Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

Part V: Why Electrons Form Shells — A Geometric Explanation

Chadwick D Rasque

Published Jan 20, 2026• 0 citations
Publisher

Abstract

This paper explains why electrons arrange themselves in discrete shells around the nucleus, using the MID/QC framework to replace the traditional orbital and probability‑cloud interpretations with a geometric, coherence‑based model. Electron shells emerge as stable standing‑wave surfaces shaped by the proton’s coherence well, with shell capacities determined by the number of phase‑compatible modes the substrate can support at each radius. This geometric interpretation unifies atomic structure, shell capacities, and charge balance under a single, intuitive framework.

DOI
10.5281/zenodo.18321927

Related papers

The Geometric Atom: Deriving Electronic Orbitals as Metric Chladni Patterns from the Recursive Standing Waves of the Vacuum

@pm14552

This monograph extends the General Static-Dynamic Recursive Information Space (GSDRIS) framework to resolve the structure of the Atomic Shell. Standard Quantum Mechanics describes electronic orbitals ($s, p, d, f$) as probability density functions derived from the Schrödinger equation, but offers no geometric explanation for why the electron does not radiate its energy and collapse into the nucleus (the Rutherford Paradox). We demonstrate that the Atom is a Recursive Resonator. Building on the identification of mass as Metric Impedance (Paper 55) and Temperature as Spectral Noise (Paper 68), we define the atomic orbital not as the path of a particle, but as a Metric Chladni Pattern. The atomic nucleus acts as a driver, inducing standing waves in the surrounding 4D-manifold. We analytically derive the quantum numbers ($n, l, m$) as topological coordinates of these standing waves. The principal number $n$ corresponds to the recursive depth (radial nodes), while the angular momentum $l$ corresponds to the surface harmonics (angular nodes). We prove that stable orbitals represent the only geometric configurations where the destructive interference of the recursive flux is zero, effectively trapping the electron in a Holographic Cage of its own resonance.

•Dec 1, 2025•0 citations

The Linear Space Theory of the Universe and the Non-Nucleus Atom Model: A Lattice-Based Interpretation of Action, Spin, and Reality

Ergun, Hamdi Bilgin

This work introduces the Phase–Lattice Atomic Model, a fundamentally geometric and dynamical reinterpretation of atomic structure. Instead of a point-like nucleus and probabilistic orbital clouds, the model proposes that mass–energy is distributed over a spherical surface through discrete, phase-coherent micro-spheres arranged along quantized spiral pathways. Each micro-sphere carries a triple-phase structure separated by 120°, and the requirement of global phase compatibility over successive Planck-scale steps enforces a natural “phase-selection rule” for propagation. This phase-coherent dynamics generates a quantized surface lattice composed of finite spiral windings, with the total number of phase-carrying nodes bounded by the experimentally observed proton–electron mass ratio (≈1836). The interaction between phase delay accumulation and phase-compatibility constraints produces amplitude envelope modulation, giving rise to a natural mechanism for electronic fine-structure splitting—without invoking continuous orbital wavefunctions. The framework provides:• a geometric foundation for spin, exclusion, and handedness;• a discrete mechanism for fine-structure variation;• a surface-dynamical alternative to point-charge quantum orbitals;• a unifying interpretation of classical and quantum behavior based on minimal phase disturbance and least-action pathways. This preprint consolidates the theoretical structure, graphical representations, derivations, and physical implications of the Phase–Lattice model and is intended to make the framework publicly accessible while formal journal submission is in progress.

•Nov 15, 2025•0 citations

The Hydrogen-Like Atom

Uri Peskin

The mere existence of stable atoms, and their arrangement into many-atom nanoscale structures, can only be explained by the laws of quantum mechanics. Indeed, the accumulated evidence regarding the internal structure of atoms motivated the formulation of quantum mechanics at the early twentieth century. Here we focus on single-electron (hydrogen-like) atoms. The dominance of the Coulomb central potential between the electron and nucleus, translates into a simple Schrödinger equation for their relative position distribution. Solving rigorously this equation reveals quantized energy levels, which explains the emission spectra of hydrogen-like atoms (Rydberg formula). The corresponding stationary probability density distributions for the electron around the nucleus are reminiscent of the Bohr model, where classical trajectories (orbits) are replaced by single-electron wave functions (orbitals). The degeneracies of different orbitals, associated with three quantum numbers, will later turn out to be critical for understanding many-electron atoms and the periodic table of the elements.

