Integrating Relativistic Quantum Mechanics, Relational Gravity and Cosmology

1. Introduction

1.1. Relativistic Quantum Mechanics, Quantum Gravity, and Cosmology

Resolving the tensions between special relativity (SR) and quantum mechanics (QM) and between general relativity (GR) and quantum gravity (QG) has proven so intractable that few theories comprehensively engage with them, and those that do often rely on complex mathematical constructs that conflict with the constraints of GR and SR.

Neither SR nor GR addresses QM, and later efforts to integrate GR and QG and SR and QM remain largely unsuccessful. Quantum frameworks, including Schrödinger’s wave mechanics, Heisenberg’s matrix mechanics, and various formulations of Feynman’s path integral rely on non-relativistic formulations, while most variations of the Copenhagen Interpretation [1], Bohmian mechanics [2], objective collapse theories (GRWf, GRWm, CSL) [3–5], MWI [6–8] and others [9–17] depend on the non-relativistic Schrödinger equation or Hilbert space representations.

Relativistic quantum field theories (QFT) typically assume a flat Minkowski spacetime, bypassing GR’s curved framework. Approaches to GR–QG, including Causal Dynamical Triangulations [18], Asymptotic Safety in Quantum Gravity [19], and the Holographic Principle [20], focus on mathematical constructs without addressing the SR–QM tension.

String Theory [21] embeds SR and GR into higher-dimensional frameworks using Hilbert and Fock spaces to describe quantum states and interactions. However, it fails to address foundational challenges such as causality, locality, and the ontological basis of 4D spacetime. In contrast, Loop Quantum Gravity (LQG) [22] primarily operates within a 4D spacetime framework, quantizing spacetime into discrete spin networks and spin foams. While more physically grounded, LQG struggles to reconcile probabilistic frameworks, physical observables, and the nature of time within 4D spacetime.

Sophisticated mathematical formulations across these theories often blur the line between physical ontology and abstract constructs. Semi-ontological approaches such as GRWf, GRWm, CSL, and Multi-Field Theories treat mathematical structures as ontic components, complicating their reconciliation with GR and SR. Frameworks like MWI lack clear mechanisms for reconciling quantum phenomena with GR and SR, while models based on Hilbert space, Fock space, matrix mechanics, or 3N configuration spaces often lead to unphysical conclusions. For example, while 4D spacetime alone cannot fully explain the dynamics of N-body quantum states, many models represent these states as evolving within non-physical, ultra-high-dimensional configuration spaces.

1.2. The DO Model

Rather than relying on mathematical constructs, the Dual Ontology model (“DO”) addresses the ontological incompatibility between quantum and classical theories. The framework integrates a discrete, background-independent 4D spacetime with a physically co-located Planck domain (the “Planck Domain”).1 Within this physical framework, each discrete spatial unit in 4D spacetime has an identical counterpart in the Planck Domain, enabling a one-to-one correspondence between the domains [23–24]. Mathematically, the Planck Domain can be expressed as a $(3\times N)$ space, where 3 represents the three dimensions of each discrete spatial unit in 4D spacetime and N represents the number of spatial units in the Planck Domain.

Taken together, the Planck Domain and 4D spacetime form a single, tightly integrated $\left(\right(3\times N)+3)$ physical structure that enables a unified, ontological, and dynamic model governed by a single set of physical laws applied consistently across scales and domains without ad hoc assumptions, fine-tuning, or perturbative techniques.23 Based on this unified structure, each quantum state is represented in 4D spacetime by an ontic energy field and in the Planck Domain by a single $(3\times N)$ point. A single energy-based operator governs both motion and gravitational response in 4D spacetime.

1.3. Theoretical Scope and Explanatory Depth

Theoretical attempts to unify one or two conflicting aspects of classical and quantum theory bear a heavy explanatory burden (see generally [25–27]). The burden is exponentially greater for theoretical models that address the SR–QM and GR–QG tensions, as well as the universe’s cosmogony and cosmology.

At a minimum, any theory that addresses the SR–QM tension in a relativistic context should include 1) quantum entanglement and nonlocality, 2) the instantaneity of quantum collapse, 3) causality, 4) the nature of time, 5) spacelike separation, 6) separability, 7) indeterminism, 8) quantum state emergence and annihilation, 9) quantum state localization, 10) unitarity, 11) quantum tunneling, 12) relativity of simultaneity, 13) total energy scaling, 14) quantum nonattenuation and quantum exclusivity, 15) the Born Rule’s relationship to SR, 16) the ontic representation of N-body quantum states in 4D spacetime, and 17) the quantum-classical divide.

Theories addressing GR–QG tensions should include 1) cosmological and black hole singularities, 2) regularization, 3) background independence, 4) the nature of time, 5) nonlocality, entanglement, and instantaneity, 6) gravity’s quantizability, 7) the equivalence principle, and 8) the black hole information paradox.

Finally, theories that address the universe’s cosmogony and cosmology should include 1) the emergence of 4D spacetime, 2) its near homogeneity and isotropy, 3) the horizon and causality problem, 4) the flatness problem, 5) gravitational entropy approaching zero, 6) the cosmological constant and dark energy problem, 7) the hierarchy problem, 8) global energy conservation, 9) the large scale structure of the universe and 10) the arrow of time.

1.4. DO Outline

Section 2 establishes the ontological framework, detailing a discrete 4D spacetime and its ontologically distinct Planck Domain. The foundation is essential for resolving the SR–QM tensions, explored in Sections 3–5. These sections reconcile the deterministic, relativistic evolution of quantum states in 4D spacetime with their instantaneous collapse in the Planck Domain. Resolving this tension underpins the model’s ability to address broader challenges, including GR–QG and relational gravity, the cosmological constant problem, the horizon problem, the hierarchy problem, and the quantum-classical divide.

The model’s ontological and dynamic foundations also provide the basis for exploring quantum path irreversibility in Section 6. Grounded in the asymmetry between relativistic quantum state evolution in 4D spacetime and instantaneous collapse in the Planck Domain, the analysis explains the physical basis for the unidirectional arrow of time. Building on these insights, Section 7 directly confronts the GR–QG tensions. Based on a discrete, background-independent 4D spacetime, the model demonstrates that gravity is a relational phenomenon, inherently non-quantizable, and provides an ontological basis for the equivalence principle as well as a theoretical resolution for the Black Hole Information Paradox.

Section 8 investigates the instantaneous transition of 4D spacetime at Heat Death to $t=0$, based on the same ontological structure and dynamics that govern quantum-state collapse in general. It re-interprets the horizon problem without resorting to inflationary, bouncing, or cyclic theories, identifies the ontological source of the cosmological constant, and resolves the hierarchy problem based upon the DO’s ontological and dynamic principles. Section 9 offers a Coda on the quantum-classical divide, Section 10 synthesizes the model’s contributions and outlines broader implications.

Section 11 provides a concise mathematical summary of the model’s core constructs. Appendix A presents the formal mathematical foundations of the DO and Appendix B provides a computationally viable proof of concept for five independent DO validations: (i) $t=0\to$ CMB (low-ℓ TT) mapping; (ii) CMB → first stars (Pop III → Pop II/I) evolution; (iii) spin-correlation / CHSH recovery; (iv) two-body orbital dynamics; and (v) three-body coupling. Appendix C outlines future tests and experiments.

2. The Ontological Framework of the DO Model

2.1. Discrete Spheres

Under the DO model, Discrete Spheres are three-dimensional units of space that form the discrete substructure of both 4D spacetime and the Planck Domain (see [28–32]). Each Discrete Sphere is structurally invariant, possesses an identical shape and volume, and represents the smallest structural quantum of space.4 Each is identified by a unique set of x, y, z coordinate identifiers, and N designates the number of Discrete Spheres that comprise the Planck Domain.

The dual presence of each Discrete Sphere in both 4D spacetime and the Planck Domain is referred to as the Planck Identity, which establishes a one-to-one identity and coordinate mapping between domains. The mapping, facilitated by the SOAN (Section 2.2), ensures that each Discrete Sphere occupies the same x, y, z coordinate identifiers in 4D spacetime and the Planck Domain, creating a single, integrated ontological framework.

2.2. The SOAN

The concept of nothingness has long perplexed philosophers and scientists. Greek and Roman thinkers struggled with the concept of zero and the void. Although these concepts no longer trouble most physicists, “nothing” is now used figuratively to mean “not anything” rather than an ontic state of nonexistence. For example, in GR, “nothing,” as in “not anything,” colloquially describes what 4D spacetime expands into after its inception at $t=0$.5 In LQG, “nothing” refers to the absence of further spatial degrees of freedom below a minimum geometric scale rather than an ontic void [33]. “Nothing” has also been used to denote the absence of space and time [34].

