable scalar sector
H	generator of projected evolution
$D_{\mathrm{bare}}$	unprojected quadratic operator
$D_{\mathrm{PTQ}}$	PTQ projected operator, Eq. (14)
G	positive metric operator selecting the physical norm
${\langle \psi \mid \chi \rangle }_G$	physical G-inner product
$j^0$	probability density, Eq. (23)
$j^{\mu }$	conserved probability current
$P_i$	Born-rule probability, Eq. (44)

With this notation fixed, the reader may regard the main text as a sequence of three reductions: from geometry to observability, from observability to metric-compatible evolution, and from metric-compatible evolution to probability. Appendix A is intended to make that sequence transparent at the level of symbols as well as at the level of concepts.

Appendix B.1. Appendix-Level Role and Logical Status

The results recorded here should be read as support for the main text, not as new independent claims. In particular:

1.

Appendix B.1 records the minimal projector properties needed to justify the use of ${\Pi }_{\mathrm{PT}}$ in Section 2;

2.

Appendix B.2 records the pseudo-Hermitian setup underlying Section 3;

3.

Appendix B.3 records the short derivation of the conserved G-current used in Section 4.

These fragments are sufficient for the present paper because its claim is not that PTQ solves the full measurement problem, but that PTQ supplies a coherent and auditable route to the Born-rule form once the projected sector, the admissible metric, and the conserved current have been fixed.

Appendix B.2. Projector Properties in the Scalar Observable Sector

We first record the basic algebraic properties of the PT projector used in Section 2. Let ${\Pi }_{\mathrm{PT}}$ act on scalar quantities $\mathcal{O}$ according to Equation (7),

$${\Pi }_{\mathrm{PT}}\left[\mathcal{O}\right]={\displaystyle \frac{1}{2}}\left(\mathcal{O}+{\mathcal{O}}^{\mathrm{PT}}\right).$$

(A57)

The projector properties relevant for the present paper are the following.

Lemma A6  

(Idempotence of the PT projector). Assuming that the combined PT action on the scalar channel is involutive,

$${\left({\mathcal{O}}^{\mathrm{PT}}\right)}^{\mathrm{PT}}=\mathcal{O},$$

(A58)

the operator ${\Pi }_{\mathrm{PT}}$ is idempotent:

$${\Pi }_{\mathrm{PT}}^2={\Pi }_{\mathrm{PT}}.$$

(A59)

Proof. 

Applying ${\Pi }_{\mathrm{PT}}$ twice gives

$$\begin{array}{cc}\hfill {\Pi }_{\mathrm{PT}}^2\left[\mathcal{O}\right]& ={\Pi }_{\mathrm{PT}}\left[{\displaystyle \frac{1}{2}}\left(\mathcal{O}+{\mathcal{O}}^{\mathrm{PT}}\right)\right]\hfill \\ \hfill & ={\displaystyle \frac{1}{2}}\left[{\displaystyle \frac{1}{2}}\left(\mathcal{O}+{\mathcal{O}}^{\mathrm{PT}}\right)+{\displaystyle \frac{1}{2}}\left({\mathcal{O}}^{\mathrm{PT}}+{\left({\mathcal{O}}^{\mathrm{PT}}\right)}^{\mathrm{PT}}\right)\right]\hfill \\ \hfill & ={\displaystyle \frac{1}{4}}\left[\mathcal{O}+2{\mathcal{O}}^{\mathrm{PT}}+{\left({\mathcal{O}}^{\mathrm{PT}}\right)}^{\mathrm{PT}}\right].\hfill \end{array}$$

(A60)

Using Equation (A58), this becomes

$${\Pi }_{\mathrm{PT}}^2\left[\mathcal{O}\right]={\displaystyle \frac{1}{2}}\left(\mathcal{O}+{\mathcal{O}}^{\mathrm{PT}}\right)={\Pi }_{\mathrm{PT}}\left[\mathcal{O}\right],$$

(A61)

which proves Equation (A59). □

Lemma A7  

(Projected scalar observables are PT-even). For any scalar quantity $\mathcal{O}$, the projected quantity ${\Pi }_{\mathrm{PT}}\left[\mathcal{O}\right]$ is PT-even:

$${\left({\Pi }_{\mathrm{PT}}\left[\mathcal{O}\right]\right)}^{\mathrm{PT}}={\Pi }_{\mathrm{PT}}\left[\mathcal{O}\right].$$

(A62)

Proof. 

