Quantal theory of gravity’s two‐entity formalism: wavefront and corpuscle constituents of the test particle coexist as per recent atom-interferometry

1 Introduction

The Quantal Theory of Gravity (QTG) (Yarman et al., 2020; Marchal et al., 2023) is conceived as a unique framework for describing the behavior of any gravitating test object in the presence of a host mass, where it is necessary to distinguish the former’s “wave-like” (quantal) characteristic from its “particle-like” (corpuscular) characteristic depending on the mode of interaction.

While the General Theory of Relativity (GTR) is often lauded for its elegant simplicity—as summarized by John Wheeler’s mantra that matter’s energy content curves space and space directs it in its path—Einstein’s framework remains fundamentally incomplete in its treatment of the internal structure of matter, which is assumed to be point-like. As it turns out, the GTR relies on the clock hypothesis (Einstein, 1956), which assumes that the rate of a clock is unaffected by its acceleration and is decided only by its instantaneous translational velocity.

However, previous research has reported both theoretical predictions and experimental indications that this hypothesis fails under high-precision scrutiny (Yarman et al., 2020; Marchal et al., 2023).

Such a failure reveals that the GTR not merely faultily approximates the behavior of high-energy photons, but faces, at its core, well-known conceptual tensions with quantum mechanics (QM) due to its space-time curvature foundation.

The QTG, on the other hand, offers an alternative perspective in regimes where the applicability of the GTR is debated by exponentially scaling the Hamiltonian frequency of the moving client body per se as f = f 0 ∞ e α (in keeping with h f = γ m 0 ∞ c 2 ), thereby providing the first-principles mechanical reason for motion that the GTR lacks (Marchal et al., 2023). The frequency f here is associated with the “internal dynamics” of the test object embedded in gravity, and α is the usual dimensionless field potential, while f 0 ∞ is the same frequency in but empty space, m 0 ∞ is the proper rest-mass of the test object, and γ is the associated Lorentz coefficient.

Despite such a foundational difference between the QTG and Einstein’s metric worldview, the former’s solutions to classical problems agree, within measurable precision, with all the experimental results so far thought to confirm the GTR. Yet, this strikingly applies to our quantal case. At any rate, the QTG offers a conceptually different, potentially simpler, and broader mechanistical framework. Thus, while the GTR remains the standard benchmark for gravitational tests in the weak-field and wave-like regimes (Will, 2014), the QTG reproduces the standard weak-field results commonly attributed to the GTR, and provides a more complete foundation in full symbiosis with QM. Thus, unlike metric theories, the QTG is directly applicable to all bound fields through the Hamiltonian scaling of rest-mass in contrast to standard geometrical approaches that often fall short in this regard.

In the current landscape of theoretical physics, various attempts were made to bridge the gap between gravitation and QM by exploring the fractal or topological nature of the vacuum (Rekik and Ouari, 2024). Alternative approaches to the unification of QM with gravity and the resolution of wave–particle duality include information-theoretical, decoherence-based, and pilot-wave frameworks. Even though these complex geometrical endeavors offer intriguing insights into many areas of modern research, including cosmic acceleration, the QTG’s two-entity formalism (Marchal et al., 2023) distinguishes itself by returning to fundamental energy and momentum conservation laws. We thereby demonstrate that the transition between quantal and corpuscular identities is not a topological emergence but a direct consequence of the Hamiltonian scaling of rest-mass within what we call Universal Matter Architecture (UMA).

The two-entity behavior as per UMA is not merely a theoretical construct but is supported by robust empirical data. As detailed in Supplementary Appendix A, UMA allows for systematization at all levels of matter and, furthermore, the quantization of the second law of thermodynamics (Yarman et al., 2018), thus providing a compelling validation of the forthcoming UMA theorem.

This entails that the same scaling rules governing the macroscopic trajectories of gravitationally-interacting photons also dictate the classification of sub-atomic particles within nuclear fields, as well as diatomic molecules—in fact, all molecules conducive to such a straight systematization—thereby establishing a unified mechanical foundation across all scales.

It should be stressed that the structural departure of the QTG from the metric tradition is predicated on the recognition that the clock hypothesis—the assumption that the rate of a clock is unaffected by its acceleration—is not a universal rule at all, but rather is a low-energy approximation in gravity that fails in high-energy regimes (Marchal et al., 2023).

By demonstrating that the rest-mass of a particle scales dynamically with the gravitational potential, we show that time dilation is a secondary effect of a primary mechanical transformation of the object’s energy content per se. UMA introduces, in effect, the Hamiltonian scaling of rest-mass as the foundational pillar of not only the QTG but also of Yarman’s Approach (YA)—with this being a generalization of the interaction of the test object with any kind of corresponding field (Supplementary Appendix A).

Rest-mass alteration vis-à-vis the field gradient clarifies how the “internal dynamics” energy levels of the entity under consideration are actually not invariant but are intrinsically coupled to the field density. This provides the rigorous link for dynamic exchange between wavefront and corpuscle constituents that any particle is composed of. The QTG thus distinguishes itself from metric theories by positing a ubiquitously common mechanistical physicality that underlies all types of interactions.

Below, in Section 2, the main features of our undertaking are excerpted from (Marchal et al., 2023). In Section 3, we recall our quantum mechanical theorems from the preceding YA (Marchal et al., 2023) that serve as the foundational pillars of the QTG. Here, we also introduce recent experimental results where wave and corpuscle behaviors are shown to be reciprocal in full compatibility with the QTG’s two-entity formalism.

Such a two-entity formalism, which will be elaborated on momentarily, provides for the first time the mechanistical explanation of how a particle can maintain both its corpuscular and wave identities, as its wavefront constituent diffractively explores multiple paths and its kernel constituent is hauled across just one of them.

Thus, a major novelty achieved through our framework is that, in a wave-like (quantal) mode of interaction with gravity, the test object splits into an accelerating wavefront constituent described by the familiar notation h f = γ m 0 ∞ c 2 operating locally—and this explains why the “proper rest-mass” m 0 ∞ appears in the given equality, which, in turn, is what should actually be meant under de Broglie’s said foundational correspondence—and a necessarily recoiling (i.e., wavefront skipping) core or (in the case of a photon) kernel constituent m 0 ∞ e ‐ α as tracked by the remote observer (Marchal et al., 2023).

In any case, hf and m 0 ∞ e ‐ α point respectively to the wavefront and core or kernel aspects of the said object, where:

• Both the local observer and the distant observer agree that hf accelerates;
• There exists for the local observer just the wavefront constituent of energy hf with regards to a wave-like mode of interaction;
• The distant observer is able to witness the core/kernel decelerate as it pumps a minute energy to hf (whereas the local observer cannot, because the local measurement apparatus is geared to detect the completion of a wave-like occurrence).

Here, f denotes, in one suitable interpretation, the frequency of the electromagnetic (EM) radiation that would be emitted were the de Broglie local relativistic energy γ m 0 ∞ c 2 to undergo annihilation at the given coordinate, with h being the Planck constant, c the velocity of light in empty space, γ the Lorentz factor associated with the proper rest-mass m 0 ∞, and α the dimensionless gravitational potential parameter.

The frequency f constitutes the fundamental ingredient of QM with regards to the derivation of the de Broglie wavelength relationship λ _*B* = h/p, with p being the momentum of the object that interacts quantally (i.e., wave-like).

In Section 4, we replicate for completeness the equation of motion derived under the QTG’s single-entity formalism with respect to the quantal (wave-like) case (as merely a plausible approximation) (Marchal et al., 2023), which agrees with all of the observations in space-time physics that have been attributed to the success of the GTR—with the difference that the QTG excludes spatial singularities.

In Section 5, as the principal novelty of this contribution, we propose to describe, vis-à-vis the QTG’s two-entity setup, a gravitating test object composed of:

• A wavefront constituent of energy hf operating locally, and
• A corpuscular constituent of mass m 0 ∞ e ‐ α as tracked by the remote observer; which eventually get torn apart from each other in the corresponding field.

Above all, we reiterate the conformity of such a setup with the recent Chinese atomic slit experiment in Figure 1.

Beyond this, we highlight our prediction as to how achieving this experiment in a vertical direction in gravity, or in a convenient field, will lead to the observation of blurred fringes on the screen at the near-field limit.

We tabulate a summary of our key equations in Section 6, and conclude in Section 7 by emphasizing the remarkable coincidence of the predictions of the QTG with those of the GTR (Marchal et al., 2023; Einstein, 1956).

However, QTG’s outstanding conformance—in regards to both its single-entity and two-entity iterations—with the end results of the GTR only stems from a wave-like mode of interaction.

Concurrently, we highlight our prediction of a controversial phenomenon: the practically null gravitational deflection of high-energy γ-quanta on the surface of the Earth (Marchal et al., 2023). It must be emphasized that this effect falls outside the scope of the GTR and other metric theories of gravity, where the propagation of EM radiation is unambiguously determined by the metric of space-time and cannot depend on the frequency of the photons.

Therefore, additional relevant experiments in differentiating the QTG from GTR become crucially important. The QTG remains poised to be an experimentally testable alternative to the GTR due to being completely compatible with QM. The QTG’s unique predictions can hence decisively differentiate it in future astrophysical and laboratory research.

In the QTG terminology, the expressions (i-a) quantal and (i-b) wave-like, and also (ii-a) corpuscular and (ii-b) particle-like or projectile-like, are hence used interchangeably.

To provide a clear illustration of the QTG’s second iteration, we present in Figure 1 an AI-generated visualization to elucidate matter behaving either wave-like or particle-like in the propounded two-entity understanding—for example, as regards the dual nature of the ambient wave train of wavefronts and the localized kernel—which is contrasted with the solely probability density understanding of standard QM. Such a picture, showing the overlay of an interference pattern onto an incoherent strike-event distribution, is particularly essential for grasping the QTG’s ability to account for the gradual shift from fringes to lumps.

2 Fundamental aspects of the QTG

The QTG originates from YA (Yarman, 2006), which is grounded upon the following energy conservation postulate.

• - The rest-mass (or, should the velocity of light c be taken as unity, the rest-energy) of a test object is decreased in the presence of gravity as much as its static binding energy E _*B* at the given location of the field.

Mathematically, this postulate is equivalent to stating that the gravitational field represents a function of the present space-time coordinates; thus, it is applicable to any non-radiative force field, such as the bound EM field for non-radiating electrically charged particles, which also represents a function of the present state coordinates (Jackson, 1998). The above postulate thereby frames the rest-mass dynamics (RMD) of a test object moving in a non-radiative field.

It must be clarified that, in the QTG framework, gravity is not an external force later added to the equations but an intrinsic consequence of the conservation of energy within UMA, as we shall soon elucidate. Where the GTR postulates curvature, the QTG posits that the field is a manifestation of the phase-density of the environment. In the QTG, no interaction is constructed from arbitrary observational data but arises because the test particle must maintain its structural equilibrium by scaling its “internal dynamics” clock frequency f against the local potential. Thus, the QTG unifies gravity with electromagnetism at the level of the Hamiltonian rather than just the metric.

A number of principal implications of RMD can be straightforwardly derived (Yarman et al., 2017). For the convenience of the reader, we will shortly reproduce them below.

To ensure mathematical clarity and address the requirement for a rigorous lexicon of the variables used throughout this formalism, we define the following quantities as the primary parametric inputs of the QTG’s two-entity setup.

• m 0 ∞ represents the rest-mass of the localized kernel at an infinite distance to the host.
• f 0 ∞ is the associated Hamiltonian “internal dynamics” clock frequency of the dual-entity configuration in, again, the absence of a field.
• α is the dimensionless field potential.

Importantly, m 0 ∞ and f 0 ∞—the respective properties of the two entities in question—are not merely mathematical abstractions but physical components, so that the ambient wave train involving the frequency f captures the phase dynamics, and the localized kernel m 0 ∞ serves as the seat of the rest-mass. As will be detailed, the interaction in a fall is driven by the scaling relationship f = f 0 ∞ e α, which dictates the evolution of the entire system.

It is, thus, essential to highlight that the relationship f = f 0 ∞ e α represents a physical increase in the “internal dynamics” clock frequency of the dual-entity configuration as it enters a higher-potential zone (or undergoes free fall). This is not an arbitrary assignment but a structural necessity, as shall be stated below.

Basically, in gravity, the wavefront of energy hf accelerates while the localized kernel m 0 ∞ conjointly experiences a mechanical recoil, which leads to the internal energy content of the gravitational setup being re-distributed. While the rest-mass scales as m = m 0 ∞ e ‐ α, the rate f corresponding to the object’s “internal dynamics” must become increased in a fall to maintain Hamiltonian equilibrium. This derivation provides a definitive mechanical reason for the failure of Einstein’s clock hypothesis. In other terms, the clock rate is not just a passive coordinate effect but an active response of the matter’s architecture to the energy–momentum exchange with the field.

2.1 Implication 1: constancy of total energy in a closed system

The total energy E _*total* of an object with the rest-mass m 0 ∞ is where E _*B* is the static binding energy that represents a function of the distance r from the source M to the test object of rest-mass m(r) at r, and hence, the rest-mass m(r) also becomes a function of r due to the constancy of the RHS of Equation 1.