Nevertheless, the concept of an ontic SOAN, devoid of space and time, remains alien to theoretical physics, partly because a physical nothingness cannot be experimentally verified. Rather than discarding the concept outright, this analysis emphasizes its explanatory depth, positioning the SOAN as a necessary ontological element that bridges 4D spacetime and the Planck Domain. Under the DO model, the SOAN provides a physical explanation for the one-to-one correspondence between each Discrete Sphere in 4D spacetime and the Planck Domain, as well as for the dynamic evolution and instantaneous collapse of quantum states.6

The SOAN’s only defining attribute is onticness; it lacks all other physical properties.7 Since it excludes positive physical attributes, the SOAN is a passive ontological entity. It cannot be observed or measured, has no structure or boundaries, and is not governed by the laws of physics. It is a non-spatial, non-temporal bridge that links 4D spacetime to the Planck Domain. Consequently, explanatory depth, rather than experimental testing, is fundamental to verifying the SOAN’s ontological role in the DO model. From an explanatory depth perspective, the SOAN is fundamental to resolving tensions between GR and QG, and between SR and QM, and provides coherent explanations for long-standing cosmological issues.

2.3. Discrete 4D Spacetime

Unlike general relativity’s conception of 4D spacetime as a continuous, differentiable manifold, the DO framework posits that spacetime consists of Discrete Spheres arranged in a dynamic nearest-neighbor structure. This structure supports background independence and stepwise evolution governed by the Unified Evolution Equation (UEE).8 The nearest-neighbor structure imposes a discrete geometric constraint from which Lorentz invariance emerges only in the continuum limit.

2.4. The Planck Domain

The Planck Domain is the second core ontological structure of the DO model. Like discrete 4D spacetime, the Planck Domain is composed of Discrete Spheres and the SOAN. Mathematically, the Planck Domain is composed of N-tuples of ordered triples,

$$Q_N=\{(x_1,y_1,z_1),(x_2,y_2,z_2),\dots ,(x_N,y_N,z_N)\},$$

where each N-tuple represents the three $(x,y,z)$ spatial dimensions of a Discrete Sphere.

The Planck Domain is composed of $(3\times N)$ dimensions, where 3 represents the spatial dimensions of each Discrete Sphere, and N represents the number of Discrete Spheres [cf. 48]. The Planck Domain fundamentally differs from 4D spacetime [cf. 49]. It has no time dimension [50], physical properties of space, and no volume, and the laws of GR and SR, the strong nuclear force, the electro-weak force, or thermodynamics do not govern it. Moreover, unlike mathematical spaces composed of mutually orthogonal vectors, the Planck Domain integrates N Discrete Spheres into a single Planck Point. For example, the $5.58\times {10}^{186}$ Discrete Spheres that comprise the observable portion of 4D spacetime form a single Planck Point composed of $(3\times 5.58\times {10}^{186})$ dimensions.

2.5. The Tightly Integrated DO Model

The $(3\times N)$ Planck Domain and the three spatial dimensions of discrete 4D spacetime form the DO model’s single, tightly integrated $\left(\right(3\times N)+3)$ physical structure [51–52]. Although the $\left(\right(3\times N)+3)$ structure may seem complex, its core is simple. Based on an ontic SOAN, which serves as a non-spatial, non-temporal bridge between the two domains, the DO framework links 4D spacetime and the Planck Domain into a single physical structure, forming a unified framework that exists in the same location as 4D spacetime. More proverbially, 4D spacetime does not exist “here,” and the Planck Domain does not exist “there”; they co-exist in the same physical space.

The explicit bijection between discrete 4D spacetime and the Planck Domain, enforced by the Planck Identity, ensures a one-to-one identity and mapping of quantum states across both domains. Imagine, for example, that 4D spacetime consists of Discrete Spheres and that the SOAN exists within the interstices of these spheres. Assume there are five Discrete Spheres: one each on Venus, Mars, Jupiter, Sirius, and Polaris. In 4D spacetime, the Discrete Spheres are spatially separated, and each is represented by a set of x, y, z coordinate identifiers. However, since the SOAN, rather than space, exists in the interstices between Discrete Spheres, from the perspective of the Planck Domain, these five spheres are not separated by time, space, or volume. The five Discrete Spheres form a single, unified 15-dimensional point $(3\times 5)$ in the Planck Domain, where 3 represents the three spatial dimensions of each sphere, and 5 represents the five spheres.

2.6. The Ontological Reality of Quantum States

The significance of the Planck Identity’s one-to-one identity and coordinate mapping extends beyond the physical integration of 4D spacetime and the Planck Domain. First, the Planck Identity ensures that N-body quantum states, which cannot be fully described in 4D spacetime alone [54], are identified and mapped in both domains. Second, dynamic changes in the physical characteristics of quantum states as they evolve in 4D spacetime are mirrored in the Planck Domain, and physical changes caused by the collapse of a quantum state in the Planck Domain are mirrored in 4D spacetime.

3. The DO and the Dynamics of Quantum States

3.1. The Dynamic Evolution of Quantum States

The analysis begins with the dynamic evolution of a single quantum state in a discrete 4D spacetime composed of Discrete Spheres and the SOAN. As a quantum state evolves in 4D spacetime, its energy occupies Discrete Spheres. The combination of a single Discrete Sphere and the portion of the quantum state’s energy within that sphere is referred to as a Bell Sphere. Building on the Planck Identity, the Bell Identity establishes a one-to-one identity and mapping between a Bell Sphere in 4D spacetime and the Planck Domain.

In 4D spacetime, all of the Bell Spheres occupied by a quantum state constitute its Bell Field, while in the Planck Domain, the same Bell Spheres form the quantum state’s single Bell Point. The quantum energy component of a given Bell Field in 4D spacetime is referred to as the Bell Energy Field, and the quantum energy component of a Bell Point in the Planck Domain is referred to as the Bell Energy Point. For example, the $1.92\times {10}^{74}$ Bell Spheres that comprise an electron in the ground state of hydrogen simultaneously form the electron’s Bell Field in 4D spacetime and its single $(3\times 1.92\times {10}^{74})$-dimensional Bell Point in the Planck Domain. As a quantum state spreads in 4D spacetime, the number of Bell Spheres it occupies increases. The Bell Identity ensures a corresponding increase in the number of Bell Spheres comprising the quantum state’s Bell Point in the Planck Domain.

Notably, both the Bell Field in 4D spacetime and the Bell Point in the Planck Domain ontologically occupy the same physical space, ensuring that the Planck Domain is not a separate, abstract domain, but a tightly integrated part of a unified ontological framework.

3.2. The Collapse of a Single Quantum State

The Bell Identity also links the instantaneous collapse of a quantum state’s Bell Energy Point in the Planck Domain with the collapse of its Bell Energy Field in 4D spacetime. Following the collapse of a Bell Energy Point, the number of Bell Spheres that comprise the quantum state’s Bell Point is reduced. Simultaneously, the Bell Identity ensures that the decrease in the number of Bell Spheres comprising the quantum state’s new Bell Point is mirrored by a reduction in the number of Bell Spheres forming the quantum state’s new Bell Field in 4D spacetime. The Discrete Spheres do not collapse. For example, assume that quantum state A is placed within an impenetrable Box A with zero potential inside. Quantum state A forms Bell Field A in 4D spacetime and Bell Point A in the Planck Domain (Figure 1).

As quantum state A spreads, the Bell Identity ensures that the increase in the number of Bell Spheres comprising Bell Field A is mirrored by a corresponding increase in the number of Bell Spheres comprising Bell Point A. The opening of Box A triggers the instantaneous collapse of Bell Energy Point A, causing an instantaneous reduction in the number of Bell Spheres that comprise Bell Point A. The Bell Identity ensures that the reduction is mirrored by an identical reduction in the number of Bell Spheres that constitute Bell Field A in 4D spacetime. Quantum state A is instantaneously generally localized within Box A, but SR has not been violated.

3.2.1. The Einstein–De Broglie Boxes Thought Experiment

The Einstein–de Broglie thought experiment further illustrates the dynamic evolution of a quantum state [54–57]. Quantum state B is generated, forming Bell Field B in 4D spacetime and Bell Point B in the Planck Domain. The quantum state is inserted into Box B. As it spreads, it occupies an increasing number of Bell Spheres in 4D spacetime and the Planck Domain (Figure 2).

An impenetrable divider is inserted into Box B, creating Box C and Box D. The quantum state now forms two equal Bell Fields in 4D spacetime, Bell Field C and Bell Field D, and a single Bell Point in the Planck Domain, designated as Bell Point CD. Box C is sent to Princeton, and Box D is sent to Copenhagen (Figure 3).

Despite their separation, the Bell Identity ensures that the quantum state continuously forms Bell Fields C and D in 4D spacetime and Bell Point CD in the Planck Domain.