Cambridge University Press eBooks•May 11, 2023•0 citations

Electrons as Spherical Membranes: A Unified Geometric Framework for Bound and Free States

Jian Zhou

We propose that wave-particle duality is not a fundamental physical phenomenon but a linguistic paradox arising from projecting three-dimensional quantum objects onto incompatible low-dimensional mental models. We propose that the electron in any quantum state is a spherical membrane: the unique SO(3)-invariant surface in three-dimensional space, whose vibration modes are the spherical harmonics. For bound states, the membrane is a standing-wave shell with discrete modes; for free states, it is a propagating wave packet with all modes continuously excited; for scattering states, each mode acquires a phase shift. A single unified definition covers all three cases, with the external potential as the sole distinguishing parameter. This identification is mathematically equivalent to standard quantum mechanics but provides a unified geometric picture that dissolves the particle-wave dichotomy. We extend the framework to multi-electron atoms as nested charged membranes and demonstrate compatibility with sixteen fundamental quantum phenomena. This work is a reinterpretation of quantum mechanics, not a new theory.

•Mar 31, 2026•0 citations

A Structural Interpretation of the Electron in Terms of Coherence and Resonance

Henrik Nilsson

This work presents a structural interpretation of the electron in which its fundamental properties—charge, spin, and interaction—are derived from coherence structure rather than introduced as intrinsic postulates. The electron is described as a stable resonance configuration within a coherence-based field framework, allowing a unified physical interpretation of electromagnetic interaction, electron correlation, and atomic structure. A key aspect of the model is that the electron is not fully confined to three-dimensional space, but extends along an internal coherence dimension. Observable behavior is interpreted as a projection of this deeper structure, providing a structural basis for delocalization, tunneling, and orbital formation. Within this framework, charge arises from asymmetry in lateral coherence, while spin is understood as an internal phase-topological property. Electron–electron interaction is described in terms of coherence interference, offering a structural explanation for repulsion, pairing, and exclusion behavior. Electromagnetic phenomena are interpreted as coherence dynamics: fields correspond to continuous structure, while photons are treated as localized propagating excitations. This provides a unified but differentiated description of electromagnetic interaction. The framework is consistent with established theoretical descriptions while providing a physically interpretable basis for electron structure and its role in electromagnetism and atomic systems. It aims to clarify conceptual foundations rather than replace existing predictive models, and establishes a basis for further work in atomic structure and chemical behavior.

•Apr 12, 2026•0 citations

Atomic Structure: Validation of the Electron Phase Transformation Theory

József Garai

The current consensus on the atomic structure was formulated about a hundred years ago. In the enlightenment of the newly developed instrumentations and theories the electronic structure of the atoms is reconsidered. It has been concluded that in order to comply with experiments the electron must go through phase transformation from point to surface charge when captured by the nucleus. The surface charge electron shell model of the atoms is consistent with all the known features of the atoms and offers a physical explanation for Schrodinger`s wave equation.