• Defined at the outset, when it is infinitely far from the host mass M that is supposed to be immensely heavy for simplicity (but without any loss of generality), and then
• Brought quasi-statically near the source of gravity M, and finally
• Launched into an orbital motion nearby M, where it must remain constant in the closed system composed of these two masses (Yarman, 2006; Yarman et al., 2017):

E t o t a l = γ m 0 ∞ c 2 1 − E B / m 0 ∞ c 2 = Constant , (1)

The equality above is unique in the sense that it separately leads to either a wave-like description or a corpuscular description, both reducing to Newton-type equations of motion. In the wave-like case, it conjointly yields

• A quasi-Newtonian—that is to say, a “projectile-like” or “ballistic”—equation of motion, and
• An independent Schrödinger-type formulation that is yet totally relativistic.

We include Table 1 for ease of understanding (Marchal et al., 2023).

TABLE 1. Unified structure of the two-entity formalism of the QTG, where our root equation E t o t a l = γ m 0 ∞ c 2 1 − E B / m 0 ∞ c 2 = Constant can concurrently engender either venue.

It is essential to recognize that, while conventional metric theories treat the gravitational potential as a geometrical property of space-time that causes, say, a Doppler shift in light, the QTG reverses this ontological priority so that, say, gravitational redshift—interpreted as a fundamental mechanical scaling of the particle’s rest-mass within the field—is the primary physical phenomenon. A substitute metric is subsequently derived as an effective mathematical placeholder for this scaling (Marchal et al., 2023). By anchoring the metric in the physical reality of the Doppler shift and the upcoming UMA theorem, the QTG remains inherently singularity-free even in the limit of intense gravitational fields.

Table 1 thus illustrates how the overall energy conservation law given by Equation 1 is tantamount to a universal root postulate governing both quantal and corpuscular occurrences. From this conservation principle, the two modes of interaction mathematically emerge.

(A) Left branch of Table 1 (viz., wave-like/quantal regime): when the interaction size is comparable to the de Broglie wavelength between the coherently and steadily broadcast wavefronts of the ambient wave train, whereby the system is composed of two phase-locked entities comprising the wavefront of energy h f = γ m 0 ∞ c 2 and the localized kernel of reduced rest-mass m 0 ∞ e ‐ α coupled to it, with both entities hence obeying their respective Newtonian-like equations of motion. The dynamic case here yields, in the appropriate limit, a fully relativistic Schrödinger-type formulation, so that this mode of interaction, if in its less-than-full weight, accounts for interference phenomena together with sporadic localized detection events (cf. Figure 1).

(B) Right branch of Table 1 (viz., particle-like/corpuscular regime) occurs when the fringe-inducing superposition of wavefronts is largely or wholly absent, where the dual-entity configuration behaves as a single corpuscular entity governed by another quasi-Newtonian equation of motion derived, once again, from the same energy conservation law. This mode of interaction in its full weight lets us make distinct predictions such as never-before-seen suppressed gravitational bending in regards to ultra-high-energy gamma quanta (Marchal et al., 2023), and even the possibility of retrograde orbital precession in the particle-like regime.

We emphasize that no additional postulates are introduced at this stage; either a standard quantum mechanical or classical behavior may arise from the same energy conservation foundation depending solely on the mode and weight of interaction.

Hereinafter, regarding the derivation of Equation 1, we imply that the source M is located at the origin of the coordinates, along with the details which will be particularized below. Furthermore, it will be convenient to introduce two different observers.

• A local observer attached to the test mass m (r), were this at rest at r.
• A distant observer situated far from the source of gravity M.

The local observer is otherwise fixed on the trajectory at r if the object is moving.

As discussed previously (see, e.g., Yarman et al., 2020; Marchal et al., 2023), the introduction of two different observers happens to be overarchingly informative from the measurement viewpoint, especially under the application of Cartesian-like coordinate systems to both.

In the present approach, the distance between m(r) and M is to be measured via the locally registered travel time of high-energy γ-quanta that are practically unaffected by gravity (see further below). Note that we will assume a non-zero rest-mass for the photon in general. While the compatibility of such an assumption with the Standard Model (SM) will be discussed elsewhere, the consideration of photons as non-zero rest-mass disturbances in the EM field greatly facilitates the approach at hand. In particular, with this assumption, Equation 1 becomes valid for both light and ordinary matter.

2.2 Implication 2: the gravitational force law

In order to obtain an explicit expression for E _*B*(r), which is the static binding energy between M and m(r), we formulate, for the distant observer, the gravitational law for the said two masses at relative rest similar to its classical form:

F Distant Observer = G M m r r 2 . (2)

However, it must be emphasized that we additionally assert a rest-mass variation for m(r) due to the energy conservation postulate in Equation 1 above.

Furthermore, the gravitational constant G—as assessed by the distant observer—entering into Equation 2 is a coordinate-dependent quantity, though it will be the same when assessed by any local observer doing its measurement. One can thereby reasonably rename it the “gravitational parameter,” inasmuch as it necessitates the separate introduction of the quantities G (that remains variable from the vantage point of other observers looking at the distant observer) and G ₀ (that every local observer assesses as being properly the same), with the latter constituting what had been measured by Cavendish in his proper frame of reference (Poynting and Chisholm, 1911). Moreover, the measured distance from m(r) to M also happens to be different for both observers.

Thus, defining r ₀ as the distance between m(r) and M as determined by the local observer, we obtain the equality (Yarman et al., 2020; Marchal et al., 2023; Yarman, 2006)which remains invariant for both observers.

G / r 2 = G 0 / r 0 2 , (3)

Based on Equation 3, we can rewrite Equation 2 in the equivalent form (Yarman et al., 2020; Marchal et al., 2023; Yarman, 2006)which still describes the gravitational force between the test object and the source at relative rest with respect to each other as again assessed by the distant observer.

F Distant Observer = G 0 M m r r 0 2 , (4)

The proper gravitational parameter G ₀ (as measured by Cavendish) is the same for all local observers situated in the midst of a static gravity as explained above, given that the QTG strictly adheres to the Strong Equivalence Principle. In the QTG, all proper frames that scrutinize compound quantities such as a particular force G ₀, the Planck constant h, the elementary charge e, the speed of light in empty space c—or any specific speed—shall measure them as remaining unaltered either outside gravity or when embedded in it (viz., in view of the local observer doing the said measurement, which can easily be checked with respect to how basic quantities are altered in gravitation) (Yarman, 2006).

We moreover show that the quantity gravitational force × the square of the distance measured between M and m(r) remains invariant for both local and remote observers (Marchal et al., 2023). We can furthermore express the gravitational force exerted by M onto m(r), in the way assessed by the local observer, as (Yarman, 2006)

F Loal Observer = G 0 M m r r 2 . (5)

The physical meaning of Equations 2–5 have been discussed in detail in (Marchal et al., 2023). In any case, in view of Equation 3, the locally assessed force F _{Local Observer} is to be expressed in terms of the coordinate r, and the same force F _{Distant Observer} as assessed by the distant observer is to be expressed in terms of the coordinate r ₀.

2.3 Implication 3: static binding energy in gravity

The static binding energy E _*B*(r) for the masses m(r) and M at relative rest is obtained through the simple equation Fdr = dm(r)c ² (Yarman, 2006) and is given by the relationshipwhere we have introduced the dimensionless gravitational potential parameter

E B r = m 0 ∞ c 2 − m r c 2 = m 0 ∞ c 2 1 − e − α , (6a)

α = G 0 M / r c 2 . (6b)

2.4 Implication 4: equation of motion under gravitation

By combining Equations 1, 6a, we obtain, along the trajectory of the test object,

E O v e r a l l = γ m 0 ∞ c 2 e − α = Constant . (7)

Hence, the equation of motion for a test particle acquires the general form

d d t γ m 0 ∞ e − α = 0 (8)

The above equation could be derived directly from Equation 1 without postulating an action principle and a space-time geometry in the presence of gravity. However, Equation 7 may be the integral of more than one differential equation, along with disparate physical significances—particularly regarding (i) the wave-like regime and (ii) the particle-like regime—and this constitutes an unequaled privilege of the current undertaking.

Nevertheless, Equations 7, 8 of the QTG—whereby we can quickly obtain the equation of motion with the latter step—are based solely on the foundational Equation 1 that is applicable to all bound fields. This predominantly distinguishes our enterprise from all other theories of gravity and makes it intrinsically consistent with the law of energy conservation and also with QM, given that Equation 1 immediately engenders the de Broglie wavelength relationship λ _*B* = h/p, as will soon be shown.

At this point, we draw attention to how the established direct link between energy conservation (Equation 7) and motion (Equation 8) belies a number of principal points that we outline as follows.

• First, Equation 8 shows that its solution does not depend on the mass of the test particle.
• The QTG is therefore fully compatible with Einstein’s Weak Equivalence Principle (Einstein, 1956) and remains in full harmony with the outcome of Eötvös-type experiments (Yarman et al., 2019).
• Furthermore, Equation 8 implies the validity of both local Lorentz invariance and local position invariance, which also makes the QTG fully compatible with the Strong Equivalence Principle.

3 Quantum mechanical theorems proposed by YA between mass–energy–frequency–size, and their implications: toward the two-entity iteration of QTG

Further development of QTG is founded on the quantum mechanical theorems proven by the first co-author in Yarman (2006) and updated in our recent paper Marchal et al. (2023).

For convenience, we copy their formulation in this section after referring the reader to our previous work Marchal et al. (2023) and Yarman (2006) for the full details of their proof (Yarman, 2009). A forthright proof of Theorem 1 is presented in Supplementary Appendix A. Related proofs with regards to the next two theorems follow immediately.

Theorem 1 Consider a quantum object composed of K particles with the rest-masses m _*k*0 (k = 1, … , K) and assume that all of these masses are multiplied by the arbitrary number ξ. Then, the following re-scalings take place:where E ₀ is the total energy of the object, T ₀ is the unit period of time of its clock motion, and L ₀ is the characteristic length of the space size in which the given motion takes place, whereby all quantities appearing under Equation 9 are ascribed to what can be referred to as the said object’s internal dynamics.

It is crucial to distinguish this result from the classical interpretation offered by the GTR. In Einstein’s metric worldview, gravitational redshift is due to the stretched curvature of space-time, where the clock under consideration is a passive observer of the geometry. In contrast, our derivation actively turns the redshift phenomenon into a quantal reconfiguration of the internal dynamics of matter in flat space-time (Theorem 2). In other words, the overall energetic content of the host, its medium, and the lowered client body is re-distributed such that the frequency shift becomes a structural necessity of energy conservation.

By deploying the de Broglie correspondence in the re-scaling of the Hamiltonian, we demonstrate that the gravitational medium is not a static background stage for, but a dynamic participant in, the particle’s intrinsic periodicity. This explains why the GTR’s foundational clock hypothesis—which assumes a point-like clock in the field—is physically untenable already at the quantum mechanical level.

It must be emphasized that, within the QTG framework, the space-time metric is not an ontological primitive as it is in the GTR. Instead, a corresponding metric emerges as a post hoc mathematical characterization—an effective representation that emulates the underlying superfluidic vacuum of fluctuating phase density. This medium imparts onto the client entity a Hamiltonian re-scaling of its rest-mass, strictly governed by the overarching law of energy conservation. By treating such a metric formulation as a subsidiary method, the QTG naturally avoids singularities that occur when geometry is taken as the fundamental generator and raison d’être of gravitation. We emphasize that the above theorem demarcates a cornerstone guiding principle christened by the first co-author as “Universal Matter Architecture” (UMA) (Yarman et al., 2021).

Theorem 2 The unit period of time T ₀ of the clock of the resting test object measured at infinity, which interacts with a non-radiative field, stretches due to the mass deficiency it undergoes thereat, and this, as much as the static binding energy it delineates in the field under consideration. Likewise, the space size L ₀ associated with this resting object stretches as much in all directions of 3D space.

It has been noted that clock retardation in the gravitational field is a well-known outcome of the GTR. However, we must emphasize a profound ontological difference with the QTG: in the GTR, redshift is the product of the metric ensuing from time dilation; in the QTG, as derived in Theorem 2, it is a purely quantum mechanical consequence of mass-energy-frequency-size rescaling and energy content redistribution within the matter architecture itself.

Thus, a direct implication of Theorem 2 is gravitational redshift, which can effectively be related to the fatiguing of the considered clock’s “internal dynamics” motion as a quantum mechanical result of the decreased energy, on par with the rest-mass deficit of the test object in the presence of the source of gravity.

As a corollary, had the quantities proper period of time T ₀ and proper length L ₀ entering into Theorems 1, 2 for a test object with the proper rest-mass m 0 ∞ been embedded in a radially symmetrical gravitational field at a resting distance r from the center of the source of rest-mass M and held there at rest, then the corresponding quantities T(r) and L(r), when assessed by a resting distant observer outside of gravity, would become a function of the static binding energy E _*B* as defined by Equation 6a:

T r = T 0 1 − E B r / m 0 ∞ c 2 = T 0 e α , (10a)

L r = L 0 1 − E B r / m 0 ∞ c 2 = L 0 e α . (10b)

We stress that the transformations Equations 10a, b with regards to the “local unit period of time” T ₀ and the “local unit length” L ₀ are valid when the object remains at rest at distance r from the gravitational source as observed by the distant observer.

