Originally written: October 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 X: (Appendix of Paper IX): MlD0 – The Quantum 0-D Light-Mass and mp (Planck mass
Eran Shimony (ORCID number – 0009-0003-5610-6736)
Physicist (B.Sc.), MBA, and Certified Real Estate Appraiser
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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)]
[Download Article VII: Introduction to the New Quantum Geometry Formalism (PDF))]
[Download Article IX: Introduction to the New Quantum Geometry Formalim (PDF))]
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Publication History & Context: This article was published on October 21, 2026 (DOI.org/10.17605/OSF.IO/X5JWN)
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) …..
ABSTRACT
In Paper IX, we successfully derived the total cosmic 0-D light-mass, achieving one of the primary foundational goals of the Theory of Spheres (ToS). Seeking to extend Paper VIII toward the derivation of electroweak boson masses (W±, Z0) using methods analogous to Paper IV – where Planck's constant was derived via the universal cosmic constant (Λ ≈ 1.44 × 10-122) acting as both the vacuum catastrophe and a classical-quantum bridge – an unexpected numerical result emerged. The calculated value matched the historical Planck mass (mp) with remarkable precision. This implies that the Planck mass – whose true physical nature remained elusive to Max Planck himself and subsequent generations – is fundamentally the quantum baryonic Mass or (√3 × Quantum 0-D Light-Mass), evaluated at the quantum scale.
Finally, we conclude in a perfect closure by realizing that the square root of the cosmological constant (√Λ) serves as the cosmic-quantum connection coefficient between the 0-D light-mass viewed from cosmic and quantum perspectives. Consequently, it also acts as the hierarchical connection coefficient to the 0-D mass across an almost infinite chain of universes (parent and sub-universes).
Part A: Introduction
This appendix reviews the derivation of the quantum baryonic Mass and the Quantum 0-D Light-Mass from the 0-D harmonic oscillator engine at the quantum scale, exploring its deep physical implications, with a particular focus on the profound impact of this relationship, regarding the position of our universe within a potentially infinite hierarchy of parent and sub-universes.
Chapter 1: From Cosmic Scaling to Quantum Phase Mass (mp) – The Mathematical Derivation
When analyzing the macroscopic-to-quantum transition, we utilize the observable baryonic mass of our universe (Mbaryonic = 1.54 × 1053kg, as derived in Paper IX) and using the square root of the dimensionless cosmic scaling constant (√D0) ≈ 1.2 × 10-61, derived from D0 ≈ 1.44 × 10-122) (Paper II):
Note: In this Appendix, from this point forward, to distinguish between macroscopic-cosmic and microscopic-quantum perspectives, the relevant mass variables are designated accordingly: Mbaryonic and MlD0, as along the series, denote the cosmic baryonic mass and total cosmic 0-D light-mass, respectively, while Mb(qua) and MlD0(qua) denote their corresponding quantum-scale.
To transition accurately back from 3-D macroscopic space into the pure 0-D dimensionless state, we apply the inverse spatial projection factor, reversing the 3-D isotropic equipartition from X, Y, Z back to the 0-D phase quantum established in Paper IX. Therefore, the reverse transition from the observed space back to the 0-D phase requires multiplication by √3 (since the forward projection required division by √3): MlD0(qua) = Mb(qua) × √3 = 1.848 × 10-8kg × √3 ≈ 3.2 × 10-8kg.
v(Note: Unlike in Paper IX, no inverse 4-quarter scaling is required here, as the 0-D quantum phase represents a fundamental pulse unit evaluated externally to the 4-quarter geometry)
Comparing this derived value – MlD0(qua) ≈ 1.848 × 10-8kg – to the officially measured Planck mass (mp ≈ 2.176 × 10-8kg) (⇒ mp ≈ Mb(qua) = MlD0(qua) / √3 – as derived here and applied hereafter), leads to a minor deviation of about 15% to 17%. The deviation primarily attributed to rounding results along the way and to standard, mainly cosmological, approximations used throughout the series – such as: the age of the universe, its mass and its radius. Undoubtedly, the major factor is attributed to the constant speed of light used by traditional physics.
Examples of such deviations accumulated along the series include: the calculations of the Hubble parameter and the cosmological constants (H0, Λ) in Paper II, the Coupling Constant (α – denoted as αs later in Paper VII) in paper III(a), Planck's constant – ℏ in paper IV, as well as external and internal structural coupling constants (αs and βs), and Gravitational Constant G0(now) in Paper VI. Even subtle, careful discrepancies appear in the lepton masses calculated in Paper VII.
