Originally written: April 2026
Theory of Spheres' ('Quantum Gravity')'
or
'E'Empty Spheres in Dimensionless Space' – The 'Fundamental Force Behind Everything
(.Q('Quantum – Gravitational Force', 'Dark Matter' / 'Dark Energy', etc.)
Paper III(a) – Updated version: The Quantum–Gravitational Force as Derived from the Spheres
Eran Shimony (ORCID number – 0009-0003-5610-6736)
Physicist (B.Sc.), MBA, and Certified Real Estate Appraiser
Download the complete research papers in PDF format for offline reading and citation:
[Download Article I: The Framework (PDF)]
[Download Article II: The Spheres Classical components (PDF)]
[ Download Article III: The Quantum–Gravitational Force as Derived from the Spheres (PDF)]
]Paper III(a) – Updated version: The Quantum–Gravitational Force as Derived from the Spheres (PDF)]
[Download Article Ⅳ: The Planck's constant – ℏ (PDF)]
i
Publication History & Context: This article was published on April 11, 2026 (DOI: 10.17605/OSF.IO/PF3BH).
It builds upon the foundational theory established in the previous work: 'Theory of Spheres – Part I', published on December 29, 2025 (DOI: 10.5281/zenodo.18081185)
and upon the 'Theory of Spheres – Part II' published on February 01, 2026 (DOI: 10.5281/zenodo.18446035).
J
ABSTRACT
The Geometric Origin of Mass and Gravity: Bridging the Quantum-Newtonian Century-Old Conflict through the zero-dimensional 0-D Harmonic Oscillator.
This paper presents a fundamental derivation of mass and gravitational force within the framework of the Theory of Spheres (ToS). We demonstrate that Newton’s Law of Universal Gravitation (1/R2) is not an independent postulate but an emergent geometric result of spherical normalized pressure distribution (P=1) within a l (0-D) harmonic oscillator.
By expanding the gravitational interaction through a Taylor series, we reveal two primary force components: a classical Newtonian term that governs macro-scale gravity, and a high-order stabilizing 'Catastrophe' term responsible for the internal stability of the observed cosmological expansion.
This work offers a geometric suppression which resolves the Vacuum Catastrophe by deriving the coupling constant as: α ≈ 5 x 10-61(Planck units) ≈ 2.3 x 10-53(m.k.s units), proving that the immense vacuum energy is suppressed through geometric interference to the minimal residual 'noise' of reality (Λ ≈ 10-122).
This framework successfully establishes the long-sought direct link between sub-quantum interference and classical Newtonian mechanics, offering a unified geometric description, linking sub‑quantum interference with classical gravitation.
Part A: Introduction
This paper is a direct continuation of the previous work, which primarily focused on the interior expansion from the center toward the surface of the sphere. However, as established in the principles of the Theory of Spheres (ToS), a second expansion exists: outward from the surface of the 'Equilibrium Sphere'. Correspondingly, there are also two modes of contraction: reduction from the outside onto the surface of the 'Equilibrium Sphere' and from the surface back into the interior. This symmetry is not merely mathematical but represents the fundamental nature of cosmic pressure and stability, consistent with the 'Twin Universe Theory' and the 'entanglement Quartet' (Part E First paper in the series).
In this work, we demonstrate that the fundamental forces of nature emerge from the structural dynamics of the 0-D harmonic oscillator. We show that the interaction between internal frequency and external invariant pressure (P=1) naturally gives rise to both cumulative gravitational effects and the high-energy stability of the vacuum.
We will demonstrate how these geometric dynamics lead to the emergence of 'Newtonian Gravity', and show that while cosmic pressure reaches equilibrium through geometric symmetry, the resulting mass-energy interactions allow Newton's equations to emerge directly from the 'ToS' framework.
Chapter 1:The Four Entangled Sides of the Sphere
To define the 'Four Entangled Sides', we categorize the velocity distribution into distinct quadrants of the harmonic cycle.