We can now relate the distance r, as assessed by the distant observer, to the same distance r ₀ as but assessed by the local observer (Yarman et al., 2020; Marchal et al., 2023; Yarman, 2006):

r = r 0 e α . (10c)

We note that, unlike in competing theories of gravity, all the foregoing QTG relationships are written in an integral form right at the outset instead of a differential form, which incontrovertibly constitutes an important advantage.

3.1 Expressing wave-like behavior through the de Broglie foundational correspondence

A major example conforming with Theorem 1—where, in the case of ξ > 1, mass increases while the size and the period of time shrink—is the scenario of an object, including a photon, that interacts with a host body in a wave-like manner. In such a case, we consider a test particle with the relativistic mass m _*i* and velocity v _*i* in view of the distant observer cruising far from any sources of gravity, and which bears the initial “internal dynamics” clock frequency f _*i* associated with its relativistic energy.

Thus, we write the de Broglie foundational equation (de Broglie, 1925):

m i c 2 = h f i , m i = m 0 ∞ , (11a)

1 − v i 2 / c 2 1 / 2 . (11b)

The characteristic frequency f _*i* = m _*i* c ²/h of a particle—such as an electron—can be interpreted through three conceptually distinct but complementary lenses.

• The interlocking phase undulation, where f _*i* represents the oscillation rate of a superluminality-compatible phase velocity that is strictly interlocked with the particle’s wavefront aspect. This periodic process ensures that the wavefront constituent of the particle and its relativistic mass-energy facet remain synchronized across all frames of reference.
• The Zitterbewegung frequency, where f _*i* may be understood as the inverse of the Zitterbewegung period (on the order of 10⁻²² s). This characterizes the particle’s corpuscular aspect as a “spatially confined diminutive vorticity”—a rapid, internal oscillation.
• The frequency of the EM radiation that would be emitted upon the total annihilation of the relativistic mass m _*i* . These, therefore, are the best ways of considering f _*i* throughout the present approach.

Recall that de Broglie did not initially know anything about annihilation; he instead associated f _*i* with the “particle’s internal clock-like periodicity”, just like recent measurements endorse (Gouanère et al., 2005; Catillon et al., 2008; Osche, 2011; Durey and Bush, 2020; Nolte, 2018). We call it “internal dynamics clock frequency” versus the usual special relativistically decreased frequency for a moving object: f _*i* ** = f* _*i* (1-v _*i* ² /c ² ).

Equation 11a is not—except for the case of a photon—equivalent to the customary de Broglie wavelength relationship λ _*Bi* = h/p _*i* , where λ _*Bi* is the de Broglie wavelength, h the Planck constant, and p _*i* the relativistic momentum of the particle. Still, it constitutes the basis for the derivation of the de Broglie wavelength relationship. Notwithstanding, in this study, Equations 11a, 11b comprise the necessary and sufficient conditions to express a wave-like behavior.

We note that the correspondence f _*i* λ _*Bi* = c ² /v _*i* between f _*i* and λ _*Bi* can be directly obtained by demanding a permanent harmony between the phases of

• A sinusoidal wave bearing the same frequency as f _*i*, and
• The sinusoidal wave of frequency f _i ** as defined immediately above.*

This necessitates that the former Zitterbewegung-induced sinusoidal phase oscillation of frequency f _*i* moves with the superluminality-permitting velocity c ² /v _*i* (as demonstrated by de Broglie) to constantly catch up with the latter wave train of frequency f _*i* ***, which in turn moves with velocity v _*i* (e.g., Supplementary Appendix A of Marchal et al. (2023)).

In this sense, the said wavefront-kernel synchronizing internal dynamics clock frequency f _i relays information about the wavefront presence of the object, which is spread in space as a coherently and steadily broadcast wave train bearing between its amplitude peaks or troughs the de Broglie wavelength λ _*Bi*.

Nevertheless, a gravitating rest-mass or the proper mass m 0 ∞ stays the same for a local observer fixed on the trajectory of m _*i* ; thus, Equation 11b is primarily written for this observer (see further below). On the other hand, if any wave-like mode of interaction were to occur (Davisson and Germer, 1927a; Davisson and Germer, 1927b; Davisson and Germer, 1928), it would have to take place with respect to the properties dominant in the particle’s proper frame—hence, in view of the local observer.

What we mean by wave-like or quantal mode of interaction is precisely a match between the de Broglie wavelength of an incoming test object and the proper frame’s interaction size, just as is the case for electrons falling into resonance with crystal lattices (Davisson and Germer, 1927a; Davisson and Germer, 1927b; Davisson and Germer, 1928). Thus, if the crystal in question were embedded in gravity, a wave-like mode of interaction shall take place in line with the locally measured wavelength of the incoming test object. This implies that, in a gravitational scenario, such a match should entail a direct alignment of the locally measured de Broglie wavelength with the local properties embodying, in particular, the local momentum and the locally measured crystal lattice size—no matter whether or not the said (i) wavelength and (ii) size would be transformed in a covariant fashion.

To rephrase, in a wave-like interactional case, when a quantal regime is realized, it is imperative to start with Equation 11b valid for the proper frame (which hence involves just local properties and, therefore, the proper rest-mass m 0 ∞ and its velocity).

Upon engagement with the host, already when referred to the local observer, the test object starts to accelerate; the initial relativistic mass m _*i* will increase to become equal to m _*hf* (see right below) in lieu with the increase of its velocity v _*i* becoming v, and by the same token, the initial frequency f _*i* will rise to become f.

We can thus rewrite Equations 11a, 11b for the gravitating object at the given instant and coordinate as

m h f c 2 = h f , m h f = m 0 ∞ , (11c)

1 − v 2 / c 2 1 / 2 . (11d)

Since lengths and time intervals stretch by the same amount for both the local and the distant observer (cf. Theorem 2), specific instantaneous velocities remain invariant (Marchal et al., 2023). This is another advantage of the QTG in contrast to metric theories of gravity.

The instantaneous “internal dynamics” clock frequency f then transpires as a local wave-like property—meaning, it would become all at once:

• the Zitterbewegung rate of the kernel vorticity in the proper frame;
• the locally applicable phase-synchronizing undulation periodicity between the coupled wavefront and the kernel;
• the frequency of the EM radiation which would be emitted if the locally measured instantaneous relativistic mass m _hf were totally extinguished, owing to the fact that it is built on the local rest-mass m 0 ∞.

All the same, the frequency f acquires by construct (cf. Equations 11c, 11d) a value that is identical to both the local and the distant observer (whether or not f is defined based on the local rest-mass). In such a case, as the test object accelerates in the presence of gravity, where wave-like properties may become manifest, Equation 11c can be considered a generalized setting of the “internal dynamics” clock frequency f defined at the given instant and coordinate.

On (i) the de Broglie speed λ_Bf = c ²/v that ensues from the “permanent harmony” constraint between the phases of the wave frequencies f and f, (ii) the annihilation equality λf* = c, and (iii) the de Broglie wavelength λ _B in terms of λ—that is, λ _*B* = (c/v)λ.

Equation 11a, written in empty space, historically constituted the foundational premise of QM that yielded the de Broglie wavelength relationship λ _*B* = h/p (Marchal et al., 2023). Different equalities coming up with this relationship ought to be retained clearly. Therefore, in order to arrive at his wavelength relationship, de Broglie had to impose a permanent harmony between the phases of (i) a wave bearing the same frequency as that of the “internal dynamics” clock frequency f (which, in the aim of keeping up with the object’s motion, is required to propagate with the incorporeal velocity c ²/v), and (ii) the usual wave associated with the special relativistically decreased frequency f* propagating with the (sub)luminal velocity v (see above, as well as Supplementary Appendix A of Marchal et al. (2023)).

The superluminality-compatible de Broglie wavelength λ _*B* can, thus, be expressed in terms of the lightspeed-travel-restricted de Broglie wavelength λ = c/f. Therefore, λ _*B* =(c/v)λ, where λ is the wavelength of the EM radiation that would be emitted if m _*hf* of Equations 11c, 11d were totally annihilated.

Note that both λ and f are measurable quantities (Davisson and Germer, 1927a; Davisson and Germer, 1927b; Davisson and Germer, 1928; de Broglie, 1925; de Broglie, 1973). The lightspeed-travel-restricted de Broglie wavelength λ is associated, as such, with the frequency f and the de Broglie wavelength λ _*B*, which are strictly linked, so λ _*B* is thereby contingent upon f.

Any confusion that might arise at this point should be eased by recollecting that:

• f in λ = c/f is the frequency of the EM radiation that would be emitted upon the annihilation of the test object in its motion;
• f appearing in λ _*B* f = c ² /v is, as stated, either (A) the Zitterbewegung-related core/kernel vortex periodicity or (B) the wavefront-kernel interlocking de Broglie’s clock-like phase undulation rate of the object at hand. The latter relays information propagating with the velocity c ²/v about the presence of the wave-like aspect of the entity as it is spread in space in the form of a wave train of wavefronts, the peaks or troughs (or slopes in the case of intermittent pulses) of which are separated by the de Broglie wavelength λ _*B* (de Broglie, 1925).

Such a framing under the QTG directly leads to the de Broglie wavelength relationship λ _*B* = h/p at the root level—the energy conservation law Equations 1 and 7, as will soon be shown in detail. Thus, the de Broglie wavelength relationship is innate to the QTG. It is, therefore, remarkable that the QTG’s framework is laid down in complete harmony with QM (Marchal et al., 2023).

Next, we notice that the proper rest-mass m 0 ∞ of the test object is decreased, as implied by Equations 1 and 7, in the presence of gravity by e ^*-α* in view of the distant observer. Since instantaneous velocities remain the same for both observers, the instantaneous Lorentz factor 1/(1-v ²/c ²)^1/2 will have to grow by the same amount—by e ^*α* (for both the local and the distant observers).

Thereby, based on Equations 11c, 11d, the wavefront of energy h f = γ m 0 ∞ c 2 associated with the test object will have to increase as much in view of both the local and the distant observer; whenever this is the case, it constitutes the key which governs the wave-like mode of interaction under gravity. We then have two conjoint instantaneous particle constituents associated with the same test object of initial proper rest-mass m 0 ∞ (at infinity): (i) a quantal constituent hf and (ii) a corpuscular constituent m 0 ∞ e ‐ α .

The law of energy conservation necessitates that energy will accumulate in the wavefront of energy hf = e ^*α* hf till the present instantaneous coordinate, which must be supplied by a minute rest-mass loss from the fatigued core/kernel m r = m 0 ∞ e ‐ α . Hence, we conclude that, in a quantal interaction, the proper rest-mass m 0 ∞ of the corpuscular entity pumps energy to the wavefront of energy hf throughout a gravitational fall as the latter experiences recoil (“wavefront skipping”, or continuous re-coupling with the preceding wavefronts in the wave train). Since the wavefront of energy hf accelerates during such a fall, the core/kernel rest-mass m 0 ∞ e ‐ α must therefore recoil due to momentum conservation (Marchal et al., 2023); all this, rigorously speaking, requires that we frame a two-entity setup for the moving object.

The conceptual details of this mechanism are presented in the Supplementary Appendix A of Marchal et al. (2023). We emphasize that h f and m 0 ∞ e ‐ α point respectively to (i) the wave-like aspect, and (ii) the particle-like aspect of the test object at hand. It is, therefore, a question of two different but interrelated characteristics for the two entities of the same client body in gravity.

We thus qualify the exchange of energy between the accelerated wavefront and the lagging core/kernel as an important feature of what we call RMD, whereby, following Theorems 1, 2, we state the final quantum mechanical theorem of the QTG that describes the behavior of a test object interacting in a quantal manner in gravity (Marchal et al., 2023).

Theorem 3 In a wave-like mode of interaction, the decrease of the initial proper rest-mass m 0 ∞ of an object in gravity is determined by factor e ^*-α* as viewed by a distant observer, while the quantity 1/(1-v _*i* ² /c ²)^1/2 must accordingly be increased as much due to the constancy of the energy quantity m 0 ∞ e ‐ α/(1-v _*i* ²/c ²)^1/2 (Equation 7). Similarly, the initial relativistic mass m _i = m 0 ∞/(1-v _*i* ²/c ²)^1/2 to be assessed locally increases as much to become m _*hf* = m 0 ∞ /(1-v _*i* ²/c ²)^1/2 e ^*α* = m _*i* e ^*α*. Concurrently, the initial “internal dynamics” clock frequency f _*i* becomes f = f _*i* e ^*α* , which induces a shortening of the lightspeed-travel-restricted de Broglie wavelength λ = λ_ie^-α by the same amount. The energy necessary to accrue the frequency f _*i* to f must, due to the law of energy conservation, then come from the decrease of the proper rest-mass m 0 ∞ by the same amount, which makes the latter necessarily lag behind hf as per the law of momentum conservation.

It is important to recall that, whenever we proceed to express the said transformations, it would vice versa mean that we are dealing with a wave-like, or just the same, quantal regime.

The mathematical details of the conceived energy exchange, together with the energy and momentum conservation equations, will be elaborated momentarily.

At this juncture, it is useful to recapitulate the main points in relation to a wave-like behavior.

• Both the local observer and the distant observer agree that hf _*i* accelerates, whereas
• There exists for the local observer just the wave-like constituent hf with regards to a wave-like mode of interaction occurring locally, while
• The distant observer can track the recoiling core/kernel (but the local observer cannot).