The opening of Box C or Box D triggers the collapse of Bell Energy Point CD, reducing the number of Bell Spheres that form the new Bell Point of the quantum state. The reduction is mirrored in 4D spacetime. If the quantum state is found in Box C, it forms a generally localized Bell Field C in Box C and Bell Point C in the Planck Domain, while Bell Field D and Bell Point D cease to exist. Conversely, if the quantum state is found in Box D, it forms a generally localized Bell Field D in Box D and Bell Point D in the Planck Domain, and Bell Field C and Bell Point C no longer exist. The process is the same, regardless of which box is opened first.

3.2.2. The Double-Slit Experiment

In the double-slit experiment, individual quantum states are directed at Wall (W), which has two narrow Gaussian slits (A) and (B). Due to the narrowness of the slits, every quantum state that passes through slit (A) or slit (B) diffracts, spreading as spherical Bell Fields toward Detector D (Figure 4).

As a quantum state diffracts through slits (A) and (B), its Bell Field splits into two separate fields in 4D spacetime: Bell Field A and Bell Field B.9 In the Planck Domain, the fields remain unified as a single Bell Point AB. A detection flash at Detector D indicates that Bell Energy Point AB has collapsed. The collapse reduces the number of Bell Spheres comprising the quantum state’s Bell Point in the Planck Domain. The Bell Identity ensures that the reduction is mirrored in 4D spacetime, localizing the quantum state’s Bell Energy Field to one of the diffracted paths. Following the quantum collapse of Bell Energy Point AB, the Bell Energy Field in 4D spacetime localizes to either Bell Field A or B; the other branch’s Bell Energy Field ceases to exist. The interference pattern observed on Detector D arises from cumulative quantum collapses, reflecting the probabilistic outcomes of individual quantum collapses.

3.2.3. A Which-Way Experiment

Which-way experiments compound the theoretical complexities of the double-slit experiment. The following which-way experiment has been modified by including a proton in an empty box at the center of Wall (W) (Figure 5).10 The proton is positively charged, and each electron fired toward Wall (W) is negatively charged. Slit (A) flashes if the proton is attracted toward slit (A) and slit (B) flashes if it is attracted toward slit (B).

Under the DO model, as the quantum state spreads in 4D spacetime, it continuously forms a Bell Field in 4D spacetime and simultaneously forms a single Bell Point in the Planck Domain. If slit (A) flashes, the Bell Energy Point instantly collapses, reducing the number of Bell Spheres that make up its new Bell Point; the Bell Identity mirrors this as a reduced Bell Field. The quantum state is localized at slit (A). The analysis remains the same whether slit (B) flashes or slit (A).

Once the quantum state is generally localized at either slit (A) or slit (B), it again spreads toward Detector (D). However, because the quantum state collapses at either slit (A) or slit (B), but not both, no interference pattern forms at Detector (D).

3.3. N-Body Quantum States and the Bohm-EPR Thought Experiment

The Bohm version of the EPR experiment highlights issues related to the dynamic evolution of an N-body quantum state in 4D spacetime and its collapse in the Planck Domain. A pair of electrons is prepared in the singlet state. The singlet state forms Bell Fields E and F in 4D spacetime and a single Bell Point EF in the Planck Domain. Quantum state E is sent to Princeton, and quantum state F is sent to Copenhagen (Figure 6). Testing equipment is configured to conduct a z-axis Stern–Gerlach experiment on either Bell Field E or F.

As Bell Fields E and F spread dynamically in 4D spacetime, the number of Bell Spheres that comprise their respective Bell Fields increases, as does the number of Bell Spheres that comprise Bell Point EF. The Stern–Gerlach experiment, conducted along the z-axis in either Princeton or Copenhagen, triggers the collapse of Bell Energy Point EF. The collapse instantaneously reduces the number of Bell Spheres that formerly composed Bell Point EF, and the reduction is mirrored by Bell Field E and Bell Field F, respectively. Bell Point EF forms two independent Bell Points designated as Bell Point E and Bell Point F. Bell Point E shares a one-to-one mapping and identity with Bell Field E, and Bell Point F shares a one-to-one identity and mapping with Bell Field F.

Following the instantaneous collapse, Bell Point E and Bell Point F form a product state rather than an entangled state. The single state descriptor, ${\mathsf{\Phi }}_{EF}$, transitions to a product state composed of two independent state descriptors, ${\mathsf{\Phi }}_E$ and ${\mathsf{\Phi }}_F$. This physical change is represented as ${\mathsf{\Phi }}_{EF}\to {\mathsf{\Phi }}_E\otimes {\mathsf{\Phi }}_F$, and the corresponding Bell Energy Fields E and F are generally localized. Whether quantum state E is found along the z spin-up or z spin-down axis, quantum state F’s spin is the opposite. SR has not been violated.

4. Physical Implications of the DO Model

4.1. Indeterminacy

Indeterminacy typically means that a quantum system has a determinable property without a specific determinate value [60, pp. 72–107]. In a singlet state along the z-axis, spin is a determinable property, with z spin-up and z spin-down as determinate values. While the conventional quantum formalism represents this entangled state abstractly as the state vector $\psi ={\displaystyle \frac{1}{\sqrt{2}}}{(\uparrow }_{z_1}{\downarrow }_{z_2}{-\downarrow }_{z_1}{\uparrow }_{z_2})$, the DO model provides a direct ontological description. The two quantum states, $z_1$ and $z_2$, form a single, unified system described by the state descriptor ${\mathsf{\Phi }}_{z_1{z}_2}$, with no determinate z-spin for either component prior to collapse; a singlet anti-correlation holds. The Bell Identity ensures that this system forms a single Bell Point in the Planck Domain and two corresponding Bell Fields in 4D spacetime.

After instantaneous collapse, the system transitions to a product state and the spins of $z_1$ and $z_2$ become determinate; for collapse along the z axis, the outcomes are opposite: one z-up, the other z-down, and which component is up is outcome-contingent. This physical transition from a unified state to two independent states is represented symbolically as ${\mathsf{\Phi }}_{z_1{z}_2}\to {\mathsf{\Phi }}_{z_1}\otimes {\mathsf{\Phi }}_{z_2}$. Each quantum state now forms its own Bell Point associated with its respective localized Bell Field in 4D spacetime.

4.2. Quantum State Emergence and Annihilation

Quantum state emergence and annihilation challenge the applicability of the non-relativistic Schrödinger equation, which is formulated for systems with a fixed number of quantum states and does not account for processes involving the creation or annihilation of quantum states [10]. Relativistic quantum field theory (QFT) addresses these variations, but the DO model offers a unique solution, representing quantum states as physical entities in both 4D spacetime and the Planck Domain.

Under the DO framework, the Bell Identity links the Bell Spheres that comprise a quantum state’s Bell Field in 4D spacetime and its Bell Point in the Planck Domain. During quantum annihilation, the collapse of a quantum state’s Bell Energy Point transfers the observables of the quantum state to another system, eliminating the Bell Energy Point and Bell Energy Field of the original state. During quantum emergence, a quantum state forms a new Bell Field in 4D spacetime and a corresponding Bell Point in the Planck Domain.

4.3. Physical Triggers

In the DO model, Physical Interactions in 4D spacetime are caused by one or more of the three traditionally labeled Fundamental Forces: electromagnetism, the strong nuclear force, and the weak nuclear force. Based on the Bell Identity, a Physical Interaction in 4D spacetime is mirrored in the Planck Domain, where the quantum state’s Bell Energy Point collapses instantaneously.

The DO model does not identify the precise Physical Interaction that induces quantum state collapse. Nevertheless, it provides a structured framework for examining how physical triggers may induce collapse. Within the DO framework, each Physical Interaction is a localized event in time and space, with its frequency influenced by factors such as temperature and spatial positioning (see generally [61]). Local temperature and position within the Sun affect the rate of quantum state collapse. Humans can initiate or influence the timing and location of Physical Interactions. For example, a scanning tunneling microscope can be used to control and precisely vary the rate of electron collapse. However, Physical Interactions are independent of human consciousness, ambient noise, and universal processes,but see [62,63].

4.4. Quantum State Localization

The Bell Identity links the collapse of a quantum state’s Bell Energy Point in the Planck Domain to a simultaneous reduction in the number of Bell Spheres that comprise its Bell Field in 4D spacetime. Because the reduction must be to a contiguous subset of the Bell Spheres that compose the quantum state prior to collapse, the Bell Identity places a strict boundary on the collapse outcome. Following the collapse of a Bell Energy Point, a quantum state’s new Bell Field cannot be generally localized anywhere in 4D spacetime; it must be generally localized to a contiguous subset of its Bell Spheres prior to collapse, yielding a discrete spatial configuration consistent with observed quantum measurements.