As a reminder, In the ToS framework, C(t) represents the physical speed of light, which is dynamic rather than constant. Unlike Einstein’s postulate c = const, here C(t) evolves from 0 to its maximal value and back to 0 across the harmonic cycle. And we emphasize that for short periods of time—even of millions of years—the change in the speed of light over time is not measurable.
The First Quadrant (our universe state): Represents the expansion phase starting from the boundary, where t=0 and the velocity C(t) is 0. This boundary condition at the origin is satisfied by subtracting the number '1' in Eq 1. As the system evolves, C(t) transitions to its maximal value at the equilibrium point, as represented by Eq 1, here and also in Papers 'A' and 'B'. These equations lay the groundwork for the structural entanglement of the subsequent quadrants.
The Second Quadrant: represents the continuation of expansion from the system's equilibrium point to the outside boundary.
In this phase, the speed of light C(t) decays from its maximal value at the equilibrium point back to zero at the opposite boundary. This behavior is governed by the same family of Eq 1 but as exponential decay:
The Third and Fourth Quadrants: While the first and second quadrants represent the expansion phases of the system, Quadrants 3 and 4 represent the phases of contraction.
The Third Quadrant: marks the transition from the boundary point—where the speed of light is zero (C(t) = 0)—back toward the equilibrium point, where it reaches its maximal value. This inward acceleration mirrors the dynamics of the first quadrant and is therefore governed by the same functional form as Equation 1:
The Fourth Quadrant: completes the harmonic cycle. It represents the contraction from the equilibrium point back to the original boundary, where the velocity returns to zero (C(t)=0). This phase is governed by the decay characteristics of Equation 2:
By defining these four quadrants, we establish a complete, symmetrical oscillation within the sphere. This geometric enclosure demonstrates how the speed of light fluctuates harmonically, forming the 'Four Entangled Sides' of the 'ToS' framework.
As derived in Paper II chapters 1 & 2 (Eq. 2 & 14), and according to the Hubble Law (v = ± H0D), the fundamental constants H0 ≈ ±48.41(km/s/Mpc) and Λ ≈ ±0.54194 10-52(m-2) ≈ ±1.44 x 10-122, represent the equilibrium state of the oscillator. In the current framework, these values are considered to remain consistent (see second concluding note for the table) but fluctuate in sign across the quadrants to maintain the harmonic symmetry of the sphere.
The changing values of H0 and Λ across the four quadrants are shown in the following table, where quadrant 1 values correspond to the present oscillation phase and the remaining quadrants represents the mirrored phases of the cycle.
Concluding Notes for the table:
- The transition into quadrants 3 and 4 represent states of contraction characterized by negative entropy (Negentropy). The reversal of the velocity vector C(t) necessitates a corresponding shift in the thermodynamic and temporal arrows, where the system transitions from expansion to contraction. This ensures the conservation of the 0-D harmonic state, allowing the sphere to complete its cycle without energy loss
- The numerical values for H0 and Λ were calculated in Paper II using a 70% correction for exponential distribution, bounded to expand at an asymptotic speed—the temporary speed of light – C(t,temporary).
ן
In reference to Einstein’s Special Relativity, which states that local objects cannot exceed the speed of light, we have extended this prohibition as an asymptotic limit as well as for the fabric of space-time itself.
In reality, as shown, the speed of light varies over time—ranging from 0 at the boundaries to a maximum at the system’s equilibrium point. Consequently, the calculated values for the Hubble constant – H0 and cosmological constant – Λ are valid in every other quadrant, universe or oscillator for the specific timeframe in which the same ratio of the temporary speed – C(t,temporary) to C(max) or to C(min) is maintained. Thus, even in a quantum or cosmic scale, there is no dependence on the zero-dimension light‑mass, denoted in Chapter 3 of Paper 2 (Very important note) as (MlDO ); rather, it acts as the engine responsible for the changes in the speed of light of the oscillator. In our universe it is equal to its baryonic mass 1053KG. Then, since the light speed varies over time, the height of the asymptote changes accordingly; therefore, the Hubble and cosmological constants, H0 and Λ, are also evolve over time.