Note: Unquestionably, these variations – along with other parameters such as the Maximum Speed of Light (Cmax), the Universe Lifespan (tmax), time-dependent Gravitational Constant (G0(t)), and others, which offer no direct parallel for comparison in standard physics – stem directly from the rounding results, approximations used throughout the series, and primarily the constraint of a constant speed of light used in traditional physics.
The Unified Quantum-Cosmic Identity
To reveal the exact deterministic mechanism underneath this numerical correlation, we cross-reference these boundary conditions with the four-quarter cosmic topology established in Paper IX (P. 3). In our framework, the cosmos consists of four symmetric entangled quarters (4π topology), where our observable universe constitutes a single quarter (25% baryonic matter):
i(where MlD0 is the accumulated potential of the total cosmic 0-D oscillator light-mass)
By substituting this geometric definition into our initial macroscopic-to-quantum transition, we obtain:
Concurrently, our direct quantum-scale extraction establishes that:
Equating the two expressions yields the final structural relationship:
Chapter 2: Chain Hierarchy Universes – Key mplicationsn
Since the 3-D spatial projection factor (√3 ) appears identically in the denominators of both sides of the preceding equation, it naturally cancels out. This elegant cancellation proves that the isotropic spatial dimensions clear away when isolating the foundational 0-D energy unit. Finally, by isolating and substituting the cosmic scaling constant (√D0) with the universal cosmological constant scale (√Λ ) and returning to MlD0 as a general 0-D harmonic oscillator mass engine, we unlock the exact, deterministic bridge:
Physical Meanings:
- The historical formulation of Planck mass in 1899 combined G, ħ, and c (mp = √(ħc/G) into a dimensional unit without a mechanistic blueprint. Whereas, within the Theory of Spheres, mp is revealed not as an arbitrary spacetime boundary or quantum uncertainty cutoff, but as the quantum baryonic Mass or √3 × Single 0-D Quantum Phase Mass – the fundamental energetic unit of the pure 0-D light engine.
- The dimensionless ratio (D0 = Λ ~ (mp/Mbaryonic)2 ~ 1.44 × 10-122) hermetically binds the micro – quantum scale to the macro-cosmic horizon.
- Particle accelerators (such as the LHC) measure only transient excitation states operating entirely within the 3-D spatial domain – probing secondary harmonics and transient fluctuations (such as W± and Z0 bosons at masses of ~ 80–91 GeV) without ever penetrating or altering the pure 0-D phase quantum itself (mp ≈ 1.22 × 1019 GeV).
- At the subatomic scale, √3 × Planck mass acts as a 0-D phase pulse projecting the particle spectrum, mirroring the cosmic mass (MlDO) across the universe hierarchy.
- Chain hierarchy example: The most stable ground-state chain spans is the parent (or anti-parent) universe → our (or anti-our) universe → the electron (or positron) → sub (or anti-sub) electron states, and beyond. Secondary, substantially lower-probability universe branches may similarly be grounded in higher-generation leptons, such as the muon and tau.
- By canceling 3D spatial projection factors, the exact relationship (√3 × mp ≈ MlD0(qua) = (MlD0 × √Λ) / 4)) reveals a pure 0-D structural law that enables complete deterministic mapping of physical constants, mass spectra, and fundamental parameters across the entire hierarchy of parent and sub-universes without spatial motion. (Note: Replacing Paper I, Chapter 9's initial estimate of ~70 orders of magnitude between adjacent hierarchy universes, the exact derivation here fixes the scale at √Λ ≈ 1.2 × 10-61 orders).
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Conclusion
This appendix bridges the foundational 1899 physics of Max Planck with the modern framework of the Theory of Spheres. By identifying the Planck mass as the quantum manifestation of the quantum baryonic Mass, related to the quantum 0-D light engine by a factor of √3, we eliminate the traditional mystery surrounding the Planck scale.
Furthermore, the exact derivation reveals that this fundamental mass unit of the 0-D oscillator framework is inextricably bound to the four-quarter cosmic topology and the vacuum scale (√Λ – cosmological constant root), establishing a precise, deterministic bridge that maps physical constants and mass spectra across the entire hierarchy of parent and sub-universes without spatial motion.
Eran Shimony, 2026 ©
All rights reserved by the author.
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Related Papers in the Theory of Spheres Series:
'Theory of Spheres – experiment-proposal'
'Theory of Spheres – Part III'
'Theory of Spheres – Part III(a)'