So, in a wave-like or quantal case, the wavefront of energy hf accelerates and the corpuscular entity of rest-mass m 0 ∞ e ‐ α drags on in view of the distant observer, whereby the latter pumps minute energy to the former in its deceleration in full conformity with the laws of energy and momentum conservation.

3.1.1 Envisioning the core/kernel rest-mass for huge bodies.

What is then the envisioned core or kernel rest-mass m 0 ∞ e ‐ α of the test object when we are dealing with gravity involving massive client bodies? (Yarman et al., 2016b). The answer is easy. For one thing, under YA and the QTG, the rest-mass drops off the equation of motion. This means, at the same time, that a client object orbiting a host star may not only be a giant planet but also an electron, and nothing would essentially change the underlying QTG conceptualization.

At that point, how should Theorem 3 be considered in order to address the “core/kernel rest-mass” of a giant planet? Fortunately, the answer here is easy, too. The planet is, after all, made up of elementary particles bound to each other via gravity—therefore, there is no harm in considering how just a single elementary particle inside the massive planet acts as the test object vis-à-vis the host star. All the ingredients of the said planet will behave the same. Thus, the core or kernel rest-mass of the client body de facto becomes the rest-mass m 0 ∞ e ‐ α of the said elementary particle, which, in this example, is gravitationally bound to the given hypothetical planet in its orbit around the ponderable source at the coordinate r.

3.1.2 Pound and Rebka experiment where photons exhibit wave-like behavior in Earth’s gravity

Theorem 3 provides, among other things, a straightforward explanation of the Pound and Rebka experiment (Yarman et al., 2016a) by putting into perspective the increase in a gravitational fall (or, in the original setup, the “decrease”, in the gamma quantum’s climb out of the gravitational potential) of the “internal dynamics” clock-frequency f (see Section 4 for details). In other words, this experiment already constitutes the basis of a determination of the “internal dynamics” clock frequency f (regardless of the fact that Pound and Rebka did not specifically aim to do so), insofar as preceding the much later experiments designed to identify it (Davisson and Germer, 1927a; Davisson and Germer, 1927b; Davisson and Germer, 1928; de Broglie, 1925; de Broglie, 1973).

We can therefore examine how Theorem 3 constitutes a concrete elucidation of the increase of the said frequency f in a wave-like regime (de Broglie, 1925), along with the decrease of the test object’s rest-mass in gravity.

3.1.3 Elaboration on the de Broglie wavelength relationship

It can be observed that the original claim of de Broglie expressed through (Equation 11a) was a long-contested topic in physics until recently (Davisson and Germer, 1927a; Davisson and Germer, 1927b; Davisson and Germer, 1928; de Broglie, 1925; de Broglie, 1973), no matter whether it correctly yielded the de Broglie wavelength relationship λ _*B* = h/p. Unfortunately, mainstream captivation led the majority of the scientific community to conclude that de Broglie was actually erroneous in his derivation and that his wavelength relationship has no merit apart from being a historical curiosity. Nevertheless, as we have already shown and will do further, it does have actual merit.

We recognize that the two-entity QTG formalism outlined herein constitutes a novel interpretation of QM (which we have already discussed in Section 7 of Marchal et al. (2023)); we furthermore note what de Broglie himself underlined some 50 years after he introduced his wavelength relationship, when he lamented that “QM developed differently than what he had imagined at the beginning” (de Broglie, 1973). Amongst other things, he criticized the established Born rule for the wavefunction (Born, 1927).

While the QTG offers the capacity to exercise the prevalent Copenhagen interpretation when the corpuscular constituent of the particle is disregarded in the analysis (Marchal et al., 2023), our understanding of QM, like de Broglie himself would approve, conjointly involves both the quantal and corpuscular characteristics of the test object, where both the wavefront entity and the core/kernel entity complementing each other are always ontic and not epistemic. Nevertheless, the end results of the QTG in the wave-like regime are computationally similar to Born’s clarification of Einstein’s “ghost field” concept (Rieckers, 2025). Despite this, we will not go into the details of such a problem, especially given that we are already able to operate easily in the manner of Heisenberg, Schrödinger, Born, and Dirac with what we have developed so far (Marchal et al., 2023).

It will, nevertheless, be helpful to provide a concrete example. Consider thus the diffraction of free neutrons observed, in classical terms, to strangely land one by one on a measurement screen, where they display fringes after their passage through a pair of slits separated from each other by a distance precisely matching the neutron’s de Broglie wavelength λ _*B* (Zeilinger et al., 1988).

Then, according to what we discussed previously, a single neutron engages this barrier both and at the same time

• As the forerunner of a wave train disturbance with an energy signature h f = γ m 0 ∞ c 2 unique to it (cf. Equation 11a), the way induced by the related de Broglie internal periodicity phenomenon, and
• As a diminutive gyrating corpuscle of rest-mass m 0 ∞ coupled to it, where the former entity passes through both slits, while the latter entity crosses only one of the slits at any given time on account of being guided to a predetermined destination by its associated wavefront.

The neutron’s de Broglie wavelength λ _*B* matching the slit dimensions thus causes the observed interference pattern in the QTG’s wave-like regime, because the diffractively clashing wavefronts of energy hf each will go through both slits simultaneously, no matter whether individual neutron corpuscles go through just one slit at a time (cf. Figure 1).

We could, especially in gravity, where we expect to have not so well-defined fringes, try to follow a given neutron’s corpuscle and possibly tell which slit it has gone through, just as we could observe where exactly it lands on the screen; except that, strictly speaking, there appears no viable way of measuring which slit it would go through without destroying its wave-like character, all the while knowing that the neutron’s wavefront of energy hf must go through both slits. The first co-author undertook the subject matter in Yarman (2010), Yarman (2011), and we shall elaborate on that below. Regardless, Theorem 3 essentially specifies the relational behavior of the instantaneous wavefront constituent hf and the core/kernel constituent m 0 ∞ e ‐ α , which engenders the conceptualization of the two-entity model of the QTG that we will soon discuss.

At this stage, it is important to stress that Theorem 3 facilitates the recognition of the full validity of the de Broglie foundational relationship (11a) as constituting a vital premise of QM that simultaneously yields in gravity the main space-time transformation analogs of GTR:

• Mass increase (m_hf = m_ie^α),
• Size decrease (λ = λ_ie^−α),
• Rest-energy decrease (m _0∞ c ² becoming m 0 ∞ c 2 e ‐ α), and
• Conjoint period of time stretching (as per Theorem 2) through the decrease of the core/kernel proper rest-mass m 0 ∞ by the factor e ^*-α* .

Each one, except rest-mass decrease (which needs the QTG’s special consideration), are amazingly relativistic outcomes of Einstein’s disc-world gedanken experiment (Einstein, 1956) that he considered along with his clock hypothesis (regarding how the time rate of a revolving clock depends only on its tangential displacement and not on its acceleration as such), as well as the equivalence principle, which implies that rotation and gravity induce identical results.

All the same, as explained in Marchal et al. (2023), such a hypothesis is not able to cover the entirety of the above transformations. Hence, the GTR fails to fulfill the interrelation we framed between the increase of the “internal dynamics” clock frequency f and the decrease of the core/kernel rest-mass m 0 ∞ e ‐ α in gravity, which we have just expounded and finally expressed in Theorem 3.

We stress that the foregoing set of first three properties encapsulated by Theorem 3 extraordinarily make the QTG compatible with the established predictions of the GTR, but only in the QTG’s wave-like case, as we will discuss further below. In other words, without the rest-mass decrease of the test object due to binding, the abovementioned set of three properties in the wave-like case would not be acquired, and this is particularly what Einstein unavoidably missed. Even more ironically, in order to derive the QTG’s wave-like description that practically leads to solutions identical to those of the GTR in the weak gravity limit, nothing is needed other than de Broglie’s foundational postulate (Equations 11a, 11b).

3.2 The mathematical basis of the two-entity iteration of QTG

We now transform Equation 7, using Equation 11c, intowhere C is a constant.

E O v e r a l l = γ m 0 ∞ c 2 e − α = m h f c 2 e − α = m i c 2 = С , (11e)

It is convenient to multiply both sides of Equation 11e by m 0 ∞ c 2:where C′ is a new constant bearing the dimension of [energy]².

m hf c 2 m 0 ∞ c 2 e ‐ α = C ’ , (11f)

The above equation can, via Equation 11c, further be written as

hfm 0 ∞ c 2 e ‐ α = C ’ . (11g)

Thus, for any test object, there can be two distinct entities, the wavefront of energy hf and the attached core/kernel of rest-mass m 0 ∞ e ‐ α , yielding two different aspects that entail (i) its wave-like or quantal mode of interaction, and (ii) its particle-like or corpuscular mode of interaction in gravity, along with the two distinct Equations 11c, 11e, leading to Equation 11g.

More precisely, in a gravitational fall (Marchal et al., 2023):

• The locally assessed wavefront of energy hf constituent of the particle undergoes a wave-like interaction, and
• The core or, in the case of a photon, kernel constituent of rest-mass m 0 ∞ e ‐ α lagging behind, as seen by a distant observer, supplies the required impetus to the accelerating former constituent via pumping to it a diminutive rest-mass.

Note that both entities are rooted in the original rest-mass m 0 ∞. The wavefront of energy hf accelerates and the core/kernel mass m 0 ∞ e ‐ α recoils (as will be detailed soon) according to the laws of energy conservation and momentum conservation (Marchal et al., 2023). As an outcome, we conclude that (i) different coordinates and (ii) different velocities must, in general, be associated with the throttled wavefront of energy hf and the straggling core or kernel of rest-mass m 0 ∞ e ‐ α. Therefore, the wavefront of energy hf _*i* and the associated corpuscular entity having the energy m _*i* c ² start as concentric in empty space, but they are no more so in gravity. When the object engages a host mass, the wavefront and the core/kernel detach and are eventually torn apart under strong gravitational influence, whereby the latter constituent exhibits a sort of “wavefront skipping”.

In particular, when the object undergoes a local wave-like interaction, the wavefront of energy hf must accelerate so as to “smell” its path, as conjectured by Feynman et al. (1964), thus dictating to the core/kernel of rest-mass m 0 ∞ e ‐ α in pursuit the direction in which to move (Supplementary Appendix B of Marchal et al. (2023)). The said constituents, thus, ultimately split apart from the standpoint of the distant observer as long as the interaction implies the acceleration of the wavefront constituent of the particle with the energy hf and the deceleration of its core/kernel constituent with the relativistic mass γ m 0 ∞ e ‐ α in view of the distant observer.

3.3 Experimental support for the two-entity formalism from quantum mechanical diffraction experiments (including recent realizations in the single-photon regime)

Recently reported high-precision single-photon diffraction experiments (cf. Figure 1) provide, through intricate quantum mechanical processes, indications consistent with the reciprocity of the posited wavefront and corpuscle aspects of the two-entity test particle under the QTG (CGTN, 2025; Na, 2025; Chen, 2025; Zhang et al., 2025). In these experiments, individual photons or massive particles are tallied one by one as miniscule impacts on a detection screen, while the cumulative distribution of many such events reconstructs the known interference pattern. The recorded images therefore display, within the same laboratory arrangement, both fringes and localized traces.

Under the framework of standard QM, this puzzle is commonly addressed through the probabilistic Copenhagen interpretation where, in terms of Bohr’s complementarity, the unobserved quantum object is in possession of both wave and particle characteristics at the same time, and it cannot be determined what it is doing in transit until it is measured. In contrast, the QTG’s two-entity formalism provides a classical determinism-compatible ontic interpretation wholly grounded in energy and momentum conservation. According to such a formalism, any quantum object propagating through space dynamically comprises two constituents coupled to each other, as discussed at length (cf. Equations 11c–11g). Consequently, due to the postulated two-entity nature of any test particle under the QTG, there is no need to consider the “measurement problem” or “wavefunction collapse” as auxiliary hypotheses; there is only adherence to the requirement of the stated UMA theorem. That is to say, as the particle’s rest-mass scales in, say, gravity, its energy must be distributed between (i) the wavefront constituent, and (ii) the corpuscular constituent in the Hamiltonian manner. Thus, the particle’s energy content is to be re-allocated to the throttled wavefront from the straggling core/kernel in a fall.

Based on such a description, the interference pattern arises from the fact that the superposed real wavefronts in a coherently and steadily broadcast wave train pilot their associated cores/kernels downstream to the measurement screen, where localized detection events correspond to the interaction of the latter corpuscular entity with the detector.

The reciprocity of wave-like and particle-like manifestations that are achievable through a gradual shift from the display of fringes to incoherent lumps on a backstop—whether the particles arrive protractedly or suddenly in series—therefore reflects the simultaneous presence of both physical entities as the misplaced underlying reality for the cumulative effects and statistical distributions, offering a more direct physical interpretation than standard probabilistic descriptions. The observed behavior therefore supports the necessity of the two-entity conceptualization at the level of individual quanta, in which wave-like propagation and particle-like localization do coexist in an entirely mechanistical manner. Moreover, no modification to the conservation laws is thus required; rather, the experiments hint at the non-mystical physicality of rest-mass exchange that would come into play between the wavefront and corpuscle constituents of the test particle during flight and detection, and particularly in a gravitational fall.