The evolution of H0 and Λ over time might be further addressed in a future paper, dealing with the boundaries of the universe (see Future Plans, Section 1, at the conclusion of this paper).
Chapter 2: Newton Gravity
Proving the Inverse-Square Law, Assuming a central force, (1/R2) within the 'ToS' Framework
To establish that the emergent geometric force is proportional to 1/R2, we present three distinct analytical paths.
Method 1: Volumetric-Pressure Equilibrium
This path derives the 1/R2 law from the physical requirement of pressure normalization. In the ToS framework, the internal energy of the 0-D oscillator is a volumetric property (V = 4/3π x R3) → (V ∝ R3). For the system to maintain a stable equilibrium (P=1), this internal pressure must be distributed across the sphere’s bounding surface (A ∝ R2). The resulting force interaction is therefore governed by the ratio of these geometric constraints, naturally emerging as an inverse-square relationship to ensure structural stability.
Method 2: Classical Potential Gradient
We begin by defining the central force as a derivative of the energy potential within the sphere's volume.
- Sphere Volume: V = 4/3 x πR3.
- Internal Energy: Proportional to the volume: E ≈ – P x V ≈ – P x R3.
- Potential (Φ): Assuming the potential is distributed across the radius R as a harmonic oscillator, we define: Φ(r) ≈ – E/r.
- Force Derivation: The force is the negative gradient of the potential: F(r) = -dΦ/dr ≈ E/r2
i
This confirms that the energy density and geometric constraints naturally yield the 1/R2 relationship.
Method 3: Spherical Surface Distribution (Flux)
In a three-dimensional framework, any property radiating, such as Starlight or Quantum Radiation Pressure — as observable macroscopic examples of flux radiation distribution — or exerting pressure from a center, must distribute itself over the surface area of a sphere.
- Surface Area: A = 4πR2
- Force as Pressure (Δp): If the force is viewed as an external pressure applied by the spheres, it is spread across the spherical shell: F = Δp x 4 π R2
- Effective Result: Therefore, the intensity of the force at any given point on the surface must be inversely proportional to the area it covers, leading to: Feff ∝ 1/R2
i
The Principle of Pressure Invariance and Energy-Volume Duality
The foundational framework of the Theory of Spheres (TOS) rests upon the equilibrium between internal and external spherical pressures. In paper 1 chapter 1, we established the total pressure acting upon the system as a continuous integral of all surrounding infinitesimal components. This relationship is governed by the Principle of Pressure Invariance, where the total external pressure (Ptest) is normalized to a constant value of unity ('1').
This invariance ensures that physical laws remain consistent across all scales—from discrete sub-quantum elements to continuous macro-systems.
Energy Derivation from Pressure Dynamics
In a spherical geometry, energy is not an abstract property but a direct consequence of the pressure acting upon a volume. Following the classical definition where Energy is the product of Pressure and Volume (E = P x V), and given that Ptest=1 from outside within our normalized framework, the energy density of the sphere is intrinsically linked to its geometric boundaries.
Within this framework, we define the energy of a 'Mass point' (or a zero‑dimension light‑mass of the localized 0-D Oscillator) as: E = MlD0x C2(t).
This definition aligns with the foundational proposition established in Paper 1, Chapter 24, where Mass is described not as a static quantity, but as the net curvature of space resulting from the aggregate of instantaneous wavefunctions in superposition.
Here, C(t) represents the time-dependent expansion velocity of the sphere's boundaries. By anchoring the energy E to the invariant pressure P, we establish that any change in the observed mass or gravity is not a loss of fundamental energy, but a result of the interplay between the internal frequency of the mass and the external pressure of the vacuum.