Using the definition (A57),

$$\begin{array}{cc}\hfill {\left({\Pi }_{\mathrm{PT}}\left[\mathcal{O}\right]\right)}^{\mathrm{PT}}& ={\left[{\displaystyle \frac{1}{2}}\left(\mathcal{O}+{\mathcal{O}}^{\mathrm{PT}}\right)\right]}^{\mathrm{PT}}\hfill \\ \hfill & ={\displaystyle \frac{1}{2}}\left({\mathcal{O}}^{\mathrm{PT}}+{\left({\mathcal{O}}^{\mathrm{PT}}\right)}^{\mathrm{PT}}\right)\hfill \\ \hfill & ={\displaystyle \frac{1}{2}}\left({\mathcal{O}}^{\mathrm{PT}}+\mathcal{O}\right)={\Pi }_{\mathrm{PT}}\left[\mathcal{O}\right],\hfill \end{array}$$

(A63)

where Equation (A58) was used in the final step. □

The point of Lemmas A6 and A7 is modest but essential. They justify the use of ${\Pi }_{\mathrm{PT}}$ as a genuine projection operator on the scalar channel and explain why the observable sector of Section 2 can consistently be defined through Equation (8). Combined with the projective-invariant residue ${\tilde{T}}_{\mu }$ of Equation (6), they support Proposition 1: the scalar quantities relevant to the probability sector are precisely those built from projectively invariant data and fixed under PT projection.

Appendix B.3. Minimal Pseudo-Hermitian Setup

We next record the proof fragment underlying the metric-compatible structure of Section 3. Let H be the generator of evolution on the projected PT-even sector, and assume that there exists a positive invertible operator G satisfying Equation (13),

$$H^{†}G=GH.$$

(A64)

We define the G-inner product as in Equation (15),

$${\langle \psi \mid \chi \rangle }_G:=\langle \psi \mid G\chi \rangle .$$

(A65)

Lemma A8  

(Symmetry of the generator with respect to the G-inner product). If Equation (A64) holds, then

$${\langle \psi \mid H\chi \rangle }_G={\langle H\psi \mid \chi \rangle }_G$$

(A66)

for states $\psi ,\chi$ in the projected sector.

Proof. 

By the definition of the G-inner product,

$$\begin{array}{cc}\hfill {\langle \psi \mid H\chi \rangle }_G& =\langle \psi \mid GH\chi \rangle \hfill \\ \hfill & =\langle \psi \mid H^{†}G\chi \rangle \hfill \\ \hfill & =\langle H\psi \mid G\chi \rangle ={\langle H\psi \mid \chi \rangle }_G,\hfill \end{array}$$

(A67)

where Equation (A64) was used in the second step. □

Lemma A8 is the minimal operator-level reason that the G-inner product is the physically distinguished one. It shows that the projected evolution generator is not freely specifiable with respect to the probability sector: it is compatible with the G-metric structure in exactly the sense required by Section 3.

We next record the standard norm-preservation fragment.

Lemma A9  

(G-norm preservation). Let the projected evolution be governed by Equation (16),

$$i{\partial }_t\psi =H\psi ,$$

(A68)

with H satisfying Equation (A64). Then the G-norm is preserved:

$${\displaystyle \frac{d}{dt}}{\langle \psi \mid \psi \rangle }_G=0.$$

(A69)

Proof. 

Differentiating Equation (A65) for $\chi =\psi$ yields

$$\begin{array}{cc}\hfill {\displaystyle \frac{d}{dt}}{\langle \psi \mid \psi \rangle }_G& =\langle {\partial }_t\psi \mid G\psi \rangle +\langle \psi \mid G{\partial }_t\psi \rangle \hfill \\ \hfill & =-i\langle \psi \mid H^{†}G\psi \rangle +i\langle \psi \mid GH\psi \rangle \hfill \\ \hfill & =0,\hfill \end{array}$$

(A70)

using Equation (A64). □

Lemma A9 is the appendix-level fragment behind Equation (17) and Proposition 3. The present paper does not need a stronger statement than this. Its purpose is only to show that, once the PT-even projected sector is fixed, the physical norm is the norm preserved by the evolution generated by H.

Appendix B.4. Probability-Current Derivation

Finally, we record the short derivation underlying the conserved current of Section 4. The current relevant to the present paper is not introduced independently of the metric. Rather, it is induced by the same G-inner product that defines the physical norm. Writing the local density as in Equation (23),

$$j^0\left(x\right)={\psi }^{†}\left(x\right)G\psi \left(x\right),$$

(A71)

we obtain the integrated norm

$${\langle \psi \mid \psi \rangle }_G=\int d^{3}x{j}^0\left(x\right),$$

(A72)

which is the same as Equation (24).

Lemma A10  

(Global conservation of the PTQ probability density). Under the projected pseudo-Hermitian evolution of Equation (A68), the total probability is conserved:

$${\displaystyle \frac{d}{dt}}\int d^{3}x{j}^0\left(x\right)=0.$$

(A73)

Proof. 

Using Equation (A71),

$$\begin{array}{cc}\hfill {\displaystyle \frac{d}{dt}}\int d^{3}x{j}^0\left(x\right)& =\int d^{3}x\left[\left({\partial }_t{\psi }^{†}\right)G\psi +{\psi }^{†}G\left({\partial }_t\psi \right)\right]\hfill \\ \hfill & =\int d^{3}x\left[i{\psi }^{†}{H}^{†}G\psi -i{\psi }^{†}GH\psi \right]\hfill \\ \hfill & =0,\hfill \end{array}$$

(A74)

where Equation (A64) was used in the last step. □

Lemma A10 is the appendix-level support for Equation (28) and Proposition 5. It shows that the probability density entering the Born-rule route is not an arbitrary density but the one induced by the metric-compatible norm.