Such experimental findings hence provide independent empirical support for the two-entity formalism advanced here. They also reinforce the QTG’s applicability beyond gravity, insofar as it is situated naturally within the broader context of quantum mechanical phenomena, where wave-particle (corpuscle) convergence, and not mere complementarity/duality, becomes apparent.

3.4 Single-entity formalism as a stepping stone towards QTG’s two-entity setup

In our previous paper (Marchal et al., 2023), for the sake of exercising a plausible approximation, we devised a simplified scenario where the motion of the straggling core/kernel with the rest-mass m 0 ∞ e ‐ α was considered not to be separate from the motion of the throttled wavefront of energy hf, insofar as the infalling test object was preliminarily conceived to move in gravity as a single-entity with a retarded velocity. We therefore worked on the equation of motion written for the wavefront of energy hf, where we expressed its velocity as v ₀ = v _*hf*, that is, in terms of the retarded velocity v _*core* as viewed by the remote observer. By the same token, we conjointly overlooked, for the time being, the accelerated motion of the said wavefront.

Such a conceptualization is merely a mathematical artifact of the single-entity approach we developed within the framework of the QTG, meaning that, as will be elucidated below, the ensuing metric also is approximate. Thus, as a more general consideration in accordance with Theorem 3, which we herein call QTG’s two-entity formalism, we instead must take into account the fact that, in the case where a test object is approaching a source of gravity, the wavefront of energy hf should run increasingly ahead of the core/kernel of rest-mass m 0 ∞ e ‐ α that will, in turn, trail increasingly behind.

Thus, there is actually no slowing of the gravitating test object; it is only that the corpuscular constituent of the particle recoils as it pumps minute energy into the throttled wavefront constituent having the energy hf. In other words, the test object is eventually torn apart, as stated before, into (i) an accelerating wavefront of energy hf, and (ii) a straggling (“wavefront skipping”) core/kernel of rest-mass m _0∞ e ^-*α*.

To realize such a two-entity model, and for inclusiveness, we shall first abridge from (Marchal et al., 2023) the equation of motion of the test object under the single-entity iteration of the QTG, along with the appropriate elaborations. Next, we will show that our exact two-entity setup, under the particular condition which will be introduced in Section 5, practically leads to the same results as those furnished by the QTG’s single-entity model (Marchal et al., 2023).

Notice, however, that the local and the distant observer in a two-entity setup shall witness different realities with respect to the same physical occurrence. In this regard, it can be especially observed that

• The precession of the orbit of a client body as measured locally via its proper frame’s tracking of the wavefront constituent of energy h f = γ m 0 ∞ c 2 and
• As measured by the distant observer tracking the core/kernel constituent of relativistic mass γ m 0 ∞ e ‐ α will differ.

Meanwhile, both the local observer and the distant observer could, of course, communicate to each other the results of their differing measurements at any time; this is analogous to how each observer will assess the opposite party’s respective stick meter and period of time differently (cf. Equations 10a, 10b).

Notwithstanding, the single-entity iteration of the QTG considered in Marchal et al. (2023) allowed us to define for the wavefront constituent the artefactual instantaneous velocity v _*core* of the recoiling core/kernel with the rest-mass m 0 ∞ e ‐ α , which was therefore imagined to be moving concentrically with the latter entity as though the whole ensemble slowed down in gravity, insofar as acquiring the velocity

v c o r e = λ r T r = λ 0 e − α T 0 e α = v 0 e − 2 α . (12)

Thereby, it is as if there were a local velocity v ₀ associated with the test object as an undivided whole, which, when assessed by the distant observer, slows by the factor e ^2*α*. This is precisely why we introduced the term v ₀ = v _*hf* directly above the velocity of the wavefront of energy hf instead of the earlier symbol v (now being properly utilized for the current two-entity model) (cf. Equation 11d). In the single-entity formalism of the QTG, Equation 12 comes to mean that the wavefront of energy hf appears to move with the velocity v ₀ = v _*core* e ^2*α* , where there is as yet no further mention of hf.

In any case, in this model and as a first approximation, we came to assert that, while the wavefront of energy hf should ordinarily accelerate in a gravitational fall, it can instead be thought to move concentrically with the test object’s corpuscular entity of rest-mass m 0 ∞ e ‐ α lagging behind, that ought anyway to slow down on account of supplying the necessary energy for the acceleration of the wavefront constituent of the particle.

Such an occurrence may be construed as further evidence for how the photon (which, in the QTG, behaves like an ordinary object) must bear a non-zero rest-mass, inasmuch as allowing us to refer to it specifically as the “kernel”, regardless of how tiny it may be. Thus, only if the photon possesses such a kernel—as proposed, in effect, in de Broglie (1925) and elsewhere (Vigier, 1997; Tu et al., 2005; Marchal, 2009; Mikki, 2023)—could the energy exchange we framed between the hf and m 0 ∞ e ‐ α constituents take place in gravity (cf. Theorem 3). We discussed all this in the light of Feynman’s insight (Feynman et al., 1964) in Supplementary Appendix B of Marchal et al. (2023).

Equation 12 thus constitutes a satisfactory approximation leading (with respect to the discussed quantal case of the QTG alone) to all the established space-time findings of the past century that have so-far been deemed to confirm the GTR.

4 Single-entity solution from earlier

We now recapitulate the single-entity solution from Marchal et al. (2023) as a useful tool for building the QTG’s two-entity iteration. The full formalism with respect to the two-entity problem will be delivered in Section 5.

Addressing, at this stage, Equation 8 and carrying out its differentiation, we obtain (Yarman, 2006)

− G 0 M m 0 ∞ e − α r 2 1 − v 0 2 / c 2 d r = v 0 d v 0 m 0 ∞ e − α 1 − v 0 2 / c 2 . (13a)

This equation embodies a number of deep insights.

• It may, as detailed in the footnote ^* (the way given in Supplementary Appendix H; just like is the case for all concomitant footnotes that appear hereinafter), pertain to the motion of the corpuscular entity of rest-mass m 0 ∞ e ‐ α .
• This directly gives rise to the de Broglie wavelength relationship λ _*B* = h/p that we propose to interpret here as allowing interaction without any energy exchange, all the while delineating a full harmony between STR and QM (Albert and Galchen, 2009).
• Equation 13a may, if the terms are re-arranged, pertain to the motion of the wavefront of energy hf alone in view of the local observer (see below).
• It can be considered as a platform to frame the QTG’s single-entity formalism, where the core/kernel of rest-mass m 0 ∞ e ‐ α was sanctioned to move together with the wavefront of energy hf, with both having the retarded velocity v _*core* as yet an approximation, to the extent that it is as though the test body propagated with the local velocity v ₀ that slows when viewed by the remote observer, along with v _*core* = v ₀ e ^−2α [cf. Equation 12], entailing furthermore that the coordinate r can hence be applied to this straggling ensemble (Marchal et al., 2023) (see below).
• It can also be taken as the basis to frame the motion of the retarded core/kernel of rest-mass m 0 ∞ e ‐ α within the forthcoming two-entity setup of the QTG, where v ₀ = v _*core* e ^2*α* should read as the speed of the wavefront of energy hf in gravity, expressed yet in terms of the velocity v _*core* of the corpuscular aspect, and where the coordinate r should accordingly be reverted to the core/kernel coordinate r _*core* in view of the distant observer.

We will now exercise the third and fourth items within the framework of the single-entity setup, and the last option will be considered with regards to the two-entity formalism.

4.1 Equation of motion for the corpuscular mass m 0 ∞ e ‐ α in the single-entity setup of the QTG

We continue by commenting further on Equation 13a. Here, with regards to the QTG’s single-entity formulation, it should first be recalled that the velocity v ₀ and the altitude r entering into this equation ordinarily point to the wavefront of energy hf, and they should be expressed in terms of v _*core* and r _*core* starting with the relationships in the form of v _*core* = v ₀ q(r) and r _*core* = rg(r) (with q(r) and g(r) denoting independent functions of the coordinate r), where v _*core* and r _*core* respectively point to the velocity and the location of the core/kernel of rest-mass m 0 ∞ e ‐ α. For the wavefront of energy hf, one can equivalently use the velocity v ₀ = v _*core* e ^2*α* (see Equation 12).

A description can, thus, be visualized in terms of the coordinate r of the wavefront and the velocity v _*core* of the core/kernel, making it as though the test body moves like a single entity with the local velocity v ₀ that slows when viewed by the remote observer in just the way stated previously. We shall follow this latter option to derive QTG’s single-entity framework.

Nevertheless, Equation 13a can be considered as a basis for describing the two-entity iteration of the QTG (see Section 5). We can further elaborate on this point by rephrasing Equation 13a in a simplified form:

− G 0 M r 2 1 − v 0 2 c 2 d r = v 0 d v 0 1 − v 0 2 / c 2 . (13b)

Here, the rest-mass m 0 ∞ e ‐ α of the core/kernel constituent of the gravitating particle drops off from the equation of motion.

While Equation 13b makes the QTG fully consistent with the classical result under the GTR for a test body, it simultaneously provides the mechanical basis for the physical separation of the wave and particle aspects of matter; this, as stated previously, is strikingly validated by the recent experimental findings reported under (CGTN, 2025; Na, 2025; Chen, 2025; Zhang et al., 2025). It is here, through high-precision atom interferometry, that a gradual shift from the wave manifestation to the particle signature becomes apparent (cf. Figure 1).

While standard QM relies on the probabilistic Copenhagen interpretation with its unresolved “wavefunction collapse” and “measurement problem” to explain this picture, the two-entity architecture of the QTG provides a deterministic resolution. This is how the ambient wave train composed of orderly, coherent, and steadily broadcast wavefronts of energy hf each creates the interference-favorable background phase configuration, where the localized kernel—the seat of the rest-mass m 0 ∞ e ‐ α—follows the corpuscular trajectory governed by the equation of motion given in Equation 13b.

Note that Equation 13b engenders the de Broglie wavelength relationship discussed in the footnote* above. Moreover, the factor 1 − v 0 2 / c 2 appearing in front of the Newtonian force in Equation 13b drastically diminishes this force when the velocity v ₀ approaches c, where it is practically suppressed to zero. This is not, however, what would be desired with respect to the description of the wavefront of energy hf in the quantal regime, which is expected to interact with gravity in the usual (e.g., general relativistic) manner. Fortunately, there is an alternative form of Equation 13b (see further below) which overcomes this issue while fully conforming with Theorem 3.

4.2 The particular form of Equation 13b suitable for the single-entity in the wave-like regime

We now multiply both sides of Equation 13b by the initial relativistic mass m _*i* (see Equations 11a, 11b ) and rewrite it as

− G 0 M m i e α r 2 d r = v 0 d m i e α v 0 . (14a)

This scalar equation is the same as Equation 13b; however, at the same time, and crucially more to the point, the Newtonian force, now being different, does not vanish as v ₀ approaches c.

Note further that Equations 13b, 14a lead to different vector equations of motion, as will be shown below. Furthermore, Equation 14a remains in full harmony with Theorem 3 (as shall soon be observed) and therefore with the wave-like regime of the QTG in its single-entity setting. Equation 14a requires that the original de Broglie relativistic mass m _*i* = hf _*i*/c (Marchal et al., 2023) measured in empty space increase as much as e ^*α* in the presence of gravity. This is ensured by a conjoint decrease of the rest-mass of the core/kernel aspect m 0 ∞.

All this naturally necessitates the coupling of

• Equation 13b written for the corpuscular behaving rest-mass m 0 ∞ e ‐ α and
• Equation 14a framed for the wave-like wavefront of energy hf _*i* = m _*i* c ², leading, as previously said, to different vector equations of motion (see below).

Thus, in the quantal treatment option of QTG, the increase of m _*i*, or similarly, the increase of the frequency f _*i* and the related contraction of the wavelength λ _*i* , are concomitant effects. It all takes place under the special wave-like conditions framed by Theorem 3, paired with Theorems 1, 2, where the quantum mechanical time-dilation and length-stretching transpire owing to the decrease of the core/kernel’s proper rest-mass m 0 ∞ in gravity (see Equations 10a, 10b). Equation 14a thus represents a capstone expression of our approach. Based on it, we can straightforwardly derive, for example, the result of the Pound and Rebka experiment (Yarman et al., 2016a), as well as the angular momentum conservation law under the wave-like option of the QTG (see the following section).

4.3 Equation of motion of the wavefront of energy hf

We return again to consider the throttled wavefront and the straggling core/kernel as two distinct aspects of the said test object in respect to its wave-like and particle-like modes of interaction. The quantities of these aspects are

• The relativistic mass of the wavefront constituent of the particle—or the de Broglie relativistic mass—given as m _*hf* = hf/c ^*2* (see Equation 11c), and
• The core/kernel rest-mass of the test object given as m 0 ∞ e ‐ α .

Thus, by combining Equations 14a and 11e, we obtainwhere C is a constant with the unit of energy specific to the motion.

e α = c 2 m h f / C , (14b)

Next, we modify Equation 14a to the equivalent formwhere v ₀ is the instantaneous velocity of the wavefront of energy hf.