In the 'TOS' framework, the relationship E= MlD0×C2 undergoes a critical phase-shift. While classical physics assumes energy scales linearly with mass, our model demonstrates that as mass increases toward the macroscopic limit, its 'Effective mass' actually decreases and so it's 'Effective Energy' actually decreases too (representing the non-linear reduction in effective mass-energy interaction). This is due to the Destructive Interference between the mass's intrinsic wavelength and the invariant pressure of the vacuum (P=1).
This mechanism is analogous to the double-slit experiment (dark fringes, for example): just as destructive interference creates zones of zero observed energy, the TOS framework shows that at macroscopic scales, the vast majority of universal energy cancels itself out. This 'cancellation' is geometrically constrained by the same principle that governs the Newton Inverse-Square Law. Consequently, the energy isn't lost; it transitions from observable kinetic form to static spherical pressure, maintaining the universal equilibrium (P=1).
The Mechanism of Effective Mass Decay
From the integral in Equation 3 above, we obtain the external pressure toward '1' based on the fact that mass — as stated in the previous paragraph — is a superposition of the internal and external pressure acting on the components that constitute space, and is therefore not a fundamental property but an emergent result of sub‑quantum interference.
For an infinitesimal or minimal mass, the attenuation is almost negligible, and therefore the effective mass is nearly identical to the actual mass, as the NAVP wavefunctions exhibit minimal overlap and thus minimal destructive interference.
As the mass increases, the attenuation becomes more significant, due to increasing destructive interference between densely packed NAVP's, causing the effective mass to decrease relative to the true mass. This inherent attenuation is the geometric mechanism that prevents gravitational collapse and maintains the long-range stability of the universe.
Therefore, we can express the effective mass — in an approximate form, assuming an exponential decay distribution — as follows (see a more detailed explanation in Chapter 24 of Article 1):
Where:
- Meff: the decayed resulting from the sum of the zero‑dimension light‑mass of the oscillator- MlD0 (as defined in Paper II, chapter 3 very important note). This mass constitutes the spatial volume of 'M', maintaining the cosmic pressure equilibrium.
- KM: the mass-normalization constant, representing the geometric scaling factor that aligns the zero-dimension light-mass with the observed effective gravitational mass.
- i'α' (The Structural Coupling): The interaction strength between the Newtonian part and the Catastrophe part. Its specific value (α ≈ 5 x 10-61(Planck units) ≈ 2.3 x 10-53(m.k.s units)). This coupling ensures that the total oscillator energy undergoes a geometric decay, where only a specific component manifests as observable gravity, while the rest of the energy density is suppressed to maintain cosmic stability and prevent vacuum Explosion/expansion.
i
The Mechanism of Effective Energy Decay
Having defined the effective mass, the effective energy follows directly from substituting into the generalized energy expression. Since energy scales linearly with mass, the decay in mass induces an identical exponential decay in the effective energy. Therefore, the effective energy can be written as:
This formulation shows that the energy inherits the same attenuation mechanism as the mass, preserving consistency across the model.
Deriving the Effective Mass and the Coupling Constant 'α' via Planck Units
To find the physical manifestation of α, we substitute known cosmic values. Using the accepted Planck mass (Mp = 2.176 x10-8 kg) and the estimated mass of the observable universe Muniverse = 1053kg, the ratio, or the 'mass of the universe' expressed in Planck units, is approximately: Muniverse / MP = 1053kg / (2.176×10-8) ≈ 4.6×1060p.
Narrative Explanation of the Derivation of α
To incorporate the cosmological constant into the energy‑distribution framework, we express the vacuum energy associated with Λ as an effective mass–energy. In Planck units, mass and energy share the same numerical value, allowing us to treat the vacuum energy contribution as a vacuum effective mass term, when Ct = 1 in Planck units, in term of – MΛ. This effective mass term is numerically equivalent to the vacuum energy density, used to derive the coupling parameter α, which governs the exponential suppression in the energy distribution.