The Bell Identity does not set a specific size for a quantum state’s Bell Field in 4D spacetime following collapse. The size of the Bell Field may be related to the physical trigger that initiated the collapse, or it may vary depending on the quantum state’s physical composition. Additionally, high- or low-energy collapses may exhibit different localization characteristics, and a quantum state’s momentum in 4D spacetime may also influence its localization.

4.5. Time and Instantaneous Collapse

Neither the Planck Domain nor 4D spacetime supports the concept of instantaneous collapse independently. The Planck Domain lacks a time dimension and, aside from collapse, does not support dynamic movement. In contrast, 4D spacetime has dynamic movement and a time dimension constrained by SR.

When a Physical Interaction in 4D spacetime occurs, the trigger is mirrored in the Planck Domain, causing the quantum state’s Bell Energy Point to collapse instantaneously. The Bell Identity ensures that the collapse is mirrored by a reduction in the number of Bell Spheres that comprise the quantum state’s Bell Field in 4D spacetime. Because the collapse of a Bell Energy Point is instantaneous, it is mirrored in the three spatial dimensions of 4D spacetime, with no movement along the time dimension.

4.6. Quantum Tunneling

Although commonly described as “quantum tunneling,” the appearance of a quantum state on the opposite side of a classically impenetrable barrier does not involve quantum tunneling in 4D spacetime (for a general overview, see [64]). In many traditional interpretations, the probability of a quantum state appearing on the other side of an impenetrable barrier is based on the Schrödinger equation and the exponential decay of the quantum state’s Bell Field within the barrier.

Under the DO model, the event is not tunneling but rather the instantaneous collapse of the quantum state’s Bell Energy Point in the Planck Domain. When a quantum state’s Bell Energy Point undergoes instantaneous reduction, the Bell Identity ensures a corresponding reduction in the number of Bell Spheres that constitute the quantum state’s new Bell Energy Field in 4D spacetime. Although the quantum state is localized on the other side of the barrier, it does not “tunnel” through, and SR is not violated.

4.7. The Born Rule Revisited

The DO model diverges fundamentally from the Born Rule and its mathematical interpretation of wave-function collapse as a probability density for continuous variables. In the DO framework, a quantum state’s presence in 4D spacetime is characterized by its Bell Energy Field. The field represents the physical distribution of a quantum state’s total intrinsic energy across the discrete Bell Spheres it occupies. Since the Bell Identity dictates that collapse must occur within a discrete subset of these Bell Spheres, the DO model posits that the likelihood of the quantum state becoming generally localized within any specific sub-region of its pre-collapse Bell Energy Field is directly proportional to the quantum state’s energy residing in that sub-region (cf. [65]). This approach grounds probability in the tangible distribution of the state’s physical energy, rather than in abstract mathematical amplitudes.

For example, in the case of quantum tunneling, the probability of the quantum state localizing on the far side of a classically impenetrable barrier corresponds to the proportion of its total energy distributed in its Bell Spheres beyond that barrier before collapse.11 Such an outcome is a discrete probability event, not the result of integrating a continuous probability density. The Bell Identity underpins the unitarity of this process and resolves issues such as the “four tails problem” [66] but leaves open whether the collapse of a quantum state is fully deterministic [67].

5. Resolving the Tension Between SR and Quantum Mechanics

The apparent incompatibility between SR and QM is often framed in terms and concepts derived from 4D spacetime. Despite their usefulness, common terms such as spacelike separated, non-separability, entanglement, instantaneous, local, non-local, and complex concepts such as the relativity of simultaneity and total energy scaling have unintentionally magnified a theoretical and experimental conflict that does not exist.

5.1. Spacelike Separated

The term spacelike separated is based on a 4D spacetime structure with three spatial dimensions and one temporal dimension. The term is directly related to the concepts of space and time, the theory of SR, and the spatial distance between two or more events outside of one another’s light cones. Nevertheless, the term loses meaning in relation to a Bell Point where time, space, and volume do not exist.

5.2. Non-Separability

Einstein was among the first to raise concerns regarding separability in theoretical physics. His primary concern related to two assumptions underlying his argument for incompleteness: that spatially separated systems are ontic states, and that physical effects in spacelike-separated systems cannot propagate faster than the speed of light.1213

In the DO model of 4D spacetime, a singlet state along the z-axis is described by the state descriptor ${\mathsf{\Phi }}_{z_1{z}_2}$. This system is non-separable, representing a real physical entity rather than an abstract mathematical concept. A non-separable DO singlet state has three key attributes: 1) the spatial separation of the $z_1$ and $z_2$ Bell Fields, 2) their temporal separation, and 3) the existence of a single system, whose corresponding single Bell Energy Point ontologically enforces non-separability.14

In the Planck Domain, where time, space, and volume do not exist, a Bell Point exists as a single, non-separable entity (but see [42, 72]). The Bell Energy Point is mirrored via the Bell Identity to the Bell Spheres that form the quantum state’s Bell Energy Field(s) in 4D spacetime. The non-separability of a Bell Energy Point in the Planck Domain does not violate SR.

5.3. Instantaneous, Superluminal, and Faster than Light

In QM, the terms instantaneous, superluminal, and faster than light often describe the collapse of a quantum state in 4D spacetime. Following quantum state collapse, these terms are used to describe the quantum state’s role in 1) communication, 2) signaling or the absence of signaling, 3) information transmission, and 4) matter and energy transfer.

However, under the DO model, terms such as instantaneous describe the physical collapse of a Bell Energy Point in the Planck Domain rather than a collapse in 4D spacetime. Following the collapse, the reduction in the number of Bell Spheres that comprise a quantum state’s new Bell Point in the Planck Domain is mirrored by a reduction in the Bell Spheres that comprise the quantum state’s generally localized Bell Field in 4D spacetime. The process is instantaneous, but SR is not violated cf. [71].

5.4. The Quantum Connection

In 4D spacetime, quantum discrimination refers to a quantum state’s ability to maintain an exclusive connection, excluding all other quantum states, and unattenuated denotes the strength (or lack of attenuation) of a quantum state’s connection [69], pp. 21–22. The terms are typically used to denote the connection between spacelike-separated entangled states. Discrimination and non-attenuation also imply an instantaneous, continuous connection that violates the maximum speed of light.

In the DO model, the Bell Identity ensures that all dynamic changes to Bell Spheres are mirrored in both 4D spacetime and the Planck Domain. The mirroring process ensures quantum discrimination and non-attenuation without violating SR.

5.5. Bell’s Theorem

Bell’s inequality theorem asserts that relativistic local-causation theories cannot account for the statistical predictions of quantum mechanics in spin experiments involving entangled singlet states [73–75]. More broadly, Bell’s theorem indicates that any theory conforming to quantum experimental results cannot be local [75–77].

The DO model locates Bell Energy Point collapse in the Planck Domain, where time and space do not apply. The Bell Identity ensures that the physical collapse of a quantum state is mirrored in 4D spacetime without violating relativity. Although collapse is extra-spatio-temporal, in 4D spacetime, the physical process of collapse nevertheless violates Bell’s local-causality condition and reproduces Bell-type correlations.15

5.6. The Relativity of Simultaneity

The SR–QM tension extends to the relativity of simultaneity. SR holds that 1) all inertial reference frames (frames moving at a constant speed relative to one another) are equally valid, and 2) the speed of light in a vacuum is invariant for all observers in these frames. Consequently, the relativity of simultaneity implies that a) whether two spatially separated events occur simultaneously depends on the observer’s frame of reference, and b) observers in different frames may conclude that the same event happened at different times.

For spacelike-separated electrons in the singlet state along the z-axis, ${\mathsf{\Phi }}_{z_1{z}_2}$, the collapse of the $z_1$ electron causes the simultaneous collapse of the $z_2$ electron. Because the relativity of simultaneity suggests that the order of cause (collapse of $z_1$) and effect (collapse of $z_2$) depends on the observer’s frame of reference, simultaneous collapse appears to challenge SR, implying a violation of Lorentz Invariance and a preferred frame [71], p. 185.

The DO model resolves the issue by treating the collapse of the system ${\mathsf{\Phi }}_{z_1{z}_2}$ as an event beyond 4D spacetime. For a singlet state along the z-axis, it is irrelevant whether $z_1$ or $z_2$ is measured first or whether they are spacelike separated. The Bell Identity ensures that an experiment on either quantum state in 4D spacetime is simultaneously conducted on the quantum state’s single Bell Point in the Planck Domain. Moreover, the identity reflects the instantaneous collapse of the Bell Energy Point as a reduction in the Bell Spheres comprising the now generally localized Bell Fields of both $z_1$ and $z_2$ in 4D spacetime. The formerly entangled quantum state becomes a product state, and although the collapse is instantaneous, SR remains intact.