For completeness, we also record the local continuity interpretation in its weakest form.

Lemma A11  

(Schematic continuity form). Whenever the projected kinetic operator admits a local current representation in the scalar sector, the conserved density $j^0$ may be completed to a current $j^{\mu }$ satisfying

$${\partial }_{\mu }{j}^{\mu }=0\mathrm{or},\mathrm{covariantly},{\nabla }_{\mu }{j}^{\mu }=0.$$

(A75)

Proof. 

This is the local version of Lemma A10. Its detailed form depends on the effective projected kinetic operator and on the background geometry. The present paper does not require a model-independent expression for the spatial current components. It requires only that the conserved density induced by the G-metric norm admit the usual continuity interpretation when a local representative exists. □

Lemma A11 is intentionally weak. Its purpose is not to supply a universal closed form for every projected PTQ realization, but to explain why Equations (29) and (30) are the natural local expressions of the global conservation law.

Appendix B.5. How the Proof Fragments Support the Main Route

We conclude by summarizing how the fragments collected above map onto the main text.

Projection fragment.

Lemmas A6 and A7 justify the use of ${\Pi }_{\mathrm{PT}}$ as a genuine projection operator on the scalar channel and support the observable-sector logic of Section 2.

Metric fragment.

Lemmas A8 and A9 justify the metric-compatible pseudo-Hermitian setup of Section 3 and support the claim that the G-inner product is the physically admissible norm on the projected sector.

Current fragment.

Lemmas A10 and A11 justify the conserved-current logic of Section 4 and support the use of $j^0={\psi }^{†}G\psi$ as the physical probability density.

Taken together, these fragments provide the minimal technical bridge between the main text’s three structural pillars and the final Born-rule statement of Section 5. They do not strengthen the scope of the claim. Rather, they make explicit that the route

$$\begin{array}{cc}\hfill {\Pi }_{\mathrm{PT}}-\mathrm{projected}\mathrm{observability}& ⟹\mathrm{physical}G-\mathrm{inner}\mathrm{product}\hfill \\ \hfill & ⟹\mathrm{conserved}\mathrm{probability}\mathrm{current}\hfill \\ \hfill & ⟹P_i=|{\langle \psi \mid {\varphi }_i\rangle }_G{|}^2.\hfill \end{array}$$

(A76)

is supported by a coherent chain of projector, metric, and conservation arguments already implicit in the broader PTQ program.

Appendix C.1. Continuity of Ingredients

The earlier phenomenology appendix already contained several ingredients that are directly relevant to the present paper: the use of a pseudo-Hermitian metric operator G, the corresponding G-inner product, the probability density $j^0={\psi }^{†}G\psi$, and the associated Born-rule expression $P_i=|{\langle \psi \mid {\varphi }_i\rangle }_G{|}^2$. In that sense, the present paper does not introduce those formulas from nothing. They were already present at appendix level within the broader PTQ program.

What was not developed there as an independent target was the probability sector itself. In the earlier phenomenology manuscript, these formulas appeared as supporting consistency material inside a paper whose primary purpose was weak-field phenomenology and SPARC-level testing. By contrast, the present paper isolates the probability question as its central topic.

Appendix C.2. What Is Different in the Present Paper

The difference is therefore not merely one of repetition, but one of scope, ordering, and argumentative emphasis.

First, the logical order is now explicit. The present paper organizes the argument as a dedicated route from projected observability, to metric admissibility, to conserved current, and finally to the Born-rule form. What previously appeared as compressed supporting material is now treated as the main object of analysis.

Second, the geometric observable sector is now part of the argument itself. The present paper makes explicit that probability is posed only after restriction to the PT-even, projectively invariant scalar sector. This observable-sector restriction is not a decorative preface to the probability formulas; it is the PTQ-specific starting point from which the later pseudo-Hermitian construction is read.

Third, the conserved current is now treated as an independent structural pillar. In the present paper, the step from metric compatibility to conserved probability current is separated and made explicit, so that the Born-rule form appears only after the relevant norm and current structure have both been fixed. This is part of what makes the present paper a dedicated probability-sector treatment rather than an appendix-level completion.

Appendix C.3. Reading Guide

The two texts should therefore be read as complementary rather than competing.

If the reader’s question concerns weak-field phenomenology, SPARC-level testing, or the broader dictionary relating PTQ to galactic and cosmological observables, the earlier phenomenology manuscript [1] remains the appropriate reference.

If the reader’s question concerns how PTQ supports a scope-limited but internally coherent probability sector in which the Born-rule form becomes admissible, the present paper should be read as the dedicated treatment.

Appendix C.4. Summary Statement

The present paper develops, in standalone form, a probability-sector line of argument that previously appeared only in compressed appendix form within a phenomenology paper. Its distinct contribution is not the isolated appearance of the probability formulas themselves, but their explicit organization into a dedicated structural route with clear scope boundaries and clear interfaces to the broader PTQ program.