Substituting into the Effective Energy Expression
When we define KM-m = G0 and work in Planck units (where C = 1 and G0 = 1), substitute values into Equation (5), the effective energy becomes:
Deriving the Structural Coupling (α)
To find the value of the structural coupling α, we examine the state where the system reaches its cosmic equilibrium, where we identify the effective energy with the vacuum effective mass term Eeff = MΛ.
Substituting this into Eq.6 gives: MΛ = MlDO X e-αMlDO
Then From Equation 6, we can isolate α by taking the natural logarithmic relation between the effective vacuum mass MΛ and the reference oscillator mass MlDO scale as follows:
Next, we express the effective vacuum mass (MΛ) in terms of the vacuum energy density ρΛ and the physical volume Vp: MΛ = ρΛ x Vp
Since the physical volume is that of a sphere of radius R, we write: Vp= 4/3πR3
Using the standard Friedmann relation or Einstein field equations, the vacuum energy density associated with the cosmological constant is: ρΛ = ΛC2 / (8πG), which in Planck units (C=1 and G=1) reduces ρΛ = Λ /(8π).
Substituting this into the expression for α gives:
So, the expression becomes:
At this point we substitute the numerical values, all expressed in Planck units:
- Radius of the observable universe: R = 46Gly ≈ 4.35 x 1026m ≈ 2.7 x 1061p (using Planck Length as 1.616 x 10-35p).
- Total mass–energy of the universe: MlDO = 4.6 x 1060
- Cosmological constant: Λ = 1.44 x 10-122
- We first compute the cubic radius: R3 ≈ (2.7 x 1061)3 ≈ 2 x 10184
Thus:
And:
Concluding the Derivation
The coupling constant α—defined as the positive coupling magnitude, with the sign absorbed into the exponential decay term (taken as an absolute value)—emerges naturally from the vacuum energy encoded in Λ, without requiring any empirical tuning.
Interestingly, the dimensionless coefficient that emerges from the logarithmic structure of the model is approximately 2.3. This value is not inserted by hand; it arises naturally from the interplay between the cosmic radius R, the cosmological constant Λ, and the mass scale MlDO.
The appearance of a number of order unity — specifically close to 2–3 — is reminiscent of similar dimensionless factors that appear in quantum field theory, particularly in QCD (Quantum Chromodynamics), where logarithmic running of the strong coupling produces comparable coefficients. In this sense, the coefficient 2.3 can be viewed as the stabilizing factor that maintains the coherence of the 'entanglement Quartet'.
Interpretation and Physical Meaning
This decay of mass/energy with increasing total mass provides a natural explanation for the vacuum catastrophe discussed in Article 4, Chapter 2.
The exponential attenuation suppresses the effective contribution of large-scale mass-energy densities, preventing the divergence predicted by naïve quantum field estimates.
The Effective Force
After determining α, we return to Equation (4) for the decay of the mass and substitute it into the expression for the effective force, consistent with the Newtonian inverse-square law established at the beginning of the chapter: Feff ≈ 1/R2.
More explicitly, we can write:
We now substitute the effective masses from Equation 4:
&
Defining KM-m = G0 (as done for equation 6), When the constant 4/3π (from the volume of the ball) is swallowed up into the constant G0, and collecting terms, we obtain:
In Planck units, where: G0 =1 ,the expression simplifies to:
The Newtonian Approximation
To substitute specific masses and evaluate the force in familiar scales, we expand the expression to a Taylor series, yielding:
Therefore, the total effective force, using the first-two-order approximation of the Taylor series, can be written as:
Therefore, the total effective force can be written as:
Physical Implications:
As established, α is an incredibly small constant (on the order of 10-61, Planck units). For any mass encountered in classical physics or even in astronomy—from an electron to stars mass—the term α x M remains significantly smaller than 1.