5.7. Total Energy Scaling

The instantaneous nature of quantum state collapse appears to challenge Einstein’s theory of total energy and momentum in SR [71]. The theory posits that the total energy of a body moving relative to an observer increases as its velocity accelerates. As an object approaches the speed of light, its relativistic kinetic energy theoretically approaches infinity, although SR limits its speed. In quantum mechanics, momentum is typically used instead of velocity. Accordingly, as the momentum of a quantum state increases, so does its associated total energy. If collapse were interpreted as a process in which a body in 4D spacetime actually reaches or exceeds the speed of light, the total energy required would be infinite.

While the collapse of a quantum state’s Bell Energy Point is instantaneous, it is a physical event external to 4D spacetime. The DO model and the Bell Identity ensure that the instantaneous collapse results in a reduction in the number of Bell Spheres comprising the quantum state’s Bell Point and Bell Field(s). Consequently, the reduction in Bell Spheres in the Bell Field is also instantaneous. However, the process does not result in an increase in the quantum state’s energy in either 4D spacetime or the Planck Domain.

6. Quantum Path Irreversibility and the Arrow of Time

The Bohm version of the EPR experiment (see Figure 6) demonstrates why quantum path reversibility following a collapse is impossible [78], pp. 150–162, [79,80] cf. [81]. Assume that two quantum states, $z_1$ and $z_2$, are entangled in the singlet state along the z-direction, forming a single, unified system described by ${\mathsf{\Phi }}_{z_1{z}_2}$. The Bell Field of $z_1$ is on Mars, and the Bell Field of $z_2$ is on Earth. The entangled system is separated by 225 million km, and its spin is indeterminate. In the Planck Domain, this single system ${\mathsf{\Phi }}_{z_1{z}_2}$ corresponds to a single Bell Point $z_1{z}_2$.

Following the instantaneous collapse of Bell Energy Point $z_1{z}_2$, the system transitions to a product state represented as ${\mathsf{\Phi }}_{z_1{z}_2}\to {\mathsf{\Phi }}_{z_1}\otimes {\mathsf{\Phi }}_{z_2}$. The Bell Identity ensures that the reduction in the number of Bell Spheres comprising Bell Point $z_1{z}_2$ is linked to the simultaneous reduction in Bell Spheres that comprise Bell Fields $z_1$ and $z_2$ in 4D spacetime. Bell Field $z_1$ is generally localized on Mars, and Bell Field $z_2$ is generally localized on Earth. The $z_1$ and $z_2$ quantum states are now separable, each forming its own distinct Bell Point (Bell Point $z_1$ and Bell Point $z_2$). If $z_1$ is spin-up, $z_2$ is spin-down, and vice versa. The spin of the respective quantum states is now determinate.

Before quantum state collapse, 1) $z_1$ is on Mars, $z_2$ is on Earth, and $z_1$ and $z_2$ are separated by 225 million km; 2) the singlet state forms Bell Point $z_1{z}_2$ in the Planck Domain; and 3) the singlet state is non-separable, and its spin is indeterminate. Following the instantaneous collapse of Bell Energy Point $z_1{z}_2$, 1) $z_1$ is generally localized on Mars, $z_2$ is generally localized on Earth, and $z_1$ and $z_2$ remain separated by 225 million km; 2) the formerly entangled singlet state is now a product state; 3) $z_1$ forms Bell Field $z_1$ on Mars and Bell Point $z_1$ in the Planck Domain; 4) $z_2$ forms Bell Field $z_2$ on Earth and Bell Point $z_2$ in the Planck Domain; 5) the product state is now separable; and 6) the spins of quantum states $z_1$ and $z_2$ are determinate even if they are unknown.

Reversing the path would require retracing the collapse, but the unified Bell Point $z_1{z}_2$ no longer exists; therefore, path reversibility is impossible. $z_1$ now forms Bell Field $z_1$ on Mars and Bell Point $z_1$ in the Planck Domain, and $z_2$ now forms Bell Field $z_2$ on Earth and Bell Point $z_2$ in the Planck Domain. Because Bell Field $z_1$ and Bell Field $z_2$ are now generally located on Mars and Earth, each quantum state must travel at least 112.5 million km before it can become entangled again. Even if there is an infinitely small chance that the $z_1$ and $z_2$ quantum states spread and once again form Bell Point $z_1{z}_2$, when the singlet state collapses, path reversibility becomes impossible.

The DO’s ontological structure, the asymmetric laws governing the dynamic motion of quantum states in 4D spacetime, and their collapse in the Planck Domain provide the physical basis for the arrow of time in 4D spacetime. Without an instantaneously reversible path, the arrow of time following collapse for all quantum states, and those of the proverbial egg, can only move in a single temporal and spatial direction.16

7. GR, Relational Gravity, Background Independence, and Black Holes

7.1. Discretization and Singularities

The DO model replaces GR’s assumption of a continuous, differential 4D spacetime manifold with a discrete 4D spacetime composed of Discrete Spheres. Critically, in the context of GR–QG, the physical discretization of 4D spacetime resolves several of the mathematical difficulties encountered by the Einstein Field Equation (EFE), including cosmological singularities, black hole singularities, and regularization.17

7.1.1. Cosmological and Black Hole Singularities

Specific mathematical solutions to EFE, including the FLRW model, predict a cosmological singularity at $t=0$, where the density, pressure, and energy density become infinite. Similarly, the Schwarzschild solution to the EFE, based on a spherically symmetric, uncharged, and non-rotating mass, mathematically defines the conditions under which a curvature singularity forms at the center of black holes. In both cases, the EFE’s lack of a physical mechanism to impose a cut-off at a minimum volume mathematically leads to infinite energy, mass, and pressure [82].

Notwithstanding the significant differences between cosmological and black hole singularities, discretization based on the minimum discrete size of Discrete Spheres provides a minimum volume cutoff that resolves the infinities in both cases. The discretization sets limits on the maximum possible frequencies and wavelengths for energy, creating a physical upper bound on energy density.18

7.2. Regularization

In QFT, zero-point energies from quantum fluctuations are formally summed over all modes, producing vacuum energy densities that diverge at high energies. Regularization introduces a mathematical cutoff to control these infinities, and renormalization redefines physical parameters to remove them. Even after this procedure, the predicted value can exceed the observed energy density by as much as ${10}^{120}$. In continuum approaches, these enormous contributions are often absorbed into a constant “vacuum term” that is adjusted so the remaining, effective energy density matches observations. In the DO model, Discrete Spheres function as a physical cutoff for ultra-high-energy modes, preventing ultraviolet energy from diverging cf. [83]. With no infinities to absorb, and with the large-scale constant that plays the role of $\mathsf{\Lambda }$ fixed rather than tuned (see Section 8), there is no vacuum term to adjust, and renormalization is unnecessary. Discreteness thereby eliminates both the divergence and its underlying cause, yielding physical energy densities far below the mathematical prediction of the zero-point energy summation.

7.3. Time

In QFT’s Hamiltonian and path integral formalisms (and the explicitly non-relativistic Schrödinger equation), time typically remains an external, fixed parameter. Most QFT theories assume a flat Minkowski spacetime, with time as an absolute Newtonian construct. In contrast, GR treats time as a dimension within 4D spacetime, where the rate at which time passes depends on local mass, energy, and pressure, as reflected in spacetime’s curvature. Time dilation, gravitational redshift, and lensing follow from this relational view of time.

The DO model’s approach to time diverges from both perspectives by integrating the Planck Domain, which lacks a time dimension. Except for quantum collapse, the Planck Domain has no independent dynamic movement. Single and N-body quantum states evolve entirely within 4D spacetime, under a discrete dynamics whose large-scale behavior is consistent with SR and GR.19 As a result, quantum state dynamics respect SR constraints, including the maximum speed of light, time dilation, and the relativistic increase in energy, with gravity expressed as curvature in a discrete 4D spacetime. While Discrete Spheres set the smallest unit of time at approximately the Planck time, time remains a dynamic concept shaped by GR and SR rather than QM.

7.4. Relational Gravity

Attempts to quantize gravity with continuum-only, background-metric programs have thus far failed [84]. The EFE describe a local, relativistic interaction compatible with SR, but quantum collapse is typically regarded as a non-local phenomenon with instantaneous changes to a quantum state’s wave function [85]. Hypothetical gravitons are problematic if they must simultaneously adhere to SR constraints and mediate instantaneous changes in location and momentum.20 Additionally, because standard QM holds that the position of a quantum state is undefined before collapse, gravitational coupling is ambiguous. In the specific case of an N-body quantum state, it is unclear whether gravity couples to the aggregate state as a whole or to each constituent quantum state.

Instantaneous collapse also appears to conflict with the relativity of simultaneity and with the limits of total energy and momentum in SR.21 Many QM interpretations represent the collapse with a Dirac delta function $\delta (x-x_0)$, implying infinite localization, infinite momentum uncertainty, and potential black hole formation [85].