G 0 M m h f r 2 c 2 d r = v 0 d m h f v 0 , (14c)

Note once again that the wavefront of energy hf is built on the proper rest-mass m 0 ∞ (see Equations 11c, 11d), which remains unchanged for the local observer. At the same time, in a two-entity setup, the corresponding velocity v _*hf* of the said wavefront, as assessed by the distant observer, remains equal to the velocity v ₀, given that lengths and durations stretch by the same amount in gravity (Theorem 2), with velocities thereby remaining the same for the local and the distant observer. Therefore, the wave-like interaction expression should be the same for both observers. Furthermore, we point out the relationshipwhich will serve in transforming Equation 14c into a vector form (by recalling that v _*hf* = v ₀ is the velocity of the wavefront as assessed by either the local or the distant observer), furnishing the vector equationwhich is very close to what the GTR leads to (cf. Eq. (iv) of Footnote^†) (Landau and Lifshitz, 1999; Yarman et al., 2014).

r · d r r d r = v 0 · d v 0 v 0 d v 0 , (14d)

− G 0 M m h f r 2 r r = d m h f v h f d t , (14e)

This equation describes the motion of the wavefront of energy hf = m _*i* c ² in gravity (see Equations 11c, 11d), and leads, in only two lines, to the result of the Pound and Rebka experiment (Yarman et al., 2016a).^‡ Note that Equation 14e strikingly implies the identity of the gravitational mass and the relativistic mass in terms of m h f = γ m 0 ∞, which, however, is an instantaneous relativistic mass under the dynamic case framed by Theorem 3 in only the quantal regime of the QTG (and where, otherwise, the rest-mass would decrease as governed by Theorem 2). This engenders Equation 13a, which clearly furnishes a different scheme that gives rise to another equation than Equation 14e, as addressed below.

Equation 14e is precisely what makes the centennial results associated with the success of the GTR fully aligned with the QTG (Marchal et al., 2023). Nevertheless, this occurs only when the test object enters into a wave-like mode of interaction in gravity with respect only to the description of the wavefront of energy hf. Even so, Equation 14e is written for the local observer, and it should be associated with the description of the motion of the recoiling core/kernel of rest-mass m 0 ∞ e ‐ α through Equation 14a to yield the whole picture regarding the remote observer; this will be addressed in Section 5. In summary, both the wavefront of energy hf and the corpuscular rest-mass m 0 ∞ e ‐ α were earlier treated within a single-entity model (Marchal et al., 2023) by considering both of them as hypothetically moving with the same velocity v _*core* of the recoiling core/kernel in accordance with Equation 12.

4.4 Approximation of where the wavefront of energy hf and the corpuscular mass m 0 ∞ e ‐ α superimposed upon each other move in terms of the velocity of the retarded core/kernel

We again consider Equation 14a, which is appropriate for the wave-like case of the QTG in full similarity with the GTR, because the QTG implies, in the given regime, the identity of the gravitational and the inertial mass. We also propose to treat the wavefront of energy hf and the core/kernel of rest-mass m 0 ∞ e ‐ α within a single-entity setting (Marchal et al., 2023), insasmuch as they are imagined to propagate together with the analogous velocity of the retarded core/kernel constituent.

Using Equation 3, and multiplying both sides of Equation 14a by e ^*−*2*α* , the latter constituent’s differential rest-energy can be written aswhich encapsulates energy conservation in the wave-like case of the QTG.

d m c 2 = m 0 ∞ v 0 e − 2 α d e α v 0 , (15a)

Identically, Equation 14a yields the equalitywhich can be read as an expression of momentum conservation in, again, the wave-like case of the QTG.

d m c 2 v 0 e − 2 α = m 0 ∞ d e α v 0 , (15b)

In effect, using Equation 14b and assuming the constant γe ^*-α* to be equal to unity (which is a rigorous assumption for the test object in, for example, a free fall (Yarman, 2006)), Equation 15b can instead be presented as^§

d m c 2 v 0 e − 2 α = d γ m 0 ∞ v 0 . (15c)

This justifies how, in a wave-like case, the core/kernel of proper mass m 0 ∞ of the test object depletes the infinitesimal rest-mass |dm| through an infinitely short altitude decrease in gravity entirely in accordance with the superluminality-permitting propagation velocity c ²/v ₀ e ^*−*2*α* , which is defined with regards to an object bearing the instantaneous fall velocity v ₀ e ^*−*2*α* [cf. Equation 12].

Such a formalism works insofar as Equation 15c governs the exchange of energy between the core/kernel rest-mass m 0 ∞ e ‐ α and the de Broglie relativistic mass hf/c ² of the wavefront, so that an infinitesimal kick forward is delivered to the entity of energy hf that makes the trailing corpuscular entity recoil just as infinitesimally. This is so much so that the core/kernel mass m 0 ∞ e ‐ α , as assessed by the distant observer, must lag ever further behind the throttled wavefront of energy hf with the velocity^**

v c o r e = v 0 e ‐ 2 α . (15d)

The above equality is precisely what we obtained using Equation 12. Moreover, it instead indicates that the wavefront of energy hf moves with the velocity v ₀ = v _*hf*, which anyway remains the same for both the distant and the local observer as implied by Theorem 2. The corpuscular mass m 0 ∞ e ‐ α then comes to trail behind the wavefront of energy hf with the retarded velocity v _*core* in accordance with the law of momentum conservation, which, in turn, is also invariant in view of Theorem 2 with respect to either the local or the distant observer. Thus, Equations 12–15d are complementary in providing the instantaneous core/kernel velocity v _*core* in terms of v _*hf*.

We can henceforth consider (i) the motion of the wavefront of energy hf and (ii) that of the corpuscular mass m 0 ∞ e ‐ α in relation to the two-entity test object distinctly on the basis of Equation 13b, coupled with Equation 12, in order to describe the motion of the core/kernel of rest-mass m 0 ∞ e ‐ α , along with Equation 14e describing the motion of the wavefront of energy hf. However, we explain the motion of the two entities separately in Section 5.

We stress that we started in Marchal et al. (2023) with Equation 15d to treat the object as a single-entity, and the entire conjectured ensemble made of hf/c ² and m 0 ∞ e ‐ α was conceived to move in lockstep with the velocity v _*core* = v ₀ e ^*−*2*α* [Equation 15d], where r entering into α represents the modulus of the tracking vector *r* of hf. This constitutes the fundamental ingredient for the single-entity iteration of the QTG.

At this point, taking d *r* /dt = *v* and using the infinitesimal rest-mass dm from Equation 3, we obtain from Equation 15b

G 0 M e α r 2 r r = d v 0 e α d t . (16)

Equation 14b confirms that this exact relationship is the same as the Newtonian-like Equation 14e, except that we keep the term e ^*α* instead of the Lorentz factor γ, and where v ₀, owing to Equation 15d, will now be expressed in terms of the velocity of the recoiling core/kernel of proper mass m 0 ∞. Nevertheless, the coordinate r should be reverted to r ₀ in view of Equation 10c.

4.5 Acceleration vector in the QTG’s single-entity iteration

At this stage, using Equation 16, we are able to derive the angular momentum conservation law (Marchal et al., 2023). For this purpose, we multiply both sides by the vector *r* :

− G 0 M e α r 2 r r × r = d v 0 e α d t × r . (17a)

The LHS of the latter equation is null and so must be the RHS:

d v 0 e α / d t × r = 0 . (17b)

From here, we derive (Marchal et al., 2023)

d r × v 0 e α / d t = 0 . (18)

The integral of this equation must then be a constant vector, which we designate as l :

r × v 0 e α = l . (19)

Since we have yet to revert to r ₀ when l is assessed by the distant observer (see Equation 10c), and having used Equation 12, we obtain for the distant observer:

r 0 e α × v e 2 α e α = r 0 × v e 4 α = l . (20)

Here is where we departed from Equation 14e, which was written for the local observer, and it is whence we consequently arrived at an expression with regards to the distant observer.

In order to compose the total angular momentum L , Equation 19 must be multiplied by the relativistic mass m _*Overall* = E _*Overall*/c ² (see Equation 7); thus, the total angular momentum *L* becomes a new constant of the motion of the test particle (Marchal et al., 2023):

m O v e r a l l e 4 α r 0 × v = L . (21)

The other constant of the motion is the total energy of the test object as transcribed in Equation 7.

Based on the stated properties, we can determine the total acceleration d *v* /dt of the test object in terms of both its tangential and radial components to obtain more information about its motion. The calculations yielding a new equation of motion with regards to the single-entity formalism of the QTG were presented in Marchal et al. (2023) and are replicated in Supplementary Appendix B.

d v d t = − G 0 M r 2 e − 4 α 1 + v 0 2 c 2 r 0 r − 4 r 0 · v v r 0 c 2 . (22)

As previously shown in Marchal et al. (2023), Equation 22 yields solutions indistinguishable from those suggested by the GTR for all cornerstone experimental and astrophysical space-time findings of the past century.

4.6 Application of the QTG to select problems

It has been demonstrated above, based on Theorem 2, that gravitational redshift represents, in essence, a quantum mechanical phenomenon. Equation 22, on the other hand, addresses two other phenomena known to be classical examples supporting the GTR:

• Gravitational deflection of visible light, and
• Precession of the perihelion of a planet (Marchal et al., 2023).

For the sake of completeness, Supplementary Appendix C features the precession solution given by the QTG for a Mercurial body as observed by a distant observer the way reproduced from Marchal et al. (2023), which remarkably amounts to an identical result furnished by the GTR (Ohanian, 1994).

The classical bending of light also appears as a triviality from Equation 22 (Marchal et al., 2023).

4.6.1 Non-bending of corpuscular high-energy γ-quanta in gravity

The QTG successfully describes projectile-behaving particles where Theorem 3 no longer holds; thus, the equation of motion reduces to Equation 13b, which can be re-written as

d v d t = − G 0 M r 2 1 − v 0 2 c 2 r r . (23)

Equation 23 yields practically zero gravitational deflection when the velocity of the test object tends to the asymptotical upper bound c. The above equation is thus applicable to high-energy γ-quanta just as we predicted (Arik et al., 2016). It is worth restating that “…such an outcome remains entirely at odds with all metric theories of gravity where the metric of space-time cannot depend on the energy of the particles” (Gharibyan, 2022). Therefore, preliminary experimental findings on the subject corroborate the QTG.

We were frequently asked how one can possibly determine the distance r from the test mass m(r) to the mass M of the host body if there was not in place an a priori metric of space-time. This finds its answer by means of the foregoing property: high-energy γ-quanta that are unaffected by a host mass (Arik et al., 2016; Gharibyan, 2022) can be used to gauge distances. This can be achieved if a local observer situated on, say, a planet sends a γ-quantum to its host star, with both the local and the distant observers assumed to be at rest with respect to each other for simplicity, and where the particle propagates in gravity with practically the upper bound velocity c without being affected at all. If we assume c to be a unity, the observer on the planet (whom we refer to as the “local observer”) can measure the period of time T the particle takes to go and return, whereby half of T constitutes the local measurement of the distance r ₀ of the planet to its host star (in units c = 1). Consequently, the distance r, the way assessed by the distant observer, is determined easily and already in an integral form. As discussed previously (Yarman et al., 2020; Marchal et al., 2023), the relationship between r and r ₀ is given by Equation 10c.

4.6.2 Possibility of retrograde perihelion precession in the corpuscular case

Equation 23 above—unlike Equation 16 written in the quantal case—excludes any wave-like mode of interaction. Even so, both equations are based on the same scalar differential equation—Equation 13b—which, in turn, is rooted in our energy conservation postulate at the outset—Equation 7. Nevertheless, Equation 23, when written for a projectile-behaving photon as discussed immediately above, implies practically null bending.

Thus, the question with regards to a further visualization of a test object in motion around a source body is: while a high energy photon grazing a source mass may not be engaged in a wave-like mode of interaction, would it be possible that an object in a closed orbit might also behave particle-like?

Equation 23 allows the possibility that, in a corpuscular engagement of a test object with the host body (if this were ever possible), a retrograde precession of the perihelion of the orbit will arise, which can be derived similarly to Supplementary Appendix C. Such a precession will consist of just one of three expected units with respect to what a Mercurial orbit would delineate. Astonishingly, however, that one unit of precession would be directed backwards as referred to the orientation of the elliptical motion as scrutinized by the distant observer. The subject matter at hand was raised in Polyachenko et al. (2007) and Capuzzo-Dolcetta and Bordoni (2023) with regards to near-Keplerian potentials, but we will avoid any more details at this stage.

That being so, it is clear that retrograde precession of the orbital motion cannot ever occur in the quantal case of the QTG or, for that matter, in the GTR. Thus, if any retrograde precession of a given elliptical orbit were to be observed in the future, we do not see any way for it to be explained other than through the QTG’s corpuscular regime option. Any such confirmation would constitute another finding in favor of the present undertaking.

Nevertheless, the scalar form of Equation 23 could well be transformed to the equation of motion of the core/kernel of rest-mass m 0 ∞ e ‐ α in its recoil in the quantal regime of the QTG, addressed in Section 5 below.

4.6.3 Ad hoc space-time metric equivalently furnishing the wave-like regime of the QTG

We have shown in Marchal et al. (2023) that the equation of motion (22), ignoring the fact that the coordinate *r* should be replaced by *r* ₀ on its RHS, could be obtained independently through the minimization of action with the ad hoc metricin spherical coordinates (Yılmaz, 1992) and, thus, for QTG’s quantal case only. This metric equation becomes indistinguishable from the Schwarzschild metric in the case of weak gravity, where e ^2*α*≈1-2α, thereby with r ₀ ² ≈r ² in front of the parentheses on the RHS.

c 2 d τ 2 = e − 2 α c 2 d t 2 − d r 2 e 2 α − r 0 2 e 2 α d θ 2 + d φ 2 ⁡ sin 2 ⁡ θ (24a)

The above metric formulation is not mandatory; it merely indicates that the findings arrived at by the single-entity iteration of the QTG can be reverse-engineered from an equivalent metric formalism, even when it is completely artificial and unnecessary. Such a metric can therefore be employed if desired, but not as a prerequisite for the QTG to function.