Consequently, the higher-order terms in the expansion are completely negligible. The first-order term provides the dominant correction, which remains vanishingly small in almost all physical cases. This explains why the effective force reduces to standard Newtonian gravity across the entire observable physical mass range. However, on cosmological scales, the exponent is no longer small, so the Taylor expansion around zero is invalid. As a result, the Newtonian approximation breaks down, and the higher‑order terms—the same terms responsible for the vacuum catastrophe—dominate the behavior of the force.
Chapter 2 Summary
We have demonstrated that the gravitational interaction is a residual structural effect of vacuum fluctuations, governed by the coupling constant │– (α x M) │ ≈ 2.32. By utilizing the exponential damping mechanism, we have shown that the gravitational potential is self-regulating, thereby resolving the divergence predicted by classical vacuum energy estimates.
Furthermore, we have confirmed that for all physically relevant mass ranges (Newton's apple example), this model converges with excellent accuracy to Newtonian gravity. So that at quantum scales (such as the electron mass), gravity remains an extremely weak force compared to other fundamental forces.
By reintroducing the time-dependent gravitational constant G0(t) (which was established in Article 2, Chapter 4 and set to unity (G0) in Planck units in Eq. 14), and substituting it into physically relevant mass range limit of equation 16, we obtain the time-dependent effective Newtonian gravitational force, for most mass scales, as follow:
It is important to emphasize that Eq 17 represents the emergence of time-dependent Newtonian gravitational force directly from the calculation of the effective mass decay, where a near-infinite density of oscillators or sub-universes occupies the volume unit where the mass exists. This decay is driven by the pressure differential between the normalized external vacuum pressure (set to 1) and the internal pressure resulting from the superposition of the characteristic wavelengths of all participating oscillators/sub-universes, which constitute the volume.
Conclusion: Closing the Loop
In this paper, we have successfully closed the conceptual and mathematical loop regarding the fundamental nature of mass and force within the Theory of Spheres (ToS).
The Geometric Foundation
We established the Entangled Quadruplet as the necessary symmetric framework for a stable, self-sustaining 0-D harmonic oscillator. This geometric distribution of spherical pressure provides the direct link between sub-quantum interference and macroscopic gravity.
Bridging a Century-Old Conflict: Linking Quantum Mechanics and Gravity
The most critical achievement of this work is the formal derivation of Newton’s Law of Universal Gravitation directly from the quantum-mechanical properties of the 0-D oscillator. We have demonstrated that gravity is not an independent fundamental force, but an emergent geometric result of spherical pressure distribution. This resolves the century-old conflict between quantum mechanics and classical gravitation.
Universal Stability and the Cosmological Constant
Our derivation reveals that the 'nothingness' of the vacuum is a dynamic, Uncertainty principle, pulsating medium. By proving that this energy of the vacuum, cancels out as mass increases, we provide a definitive solution to the Vacuum Catastrophe. Through the coupling constant │– (α x M) │≈ 2.3, the mechanism allows for a stable universe where the effective force reduces to Newtonian gravity across all observable scales—from the electron mass up to the falling apple—only deviating at the ultimate cosmological boundary.
Final Summary:
The Theory of Spheres provides a unified framework where the speed of light, the gravitational force, and the cosmological constant are all reconciled. Gravity is revealed to be the macroscopic manifestation of sub-quantum interference, finally offering the 'direct link' that physics has sought for decades.
Future Prospects
The framework established in this paper provides a robust foundation for ongoing development. For example, our future work might focus on two key domains:
- Dynamic Implications: We might investigate the lifetime of the universe, the maximum velocity of vacuum oscillations and its fundamental role in relativistic coupling.
- Relativistic Gravity: While the current derivation of Newtonian gravity is highly accurate for static or slow-moving bodies (such as Newton's apple), our upcoming research might expand this model in accordance with the Theory of Spheres, generalizing the gravitational force to describe relativistic interactions and high-velocity systems.
Eran Shimony, 2026 ©
English translation based on the original Hebrew version of the 'Theory of Spheres' (ToS) (2017).
All rights reserved by the author.