Under the DO model, however, all quantum states evolve in a discrete, 4D spacetime within which large-scale behavior is compatible with SR. Bell Energy Points collapse instantly in the Planck Domain, where the laws of 4D spacetime do not apply. The collapse of a Bell Energy Point localizes the quantum state in 4D spacetime rather than collapsing to a delta function, ensuring finite localization and avoiding the issues raised by $\delta (x-x_0)$. Because the Bell Identity mirrors a quantum state’s Bell Spheres across both domains, changes in energy and sphere count are reflected instantaneously by a quantum state’s Bell Energy Point and Bell Energy Field(s). Consequently, the Bell Energy Fields of quantum states in 4D spacetime and their Bell Energy Points in the Planck Domain both carry the information (spin, momentum, and position) needed to reproduce the observed quantum behaviour associated with the strong nuclear force, electromagnetism, and the weak force.22

Unlike these forces, gravity is not encoded in the quantum states of Bell Energy Points, nor does it exist in the Planck Domain. Gravity is a 4D spacetime phenomenon, arising from large-scale relational configurations of Bell Spheres.2324 The interaction of Bell Spheres, governed by the UEE, ensures that spacetime geometry is determined by the collective relational configurations rather than by the intrinsic properties of quantum states. Gravity does not arise from intrinsic Bell Sphere properties; it is purely relational among Bell Spheres in 4D spacetime and is governed by the UEE’s coupling of energy to curvature.

Accordingly, gravitational effects depend only on the Bell-Sphere energy distribution in 4D spacetime, not on quantum state observables in 4D spacetime or the Planck Domain. While quantum observables depend on a quantum state’s intrinsic properties, gravity is governed by localized relational structures in 4D spacetime. Under the DO approach, gravity cannot be quantized as an independent field, because its effects arise from relational configurations rather than from intrinsic quantum properties at the Bell Energy Point.

7.5. Background Independence

In GR, spacetime geometry is dynamic, but it is still modeled on a continuous manifold. Most quantum field theories, on the other hand, assume a fixed background geometry, such as flat Minkowski space. This fixed framework imposes a separation between geometry and the distribution of matter and energy, requiring the geometry to be specified in advance.

The DO model is background-independent from the outset [86]. From $t=0$, discrete 4D spacetime exists and is formed from the relational configuration of Discrete Spheres, with no external metric or coordinate structure imposed from outside the model. Its geometry is determined internally by the system’s physical state, and because geometry and energy distribution co-evolve, there is no fixed metric structure, and no external background is needed to define motion or curvature.

7.6. The Black Hole Information Paradox

The black hole paradox is based on the apparent conflict between two principles: 1) the QM principle that information is never lost, and 2) Stephen Hawking’s 1975 semi-classical premise that information falling into a black hole is eventually lost through Hawking radiation. The QM principle is closely related to the unitarity and reversibility of quantum processes. Hawking’s key premise is that a black hole loses mass and energy by emitting thermal radiation at its event horizon. Because the quantum state of matter falling into the black hole is inaccessible beyond the event horizon, and thermal radiation is presumed to carry no specific information, the process implies that the information is permanently lost.25

Under the DO framework, when a quantum state’s Bell Field(s) falls into a black hole, the quantum state and its information are no longer accessible in 4D spacetime. However, as long as the quantum state does not collapse, it is still represented by its Bell Field(s) within the black hole and by its single Bell Point in the Planck Domain. As the quantum state dynamically evolves within the black hole, any changes to the content or number of Bell Spheres in the Bell Field are instantaneously mirrored by its Bell Point.

The transition to thermal radiation, triggered by a Physical Interaction inside the black hole, causes the instantaneous collapse of the quantum state’s Bell Energy Point. In 4D spacetime, the collapse causes the quantum state’s transition to radiation via the Hawking process. Although the collapse is irreversible, the Bell Identity preserves the quantum state’s conserved quantities across the transition to Hawking radiation. Detailed quantum state information is not preserved after collapse, but its conserved quantities persist, so no fundamental information loss occurs.

8. Quantum Cosmology, the Cosmological Constant, and the Hierarchy Problem

The DO model departs significantly from current cosmological theories of 4D spacetime (see, generally, [88]). However, ad hoc assumptions, fine-tuning, or perturbative techniques are not required to explain the instantaneous collapse process that transforms a widely dispersed 4D spacetime at or near Heat Death to a generally localized 4D spacetime at $t=0$. The collapse explains 4D spacetime’s extreme, but not infinite, energy, pressure, and temperature at $t=0$, its nearly isotropic and homogeneous state, and its extremely low gravitational entropy. The framework also provides a unified physical account of the horizon and flatness problems while clarifying the origin of $\mathsf{\Lambda }$ independently of Dark Energy (see [89]).26

8.1. The FLRW Model of 4D Spacetime at Heat Death

4D spacetime’s status as open, closed, or flat is based on the FLRW derivation of the EFE. Experimental data currently suggests that $k=0$, indicating that 4D spacetime is flat or very nearly flat. In turn, flatness implies that 4D spacetime’s total energy density equals the critical density $(\rho ={\rho }_c)$. Based on the FLRW model and current datasets, a spatially flat 4D spacetime at Heat Death approaches a state of near-maximal entropy; as time passes, the total energy density tends toward ${\rho }_{\mathsf{\Lambda }}$, and the temperature asymptotically approaches zero.

At Heat Death, 4D spacetime’s spatial geometry, energy density, pressure, and temperature are very nearly homogeneous and isotropic, and, as a result, 4D spacetime is very close to thermodynamic equilibrium. 4D spacetime has no large-scale structures, is extremely widely dispersed, and its spatial curvature is flat or nearly flat. Even though the negative pressure of $\mathsf{\Lambda }$ dominates, 4D spacetime’s energy density equals the critical density $\rho ={\rho }_c$. Because little heat can flow near a thermal equilibrium approaching zero, little work or gravitational clumping can occur. At the macrostate level, no additional physical changes occur without work, and in the absence of work, 4D spacetime is in a state of near-maximal gravitational entropy.27

8.2. The FLRW Model of 4D Spacetime at $t=0$

Mathematical attempts to describe 4D spacetime at $t=0$ support different conclusions regarding its physical status. Under specific conditions, the EFE and Friedmann equations describe a singularity at $t=0$ caused by the divergence of the energy density, pressure, and spacetime curvature. Specific modifications to the Friedmann equations allow the FLRW model to describe a homogeneous and isotropic 4D spacetime at very large scales, where the behavior of matter, radiation, and the cosmological constant governs its dynamics.

The $\mathsf{\Lambda }$CDM model, based on the FLRW metric, does not describe 4D spacetime’s physical status at $t=0$. Nevertheless, based on a continuous, differentiable 4D spacetime framework, the $\mathsf{\Lambda }$CDM model, supported by experimental data from the CMB and other datasets,28 indirectly indicates that approximately 13.8 billion years ago, immediately after $t=0$, 4D spacetime was in a hot, dense state characterized by extreme energy densities, pressures, and temperatures [90–91]. Approximately 380,000 years after $t=0$, the temperature anisotropies of the CMB across the sky varied by approximately 1 part in ${10}^5$. The temperature variations indirectly suggest that, very near $t=0$, 4D spacetime’s energy density and pressure were nearly isotropic and homogeneous, with very small anisotropies and inhomogeneities. When the angular power spectrum around the first peak of the anisotropies is extrapolated backward to $t=0$, the CMB and related data indirectly support a nearly flat 4D curvature [92].

8.3. The DO Model

8.3.1. 4D Spacetime

Under the DO framework, discretization resolves the infinities created by the EFE’s continuous differentiable 4D spacetime manifold by creating an upper bound on the maximum possible frequencies and wavelengths for energy. Consequently, under the model, at $t=0$, 4D spacetime is characterized by a discrete, generally localized 4D spacetime with extreme, rather than infinite, energy densities, pressures, and temperatures.

Although the metric expansion of 4D spacetime, described by the scale factor $a\left(t\right)$, colloquially represents an “internal stretching” of 4D spacetime rather than a stretching into nothing, the DO model posits that 4D spacetime expands into a pre-existing substructure comprised of Discrete Spheres and the SOAN.

As we will see in Sections 8.4 and 8.7 below, this distinction provides a physical basis for a uniform cosmological constant $\left(\mathsf{\Lambda }\right)$ derived from the inherent energy density per unit volume of evenly distributed, pre-existing Discrete Spheres ${\rho }_{\mathrm{DS}}={\rho }_{\mathsf{\Lambda }}$. Moreover, the existence of a pre-existing substructure space represented by Discrete Spheres 1) explains the physical source of $\mathsf{\Lambda }$, 2) resolves the increasing total energy budget of 4D spacetime as it expands over time (cf. [93] and [3]) and 3) obviates the need for Dark Energy based on a vacuum energy density that exceeds the observed $\mathsf{\Lambda }$ by a factor of ${10}^{120}$.