Regardless, Equation 22, and thereby Equation 24a, entail no singularities or event horizons; hence, any particle gathering sufficient energy inside a “black hole of the QTG type” can break out. We thus resolve the information loss paradox (Yarman et al., 2016b).

It is striking how the results of the GTR for weak-to-moderate gravity can be tapped under the QTG, but only in regards to our wave-like case. Meanwhile, the GTR is known to be irreconcilable with QM, whereas QTG has no such shortcoming.

4.6.4 Virtual equivalency of orbits in GTR and QTG

It is remarkable that.

• The QTG’s quantal Equation 22 and
• The equation of motion that would be derived within the GTR framework based on the Schwarzschild metric are virtually the same for ordinary orbits.

An example is the simplest case—a circular orbit—to compare the two theories. For the GTR, the Schwarzschild metric furnishes the approximate relationship for the velocity v of the test object in orbit, along withwhereas the QTG yields in the quantal case (cf. Equation 22)

v 2 c 2 = α 1 + 3 α = α 1 − 3 α + 9 α 2 , (24b)

v 2 c 2 = α e − 4 α 1 − α e − 4 α = α 1 − 3 α + α 2 . (24c)

The difference is then as little as 8α ². It just so happens that the orbital velocity is just a little greater in the GTR for a given altitude than it is in the QTG; or conversely, the orbits in the QTG are situated a little farther than those of the GTR for the same velocity. These occurrences ordinarily remain, as of now, undetectable. But the difference evidently becomes important in strong fields.

4.6.5 An altogether different space-time metric under the corpuscular regime of QTG

In the particle-like case of the QTG, an altogether different metric, which remains in full conformity with Theorem 2, can be derived straightfowardly in an integral form in view of Equations 11a, 11b (Yarman et al., 2020):

s 0 2 = c 2 t 0 2 − r 0 2 , (24d)

s 2 = s 0 2 e 2 α . (24e)

This set of equations constitutes a Minkowskian metric that is consistently harmonious with QM.

4.6.6 Full quantum mechanical deployment of the QTG

A full quantum mechanical deployment can be straightforwardly associated with the present undertaking, too (Marchal et al., 2023). Its point of departure simply consists of squaring Equation 1 and replacing the ensuing overall momentum *p* with the quantum mechanical momentum operatorwhich is nothing other than the expression of the de Broglie wavelength relationship in terms of quantum mechanical operators. Note that this procedure can be applied to any non-radiative field. In addition, while standard quantum mechanics reproduces the expected statistical outcomes, it does not assign, unlike the QTG’s two-entity formalism, distinct dynamical roles to the quantal and corpuscular constituents that are posited to comprise an individual particle.

p = − i ℏ ∇ , (24f)

5 Complete solution to the QTG’s two-entity setup in the wave-like regime

A primary result of the extended QTG herein—representing a significant milestone in comparison to earlier incremental developments of the UMA scaffolding—is the mechanistical resolution of the wave–particle duality through the posited two-entity formalism. This leads to the prediction of how the wavefront and corpuscle constituents of a test particle should physically separate under strong gravity. Figure 2 visualizes this by depicting a cross-section of the internal structure of the gravitating particle composed of

• A wave train of accelerating wavefronts and
• An associated core/kernel vorticity that undergoes “wavefront skipping” as it gradually trails behind.

Thus, in a field-like gravity, as discussed above, the wavefront of energy hf is throttled forward while the core/kernel of rest-mass m 0 ∞ experiences a physical recoil (cf. Theorem 3), insofar as this reflects the energy re-distribution necessitated by the field. Such an architectural re-arrangement of the energy content of the gravitational system’s components is achieved via exponential scaling, where the wave train’s frequency increases as f = f 0 ∞ e α , while the corpuscular rest-mass reduces as m = m 0 ∞ e ‐ α. This clarifies that gravity does not arise from a passive background curvature but is the action of a dynamic agent (e.g., superfluidic vacuum) that physically alters the dimensions and the clock rate of the material body.

While the outcome of the new atom-interferometry experiment (CGTN, 2025; Na, 2025; Chen, 2025; Zhang et al., 2025) appears to be sufficiently convincing in this regard, we still propose that, if the setup was oriented in the vertical plane on the surface of the Earth, or in any convenient field for that matter, the present approach predicts a “divorce” between the blurred wave-haze and the corpuscle-rain (cf. Figure 1).

This geometric bifurcation is a unique signature of the QTG that would provide a clear experimental target for the next generation of interferometric tests using atoms, whereby we sketch the expected distribution on the measurement screen as imagined by an AI tool in Figure 3.

Therefore, within the framework of the QTG, an either free-cruising or gravitating quantum object would be dynamically described by the aforementioned two physical entities coupled together in accordance with Theorem 3. In that case, the wave-like mode of interaction engendered by the clash of accelerating wavefronts of energy hf would each physically culminate with an interference panorama formed by the totality of all strikes to the designated backstop target of the corpuscular entity m 0 ∞ e ‐ α , whose total relativistic energy remains conserved through the rest-mass reduction factor e ^*-α* when under the influence of a host body.

Here, the two entities of concern

• Obey distinct equations of motion,
• Propagate with different velocities, and
• Are associated with different observational frames (viz., local and remote) while remaining strictly coupled to each other through the laws of energy and momentum conservation; as such, a two-entity description constitutes the conceptual and dynamic basis for all subsequent developments under the QTG.

Such a “dissociation of the entities”—meaning the physical separation of the recoiling core/kernel from its parent wavefront that together constitute the particle in and of itself—provides a concrete mechanism by which the appearance of a quantum mechanical interference panorama, alongside localized singular hits, can be made to sensibly coexist without invoking any mystical “collapse/update of the wavefunction” or being confused by the so-named “measurement problem”.

The whole process is thence governed by energy and momentum conservation laws in line with the two independent ballistic equations below, whereby a minute rest-mass is pumped to the throttled wavefront of energy hf—ascribed the archetypal de Broglie relativistic mass m _*hf* = hf/c ²—from the core/kernel of mass m 0 ∞ e ‐ α that keeps increasingly trailing as seen by the distant observer: and as written from the vantage point of the local observer (cf. Equation 5).

• Equation 14e obtained for the locally-posed wave-like mass m _*hf* = hf/c ² copied here for simplicity,

− G 0 M m h f r 2 r r = d m h f v h f d t , (24g)

• 2. Obtained in view of Equation 14d, for the straggling core/kernel of rest-mass m 0 ∞ e ‐ α and generalized to a vector form,

− G 0 M m 0 ∞ e − α r 2 1 − v 0 2 c 2 r r = m 0 ∞ e − α 1 − v 0 2 c 2 d v 0 d t , (24h)

The latter equation will also be suitable for the remote observer if the velocity v ₀ is firstly reverted to the retarded velocity v _*core* of the core/kernel constituent through Equation 15d. In that case, it is also necessary to therefore depending on the new term g (r ₀), which we hereby introduced to bridge r ₀ and r _*core* in a manner soon to be established (with the caveat that g (r ₀) must be smaller than unity).

• Express r in terms of r ₀ as imposed by Equation 4 in order to revert the measurement to the distant observer’s reference frame, and
• Explicitly transcribe the coordinate r _*core* of the core/kernel of rest-mass m 0 ∞ e ‐ α lagging behind hf as a function of r ₀ using a function such as

r c o r e = r 0 g r 0 , (24i)

Equations 13b, 14e are separately presented in vector form, as they are both derived from our root energy conservation Equation 7; however, we could also write this latter equation as Equation 11e—that is, as hfe ^*-α* = Constant = m _*i* c ²—to describe the wave-like interaction.

Therefore, the coordinates must, as stated, be differently assigned to the two distinct entities hf and m 0 ∞ e ‐ α , despite hf essentially being assessed by the local observer and m 0 ∞ e ‐ α being tracked by the remote observer throughout the motion of the gravitating test object in the wave-like regime. Hence, the outcomes of Equation 14e are, in principle, to be observed locally, and those of Equation 24g are to be scrutinized by the remote observer with regards to the two different constituents of the same object.

Nevertheless, it appears not to be a simple task to theoretically calculate the coordinate r _*core* of the corpuscular constituent of the particle (which is, in principle, possible in the QTG) when we start with the coordinate r of the wavefront of energy hf and the core/kernel assigned the velocity v _*core*.^†† Instead, we will follow a practical shortcut with the introduction of a free parameter k; until then, we will continue to use r instead of r _*core* in Equation 24g.

5.1 Acceleration vector of the test object’s hf constituent in QTG’s two-entity setup

As observed from the preceding analysis, Equation 14e is analogous to the Newtonian-like equation adapted to the relativistic case. We can further derive the angular momentum conservation law on the basis of this equation by multiplying both sides by the vector *r*:

− G 0 M γ r 2 r r × r = 0 = d γ v h f d t × r + γ v h f × d r d t = d γ v h f × r d t . (25)

Note that d *r* /dt is the velocity *v* _*hf* of the wavefront of energy hf, and *v* _*hf* × d *r* /dt = 0.

Equation 25 shows that, for the locally applicable wavefront mass hf/c ^*2* of the considered test object, we obtain the angular momentum constancy per a unity of mass:

γ v h f × r = l h f . (26)

Using Equation 11e, this can also be expressed aswhere C is a constant close to unity in most cases.

e α v h f × r = l h f C , (27)

The detailed derivation of the acceleration d *v/dt* of the wave entity having the energy hf is presented in Supplementary Appendix D. It leads to the one unit of Newtonian bending for light out of the QTG’s total output of two. This equation conjointly brings one unit of precession out of the expected overall three units when assessed by the distant observer, as demonstrated in Supplementary Appendix E. Once again, though, all this is just for the wavefront of energy hf as assessed, essentially, by the local observer.

To simplify the notation, we equate *v with the velocity v* _*hf*. Then, the acceleration vector of the wavefront of energy hf becomes

d v d t = − G 0 M r 2 r r − r · v v r c 2 . (28)

Here, the force per unit test mass is defined for the local observer fixed on the trajectory, as the interaction of hf with the host mass will take place locally (see Equation 2d).

Equation 28 is valid only for the local observer resting on the path. This appears to be profound, for it actually says that a wave-like scenario occurring in the vicinity of a host mass should involve only one unit of bending out of the two expected by the remote observer when the gravitating body is scrutinized by the fixed local observer (cf. Supplementary Appendix E), whereas the same bending phenomenon should instead be measured to be two units out of two when assessed by the remote observer (see Equation 22). Such a prediction of the QTG could be considered attractive to garner the attention of the scientific community towards the achievement of experiments similar to that described in (Gharibyan, 2022) but involving wave-like behaving EM radiation, such as radio transmissions or laser beams.

It is notable that Equation 14e, which we considered in this sub-section for the two-entity iteration, has been the pivotal equation of our single-entity gestalt dealt with in Section 4. Note further that, in the QTG’s single-entity formalism, the following workarounds have been applied (Marchal et al., 2023).

• We have considered both hf and m 0 ∞ e ‐ α to be altogether fixed to the motion of m 0 ∞ e ‐ α .
• We have supposed that the latter moves, in turn, with the retarded core/kernel’s velocity v _*core* = v ₀ e ^*−*2*α* (cf. Equation 15d).
• We have thereby assumed that it all behaves in a wave-like fashion as implied by Equation 14e.

This is how we have arrived at the single-entity quantal solutions of the QTG that perfectly match the end results of the GTR in the weak field, all the while remembering that the GTR is incompatible with QM.

5.2 Acceleration vector of the straggling corpuscular mass m 0 ∞ e ‐ α of the test object in the QTG’s two-entity setup

The core/kernel of rest-mass m 0 ∞ e ‐ α constituent of the test object moves, within satisfactory precision, with the retarded velocity v _*core* = v _*hf* e ^*−*2*α* (see Equations 12, 15d), where we then have v _*hf* = v ₀. In order to determine its acceleration, we first transform Equation 13a into a vector form using Equation 14e:

− G 0 M r 2 1 − v 0 2 c 2 r r = d v 0 d t . (29)

We can hence derive the angular momentum conservation of the core/kernel by multiplying both sides of Equation 29 by the vector *r*:

− G 0 M r 2 1 − v 0 2 c 2 r r × r = 0 = d v 0 d t × r + v 0 × d r d t = d v 0 × r d t . (30)

Thus, the vector quantity *v* ₀ × *r* must remain constant.