8.3.2. The Planck Domain

In contrast to 4D spacetime, the Planck Domain does not have a time dimension, lacks the physical properties of space and volume, and the physical laws of 4D spacetime do not apply to it. Aside from a collapse mechanism, the Planck Domain does not support dynamic movement. Nevertheless, the Bell Identity ensures that the physical characteristics of the individual Bell Spheres occupied by quantum states in 4D spacetime, including the energy densities, pressures, and temperatures of individual Bell Spheres from $t=0$ to Heat Death, are continuously mirrored in the Planck Domain. Critically, because the Bell Spheres that comprise 4D spacetime’s energy density, pressure, and temperature at Heat Death are very nearly homogeneous and isotropic, the Bell Spheres that comprise the Planck Domain are as well.

In addition, 4D spacetime’s total energy density per Discrete Sphere $\left(\rho \right)$ at any instant in time, including $t=0$ and Heat Death, equals the Planck Domain’s total energy density per Discrete Sphere $\left({\rho }_{\mathrm{pd}}\right)$$(\rho ={\rho }_{\mathrm{pd}})$ for the Planck Domain subset that mirrors 4D spacetime. Because 4D spacetime’s total energy density equals the critical density at $t=0$ and at Heat Death $(\rho ={\rho }_c)$, the Planck Domain’s total energy density at $t=0$ and Heat Death for the same subset of Discrete Spheres also equals the critical density $({\rho }_{\mathrm{pd}}={\rho }_c)$. Finally, because the quantized energies of the quantum states that comprise 4D spacetime (the Universal Bell Energy Field) and the Planck Domain (the Universal Bell Energy Point) are identical, their total energy density budgets are also identical.

8.4. Heat Death and the Collapse of the Universal Bell Energy Point

The instantaneous collapse process, which applies to single and N-body quantum states, also applies to the Universal Bell Energy Point at Heat Death. Although the physical event that triggers the collapse of the Universal Bell Energy Point is unknown, it is based on a 4D spacetime Physical Interaction.

The significance of the Universal Bell Energy Point’s collapse cannot be overstated. The Bell Identity ensures that the collapse is mirrored by the generalized localization of 4D spacetime’s Universal Bell Energy Field at $t=0$.29 Although the precise size of generalized localization is unknown, because the Universal Bell Energy Point was very nearly homogeneous and isotropic at Heat Death, the Bell Identity ensures that 4D spacetime also remains very nearly homogeneous and isotropic at $t=0$.30

The instantaneous transition from Heat Death to $t=0$ radically alters the energy density, pressure, and temperature of 4D spacetime. At Heat Death, 4D spacetime’s energy density, pressure, and temperature asymptotically approach zero. However, at $t=0$, the generalized localization of the Universal Bell Energy Field causes an instantaneous collapse of 4D spacetime to extreme but finite energy density, pressure, and temperature. Despite the extreme conditions of 4D spacetime at $t=0$, the CMB indirectly confirms that the energy density, pressure, and temperature of the Universal Bell Energy Field at $t=0$ continue to exhibit near homogeneous and isotropic conditions of 4D spacetime at Heat Death.

The collapse of the Universal Bell Energy Point at or near Heat Death also explains 4D spacetime’s instantaneous transition from a state of near-maximal gravitational entropy at Heat Death to near-zero gravitational entropy at $t=0$. At Heat Death, as energy density, pressure, and temperature asymptotically approach zero, no additional work occurs at the macrostate level, notwithstanding the continuing expansion and the presence of anisotropies and inhomogeneities. However, following the collapse of the Universal Bell Energy Point and the generalized localization of the Universal 4D Field, the extreme energy density, pressure, and temperature, along with future expansion, reset 4D spacetime’s gravitational entropy to near zero. Finally, Discrete Spheres do not collapse during the transition from Heat Death to $t=0$. Their shape, size, intrinsic energy density, and total energy budget remain invariant, forming the stable, underlying fabric of both 4D spacetime and the Planck Domain.

8.5. The Horizon Problem and Causality

The inability to explain the exceptionally high homogeneity and isotropy of 4D spacetime at $t=0$ is the source of the horizon problem. Given a singularity premised on an infinite density, pressure, curvature, and temperature, and the constraints of the speed of light c, spacelike separated regions of 4D spacetime following $t=0$ could not have been in causal contact [95]. The problem is exacerbated by CMB data, which indicate that approximately 380,000 years after $t=0$, 4D spacetime’s temperature variations were approximately 1 part in 100,000. The amplitude of the fluctuations of the angular power spectrum of the CMB indirectly supports the conclusion that 4D spacetime was nearly homogeneous and isotropic at or near $t=0$. The near-zero curvature parameter $({\mathsf{\Omega }}_k=-0.0005\pm 0.0005)$ measured at the last scattering surface supports the conclusion that 4D spacetime was within 0.1% of being flat and is consistent with the DO model’s explanation that the collapse process at Heat Death preserves the flatness of 4D spacetime at $t=0$.

The DO’s resolution of the causality problem is premised upon four critical factors. First, the DO replaces the concept of an initial singularity at $t=0$ with a discrete 4D spacetime. Second, the near-maximal homogeneity and isotropy of the Universal Bell Energy Field at Heat Death is an internal physical process caused by the expansion and cooling of 4D spacetime over extremely long-time scales. Expansion and cooling are intrinsic physical processes governing 4D spacetime. Third, the instantaneous collapse of the Universal Bell Energy Point at Heat Death causes the generalized localization of the Universal Bell Energy Field at $t=0$. The instantaneous nature of collapse in the Planck Domain ensures that localization occurs simultaneously across all of 4D spacetime. Any deviation in timing would introduce anisotropies or break causal continuity, undermining isotropy and homogeneity at $t=0$.31 This invariance across cosmological transitions is why ${\rho }_{\mathrm{DS}}$, and hence $\mathsf{\Lambda }$, remain constant. The uniformity ensures that no directional biases or density variations disrupt the near homogeneity and isotropy of 4D spacetime at $t=0$.

8.6. The Flatness Problem

The flatness of 4D spacetime’s spatial curvature and its sensitivity to minor deviations from flatness (from $k=0$ to either $k=1$ or $k=-1$) at or near $t=0$ is known as the flatness problem. Because the value of k is calculated by $\mathsf{\Omega }$, and $\mathsf{\Omega }$ is defined as

$${\mathsf{\Omega }}_{\mathrm{total}}={\displaystyle \frac{{\rho }_{\mathrm{total}}}{{\rho }_c}},$$

the total energy density of 4D spacetime at or near $t=0$ must be extraordinarily close to the critical density ${\rho }_c$ to ensure spatial flatness.

While the DO model does not provide a theoretical basis or physical data explaining why the total energy density of the universe has the particular values it does, it avoids reliance on ad hoc assumptions, fine-tuning, or perturbative techniques to maintain flatness (cf. [100]). The homogeneity and isotropy of 4D spacetime at $t=0$, along with its intrinsic flatness, are functions of 1) the DO’s tightly integrated $\left(\right(3\times N)+3)$ framework, 2) the uniform size, shape, and energy of Discrete Spheres, 3) the Bell Identity, 4) the near-maximal homogeneity and isotropy of 4D spacetime at Heat Death, 5) the instantaneous collapse of the Universal Bell Energy Point in the Planck Domain and 6) the generalized localization of the 4D spacetime’s Universal Bell Field at $t=0$. Because the total energy density per Discrete Sphere in 4D equals that in the Planck Domain and both equal the critical density at Heat Death, ${\mathsf{\Omega }}_{\mathrm{total}}=1$ is preserved through collapse to $t=0$.

8.7. The Cosmological Constant and Global Energy Conservation

Under the DO model, each Discrete Sphere is structurally inert, has an identical size and shape, and contains a very small amount of constant intrinsic energy (CIE). Discrete Spheres do not interact with matter or radiation, are unaffected by gravity or the collapse of the Universal Bell Energy Point $\left({\mathsf{\Omega }}_{\mathrm{QS}}\right)$, and their CIE is invariant across 4D spacetime.

Cumulatively, the CIE of Discrete Spheres is the ontological source of $\mathsf{\Lambda }$$\left({\mathsf{\Omega }}_{\mathrm{DS}}\right)$, and its uniform energy density is expressed under the EFE as

$${\rho }_{\mathsf{\Lambda }}={\displaystyle \frac{\mathsf{\Lambda }{c}^2}{8\pi G}}.$$

32 Consequently, the energy density per unit volume of Discrete Spheres equals the energy density per unit volume of $\mathsf{\Lambda }$$({\rho }_{\mathrm{DS}}={\rho }_{\mathsf{\Lambda }})$. The equation of state for Discrete Spheres is $w=-1$, representing the negative pressure of $\mathsf{\Lambda }$. Accordingly, Discrete Spheres do not curve 4D spacetime locally; their uniform background and intrinsic energy set 4D spacetime’s overall rate of expansion.

Together, the gradual evolution of 4D spacetime into a pre-existing substructure of Discrete Spheres and the incremental increase in 4D spacetime’s total energy budget provide an ontological alternative to the EFE’s “internal stretching” into “nothing.” As 4D spacetime expands into pre-existing Discrete Spheres, each sphere’s CIE increases 4D spacetime’s total energy budget without violating global energy conservation.33 Instead, it reflects the expansion of 4D spacetime into a pre-existing space. Moreover, the expansion into a pre-existing space clarifies why 4D spacetime’s expansion rate has increased over time. Because the equation of state for Discrete Spheres is $w=-1$, implying a negative pressure, their share of 4D spacetime’s total energy budget grows as more Discrete Spheres come within 4D spacetime’s bounds.

Without reliance on QFT, the ontic status of Discrete Spheres inherently aligns with experimentally derived values of $\mathsf{\Lambda }$, negating the need for zero-point energy, often linked to dark energy (cf. [101]). The independence from QFT further underscores the DO model’s ability to resolve $\mathsf{\Lambda }$ without invoking ad hoc assumptions or fine-tuning, providing a natural transition to broader considerations of energy scale stability in Section 8.8.

8.8. The Hierarchy Problem

The vast energy differences between the electroweak scale ($\sim {10}^2\mathrm{GeV}$) and the Planck scale ($\sim {10}^{19}\mathrm{GeV}$) pose a challenge to conventional frameworks, as large quantum corrections destabilize the electroweak scale. The DO model resolves the issue through two elements: (1) a discrete 4D spacetime structure governed by the ontic properties of Discrete Spheres, and (2) a relational gravity that decouples gravitational energy from the electroweak scale. Discretization imposes a natural ultraviolet cutoff at the Planck scale, precluding quadratically divergent contributions in quantum field theory that otherwise require large renormalization corrections to scalar masses. Gravitational decoupling prevents Planck-scale effects from coupling into the intrinsic quantum properties of electroweak-scale states, thereby removing the source of the high-energy loop corrections to the Higgs mass. Moreover, because the energy density of Discrete Spheres ${\rho }_{\mathsf{\Lambda }}$ is uniform, it drops out of the relational operator locally and appears only in the global expansion term with an equation of state $w=-1$. Since ${\rho }_{\mathsf{\Lambda }}$ is not a component of a Bell Energy Point in the Planck Domain, it exerts no influence on the Higgs scale or other electroweak parameters.

The observed ordering of gauge strengths and gravity reflects the ontological nature of the hierarchy problem. Electromagnetism is sourced at the Bell Energy Point in the Planck Domain and expressed as a Bell Energy Field in 4D spacetime; as that field spreads over Bell Spheres in 4D spacetime, the amount of energy at any one location falls as $1/r^2$. Similarly, the strong and weak interactions are also expressed by their respective Bell Energy Fields. However, unlike electromagnetism, energy exchanges are extremely short-range and adjacency-limited, so there is no geometric dilution, and the local intensity at contact remains high. Gravity, by contrast, is purely relational: it responds only to differences in the 4D energy distribution and averages over neighboring regions; a uniform background cancels. The influence measured at distance r also follows a $1/r^2$ falloff, but the per-site effect is smaller than that of the gauge fields.

9. Coda: The Quantum-Classical Divide

Finally, the DO framework highlights the ontological and dynamic separation between physical systems in the Planck Domain and a discrete 4D spacetime. The Planck Domain lacks a time dimension and spatial volume and is not governed by the laws of 4D spacetime. Its only source of dynamic movement is collapse. In the Planck Domain, a quantum state’s single Bell Energy Point represents its quantum observables, but a Bell Energy Point does not support relational properties, such as the chairness of a chair or the aliveness of a cat.

In contrast, 4D spacetime has a time dimension, three spatial dimensions, and volume, and is fully governed by its physical laws. Although a quantum state’s observables (regardless of the state’s size) appear in its Bell Energy Field, 4D spacetime also supports relational properties, such as the solidity, shape, and color of a chair or cat, as well as aliveness. The relational properties between Bell Spheres in 4D spacetime emerge from 4D spacetime’s ontology, dynamics, and laws, rather than from the observables of a quantum state’s Bell Energy Point. Although a quantum state’s observables in 4D spacetime can be determined under the proper conditions (a Stern–Gerlach experiment), 4D spacetime’s relational properties are not determined by the quantum state’s observables but by 4D spacetime’s ontology, dynamics, and physical laws.

For instance, in the Schrödinger cat experiment, assume that a radioactive atom’s quantum state, represented by its Bell Energy Point and Bell Energy Field, is a superposition of states where it has decayed and has not decayed. The instantaneous collapse of the atom’s Bell Energy Point and the Bell Identity’s mirroring of the collapse causes the localization of the atom to a decayed status in 4D spacetime. The atom’s decayed state triggers a Physical Interaction with a lethal device. Through a series of additional, interrelated Physical Interactions and Bell Energy Point collapses, the Physical Interaction between the cat and the lethal device localizes the cat’s status to “Dead” in 4D spacetime.

The cat’s status is now irreversible; there is no physical path back to “alive.” Conversely, if collapse never occurs, the cat remains alive. The cat’s status is relational, based on the status of its constituent quantum states’ interactions under the laws of 4D spacetime before, during, and after the experiment.

Moreover, in a discrete 4D spacetime, all quantum states evolve deterministically between collapses. Determinism extends to both quantum state superpositions in the case of quantum observables and to the special case of quantum state location, where the concept of superposition does not apply.34 However, in all cases, the collapse of a Bell Energy Point introduces probability by determining which subset of Bell Spheres becomes localized in 4D spacetime.

The inability to quantize gravity reinforces this principle. Like other relational properties, gravity arises purely from relational configurations in 4D spacetime. Its curvature depends on the relational properties of mass, energy, and pressure, and not on the observables of a quantum state’s Bell Energy Point.35

10. Conclusion

Ontological and dynamic premises have profound theoretical implications. The DO model is based on a physical and dynamic framework that systematically unifies the most challenging problems in modern physics across scales and domains. Built on a substructure of Discrete Spheres and the SOAN, the DO’s integrated $\left(\right(3\times N)+3)$ ontology replaces GR’s continuous 4D spacetime manifold with a discrete 4D spacetime with relational curvature and replaces high-dimensional state spaces with an ontic $(3\times N)$ Planck Domain. Without the aid of ad hoc assumptions, fine-tuning, or perturbative techniques, the DO model supports a single ontological and dynamic framework and a set of physical laws that unify GR, SR, and QM, resolve the GR–QG tension, recover the arrow of time, identify the cosmological constant $\mathsf{\Lambda }$, resolve hierarchy tensions, and account for 4D spacetime’s Heat Death and instantaneous emergence at $t=0$.

The explanatory depth of a non-temporal, non-spatial SOAN supports the Planck Identity’s one-to-one mapping between the Discrete Spheres that form 4D spacetime and the Planck Domain. In turn, the Bell Identity and Bell Spheres ensure that the physical attributes of a quantum state, including its energy, are dynamically represented in both domains. Significantly, the Bell Identity’s mirroring function supports the deterministic, relativistic evolution of quantum states in 4D spacetime and their instantaneous collapse in the Planck Domain [cf. 102].

The DO framework provides new insights into unresolved questions in physics. It employs a single energy-based operator that governs both motion and gravitational response, identifying inertial response and gravitational source at the operator level through $m=E/c^2$. The model eliminates infinities associated with singularities through discretization, resolves the black hole information paradox by preserving quantum state information in the Planck Domain, and explains gravity as a local, relational phenomenon generated by the same energy that sets inertia. The model provides a physical explanation for the irreversible arrow of time in 4D spacetime and redefines the cosmological constant $\mathsf{\Lambda }$ as the intrinsic energy density of Discrete Spheres. The model also resolves 4D spacetime’s homogeneity, isotropy, horizon, and flatness problems based on its instantaneous transition from Heat Death to $t=0$.

While quantum states are mirrored across domains, classical properties such as a cat’s aliveness are exclusively governed by the relational ontology and dynamics of 4D spacetime. Classical determinism and quantum probability are not mutually exclusive; rather, they arise from a single, unified framework. Discretized physical laws support the DO framework while preserving GR’s background independence and causal structure as well as SR’s physical constraints. The DO’s ontological, dynamic, and mathematical framework bridges the quantum, relativistic, and cosmological domains, inviting a fundamental reassessment of foundational assumptions in theoretical physics.

Constructed from the core principles of Energy and the SOAN, the DO framework’s explanatory depth points toward a unified understanding of reality, suggesting avenues for physics to transcend current theoretical limitations and uncover deeper insights into the structure and dynamics of 4D spacetime and the Planck Domain.36