Note that d *r* /dt is the velocity *v* ₀ = *v* _*hf* of the throttled wavefront constituent of the test object, and the vector product of the two is, evidently, null. All the same, Equation 29 pertains to the motion of the corpuscular constituent under the condition that we revert to the necessary coordinates. Therefore, using Equations 10c–15d, we find the angular momentum constancy per unit mass:

v c o r e e 2 α × r 0 e α = l c o r e . (31a)

As was prescribed before, we had to revert from *r to r* ₀ here (see Equation 10c) and from *v to v* _*core* (see Equation 15d), which is equivalent to stating

v c o r e × r 0 = l c o r e e − 3 α . (31b)

Next, we have to revert to r _*core* from r ₀. However, as mentioned in Footnote§§, this is not an easy task. To make it simpler, we propose to introduce a post hoc parameter 1 ≥ k ≥ 0 that, in the known classical cases, ought to be close to unity. This would mean that k = 0 entails no recoil, while k = 1 will lead to the solution of QTG’s single-entity formalism, which will be discussed below. We thus pose (see Equation 24e):With this, we rewrite Equation 31b:

r c o r e = r 0 e ‐ k α . (31c)

v c o r e × r c o r e = l c o r e e − 3 + k α . (31d)

The detailed derivation of the acceleration d *v* _*core* /dt of the core (or “kernel” in case we have a photon) is provided in Supplementary Appendix F.

Therefore, the acceleration vector of the core/kernel of rest-mass m 0 ∞ e ‐ α as derived in Supplementary Appendix G finally becomes

d v c o r e d t = − G 0 M r c o r e 2 1 + k v c o r e 2 c 2 r c o r e r c o r e − v c o r e 3 + k c 2 v c o r e . r c o r e r c o r e . (32)

Thus, for k = 1, and in weak gravity, we arrive exactly at Equation 20 of Supplementary Appendix B as stated, which is the same as Equation 22 that we obtained on the basis of the single-entity iteration of the QTG (Marchal et al., 2023). Otherwise, the ad hoc metric Equation 24a will no longer be applicable.

Therefore, should k be even just a bit smaller than unity, we would need to give up the set of results derived in the preceding section that were conformal with our single-entity approximation. Nevertheless, if gravity was not weak, even Equations 22, 32 would differ from each other, and the whole metric conception of the last century would be rendered useless; so much so that Equation 15d—in regards to the retardation of the core velocity—could be obtained only approximately based on Equation 31c under the given circumstances. This is one main contribution of the present study.

Note further that the recoil of the core/kernel constituent in gravity can, in principle, be computed through cumbersome calculations as discussed in Footnote§§. It nevertheless indicates a complicated task! This is precisely why we introduced the coefficient k and tried to gauge its value in comparison to recently collected data. We anyway know that the assertion k = 1 is as approximate as our single-entity approach (cf. Equations 15b, 15c). For instance, given the high-precision Shapiro delay measurements at the level of 10⁻⁵ to 10⁻⁴, centennial astrometric observations constitute a particularly sensitive probe to discern subtle deviations of QTG’s exact two-entity formalism prediction from the corresponding calculation under GTR (Will, 2014; Bertotti et al., 2003).

The recoil of the corpuscular entity in the presence of gravity can, in essence, be theoretically evaluated in the QTG; even so, the related calculations—as already discussed—represent a cumbersome mathematical problem.

To recapitulate:

• We embarked from Equation 24e;
• This, as shown in Footnote*, independently yields the de Broglie wavelength relationship, allowing us to ensure direct and full compatibility with QM;
• The coordinate r is to be reverted first to r ₀ as implied by Equation 4, and as tracked by the remote observer, and then to the recoiling core/kernel coordinate r _*core* as implied by Equation 31c;
• The velocity v ₀ is similarly to be reverted to v _*core* as implied by Equation 15d.

6 Summary tables

It is useful to summarize the foregoing equations in Table 2 with regards to the QTG’s single-entity setup, and in Table 3 with regards to its two-entity setup.

TABLE 2. Synopsis of the single-entity framework of the QTG.

TABLE 3. Synopsis of the QTG’s two-entity framework.

7 Conclusion

The derivational milestones presented in this work demonstrate that the Quantal Theory of Gravity (QTG) is not merely another alternative to the geometric gestalt of the General Theory of Relativity (GTR) but is a more fundamental framework that resolves the decades-long issues with singularities inherent in metric-based approaches, all the while accounting for centennial space-time findings so far thought to confirm Einstein’s curvature-based worldview.

By grounding gravitation in the mechanical conservation laws of the Universal Matter Architecture (UMA), we have shown that macroscopic interactions between bodies are manifestations of the dynamic scaling of mass-energy (e.g., by means of a “superfluidic vacuum” physically acting on both the host mass and the client body) rather than the components of the gravitational system being independent entities subject to a background curvature of space-time.

Within the QTG’s two-entity setup, a gravitating test object is thus torn apart into two separate constituents:

• An accelerating wavefront of energy hf and
• A corpuscular vorticity demarcated by m 0 ∞ e ‐ α , the relativistic energy of which remains conserved throughout the gravity-induced reduction of its rest-mass.

These two entities not only obey different equations of motion but are associated with, respectively, local and distant observers, while propagating with different velocities in distinct coordinates when assessed by the distant observer. More specifically, the wavefront of energy hf accelerates and the core/kernel of rest-mass m 0 ∞ e ‐ α recoils, as described by Equations 13b, 14e. Their coupled dynamics are then governed strictly by energy and momentum conservation without the need to introduce any space-time metric as a primary construct.

This formulation, as first introduced in Marchal et al. (2023), clarifies the physical origin of wave-particle coexistence. The wavefront of energy hf thence accounts for interference and wave-like propagation, while the recoiling corpuscular mass m 0 ∞ e ‐ α accounts for localized detection events and projectile-like impact behavior.

Thus, wave-like and particle-like aspects are not mutually exclusive descriptions but correspond to two simultaneously existing physical entities associated with the same quantum object. Mathematically speaking, the root conservation equation of the QTG—Equation 1—yields, via differentiation, a scalar equation that can be framed in two distinct ways (Table 1), leading to two different equations of motion (with both being reducible to the classical Newtonian equation of motion in the weak field and low velocity regime).

In the first equation of motion (Equation 13b), Newton’s gravitational force term is multiplied by the suppression factor 1-v ²/c ², so that, at high-to-ultra velocities, the test particle practically senses no force, thereby engendering a corpuscular description as per the QTG. In the second equation of motion (Equation 14e), there appears no suppression factor in front of Newton’s gravitational force term, and, remarkably, the masses associated with the test particle that enter into both the gravitational law and the force law become identical.

In view of Theorem 3, one can immediately qualify the latter equation of motion as the description of the wave-like behaving object. On the other hand, our root energy conservation formulation in Equation 1 leads—via the use of a new momentum operator (Equation 24f)—to an independent quantum mechanical deployment (Marchal et al., 2023), where the overall relativistic mass in the operator of concern is the mass at infinity decreased by the static binding and then increased in motion as much as the Lorentz coefficient coming into play the way framed under Equation 1. Thus, even in a stationary wave-like case, the behavior of the entity at hand is governed by two separate equations:

• An equation for the quantum mechanical probabilistic spread and
• An ordinary projectile-like equation of motion.

Recent high-precision single-photon diffraction experiments provide direct experimental support for this picture (CGTN, 2025; Na, 2025; Chen, 2025; Zhang et al., 2025). In these experiments, individual particles are detected as localized impacts, while the cumulative distribution of many such events reconstructs the known interference pattern (cf. Figure 1). The reciprocity of such a gradually unfolding interference pattern and localized detection events at the single-particle level is naturally explained by the QTG’s two-entity formalism, which anticipates the simultaneous presence of wave-like propagation towards, and corpuscular hits on, the designated target without invoking complementarity, which just so happens to be a purely philosophical quandary in standard QM that became a topic of many odd interpretations over a century.

The propounded two-entity iteration of QTG can further be tested by future quantum diffraction experiments to be performed in the vertical plane in Earth’s gravitational field. In such a setting, the wavefront entity of a single particle having the energy hf is associated with the delocalized quantum behavior that propagates through both slits, while the corpuscular constituent of mass m 0 ∞ e ‐ α engages only one slit and remains ballistic even when guided Brownian-like along to the fringes downstream, so as to ultimately register at a single point on the detection screen, to the extent that the resultant distribution will be perceptibly different from the experiment carried out in the horizontal plane, unlike standard QM predictions (cf. Figure 3).

Although no measurement can strictly reveal which slit the core/kernel traverses without destroying the interference-conducive phase conditions of the environment, the reciprocity of these two behaviors naturally follows from the QTG’s two-entity structure imposed by the laws of energy and momentum conservation. Such a mechanistical interpretation compatible with classical determinism therefore aligns the gravitational domain with recent high-precision diffraction experiments, inasmuch as they provide a direct empirical context for the formal developments presented herein (CGTN, 2025; Na, 2025; Chen, 2025; Zhang et al., 2025).

Hence, the two-entity formalism of the QTG provides the first physically comprehensible explanation of how a single particle’s core/kernel constituent can maintain its corpuscular identity while its wavefront constituent explores multiple paths. In which case—with respect to both classical and quantum experiments—the core/kernel entity goes alone through one of the slits with no interaction whatsoever, while it is the ubiquitously broadcast coherent, steady, and orderly wave-train’s associated wavefront entity that interacts with the slits and prepares the downstream phase conditions in the superfluidic environment conducive to the creation of fringes at the measurement destination.

Such a two-entity framework also leads to empirically distinguishable predictions beyond metric theories of gravity. In particular, the QTG anticipates the suppression of gravitational bending for ultra-energetic γ-quanta in contrast to the GTR and other metric theories. This prediction has been experimentally supported by recent measurements on the surface of the Earth (Arik et al., 2016; Gharibyan, 2022), which can further motivate high-precision tests aimed at probing gravitational interactions in regimes where the wave-like mode of interaction is absent.

The predictive power of the QTG, especially in regards to the recent atom-interferometry benchmarks (Zhang et al., 2025), is further reinforced by the fact that the underlying UMA scaffolding was used to rigorously formulate the systematization of matter at all levels, from nuclei to molecules, and even beyond to gases, as well as quantizing the second law of thermodynamics (cf. Yarman et al., 2018, and Supplementary Appendix A).

These successes confirm that the Hamiltonian scaling of rest-mass, in the form of rest-mass dynamics (RMD), is a universal principle from the sub-atomic realm to cosmological scales.

Moreover, deviations from unity with respect to the recoil parameter k [cf. Equation 31c] would immediately invalidate the metric paradigm; we thus emphasize the importance of future theoretical and experimental determinations of this quantity.

Beyond gravitation, the QTG universally applies to bound-field systems.

For instance, it offers a consistent explanation for the retardation of the bound muon decay-rate in atomic configurations (Yarman et al., 2011), along with gravitational redshift and spectroscopic phenomena through RMD (Marchal et al., 2023; Yarman, 2006).

In addition, the QTG naturally explains the observed accelerated expansion of the universe without invoking a cosmological constant, to the extent that we had predicted the latterly established waning of cosmic acceleration as being proportional to the inverse square of the distance (from the center of a plausible exfoliating cosmic egg) in agreement with recent astrometric data (Yarman and Kholmetskii, 2013; Abbott et al., 2024; Gueorguiev and Maeder, 2025; Xu, 2025).

In summary, the two-entity formalism of the QTG provides a unified framework contingent upon the laws of energy and momentum conservation that naturally reconciles deterministic gravitational dynamics with probabilistic quantum mechanical behavior, all the while positing wavefront–corpuscle coexistence at the level of individual quanta, yielding experimentally testable predictions across microscopic and macroscopic domains. The QTG’s conceptual simplicity and corroborating empirical evidence position it as a compelling alternative to purely metric approaches to gravitation.

The results presented herein hence require a fundamental re-evaluation of the clock hypothesis that forms the basis of the GTR, which treats the clock—and, by extension, the particle—as a passive observer of a curved space-time.

However, we have shown that gravitational redshift and motion are not mere geometric effects, whereby gravity is thence not a stage or fabric upon which the particle obeys top-down imposed geodesics; instead, a superfluidic physical background is the active participant in the particle’s “internal dynamics mechanical equilibrium”.

The centennial unawareness of the possibility of the “two-entity solution” in efforts to understand both classical physics and QM have brought about the acceptance of point-mass approximations that can no longer suffice to account for the precision achieved in modern atom-interferometry. In gravitation, the “divorce” of the core/kernel constituent from the wavefront constituent, or in any field with which the test object interacts—which we call the “dissociation of the entities”—is the actual deterministic mechanism of gravity that the standard metric description simply cannot discriminate.

While the centennial benchmarks of Einstein’s formalism documented by (Will, 2014) remain essential for the weak-field limit, the future of gravitational physics lies in the transition toward a fully mechanistical understanding that encapsulates both classical and quantal behavior. To this end, the QTG provides a clear path forward based on the UMA and through RMD. When the QTG’s abovementioned milestones are taken together with the recent quantum mechanical diffraction experiments that demonstrate the reciprocity of interference and particle localization at the atomistic level (Zhang et al., 2025), it can be clearly stated that the two-entity interpretation advanced with this contribution finds very strong observational support.

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Supplementary material

The Supplementary Material for this article can be found online at: https://www.frontiersin.org/articles/10.3389/frqst.2026.1780610/full#supplementary-material

Supplementary material Supplementary material at the source

Footnotes

Edited by: Peng Xu, Chinese Academy of Sciences (CAS), China

Reviewed by: Ophir Flomenbom, FLOMENBOM BPS, Israel

Najeh Rekik, King Faisal University, Saudi Arabia