Chemical Science International Journal•Nov 29, 2023•3 citations
This paper explains why electrons arrange themselves in discrete shells around the nucleus, using the MID/QC framework to replace the traditional orbital and probability‑cloud interpretations with a geometric, coherence‑based model. Electron shells emerge as stable standing‑wave surfaces shaped by the proton’s coherence well, with shell capacities determined by the number of phase‑compatible modes the substrate can support at each radius. This geometric interpretation unifies atomic structure, shell capacities, and charge balance under a single, intuitive framework.
This monograph extends the General Static-Dynamic Recursive Information Space (GSDRIS) framework to resolve the structure of the Atomic Shell. Standard Quantum Mechanics describes electronic orbitals (s,p,d,fs, p, d, fs,p,d,f) as probability density functions derived from the Schrödinger equation, but offers no geometric explanation for why the electron does not radiate its energy and collapse into the nucleus (the Rutherford Paradox). We demonstrate that the Atom is a Recursive Resonator. Building on the identification of mass as Metric Impedance (Paper 55) and Temperature as Spectral Noise (Paper 68), we define the atomic orbital not as the path of a particle, but as a Metric Chladni Pattern. The atomic nucleus acts as a driver, inducing standing waves in the surrounding 4D-manifold. We analytically derive the quantum numbers (n,l,mn, l, mn,l,m) as topological coordinates of these standing waves. The principal number nnn corresponds to the recursive depth (radial nodes), while the angular momentum lll corresponds to the surface harmonics (angular nodes). We prove that stable orbitals represent the only geometric configurations where the destructive interference of the recursive flux is zero, effectively trapping the electron in a Holographic Cage of its own resonance.
This work introduces the Phase–Lattice Atomic Model, a fundamentally geometric and dynamical reinterpretation of atomic structure. Instead of a point-like nucleus and probabilistic orbital clouds, the model proposes that mass–energy is distributed over a spherical surface through discrete, phase-coherent micro-spheres arranged along quantized spiral pathways. Each micro-sphere carries a triple-phase structure separated by 120°, and the requirement of global phase compatibility over successive Planck-scale steps enforces a natural “phase-selection rule” for propagation. This phase-coherent dynamics generates a quantized surface lattice composed of finite spiral windings, with the total number of phase-carrying nodes bounded by the experimentally observed proton–electron mass ratio (≈1836). The interaction between phase delay accumulation and phase-compatibility constraints produces amplitude envelope modulation, giving rise to a natural mechanism for electronic fine-structure splitting—without invoking continuous orbital wavefunctions. The framework provides:• a geometric foundation for spin, exclusion, and handedness;• a discrete mechanism for fine-structure variation;• a surface-dynamical alternative to point-charge quantum orbitals;• a unifying interpretation of classical and quantum behavior based on minimal phase disturbance and least-action pathways. This preprint consolidates the theoretical structure, graphical representations, derivations, and physical implications of the Phase–Lattice model and is intended to make the framework publicly accessible while formal journal submission is in progress.
The mere existence of stable atoms, and their arrangement into many-atom nanoscale structures, can only be explained by the laws of quantum mechanics. Indeed, the accumulated evidence regarding the internal structure of atoms motivated the formulation of quantum mechanics at the early twentieth century. Here we focus on single-electron (hydrogen-like) atoms. The dominance of the Coulomb central potential between the electron and nucleus, translates into a simple Schrödinger equation for their relative position distribution. Solving rigorously this equation reveals quantized energy levels, which explains the emission spectra of hydrogen-like atoms (Rydberg formula). The corresponding stationary probability density distributions for the electron around the nucleus are reminiscent of the Bohr model, where classical trajectories (orbits) are replaced by single-electron wave functions (orbitals). The degeneracies of different orbitals, associated with three quantum numbers, will later turn out to be critical for understanding many-electron atoms and the periodic table of the elements.
We propose that wave-particle duality is not a fundamental physical phenomenon but a linguistic paradox arising from projecting three-dimensional quantum objects onto incompatible low-dimensional mental models. We propose that the electron in any quantum state is a spherical membrane: the unique SO(3)-invariant surface in three-dimensional space, whose vibration modes are the spherical harmonics. For bound states, the membrane is a standing-wave shell with discrete modes; for free states, it is a propagating wave packet with all modes continuously excited; for scattering states, each mode acquires a phase shift. A single unified definition covers all three cases, with the external potential as the sole distinguishing parameter. This identification is mathematically equivalent to standard quantum mechanics but provides a unified geometric picture that dissolves the particle-wave dichotomy. We extend the framework to multi-electron atoms as nested charged membranes and demonstrate compatibility with sixteen fundamental quantum phenomena. This work is a reinterpretation of quantum mechanics, not a new theory.
This work presents a structural interpretation of the electron in which its fundamental properties—charge, spin, and interaction—are derived from coherence structure rather than introduced as intrinsic postulates. The electron is described as a stable resonance configuration within a coherence-based field framework, allowing a unified physical interpretation of electromagnetic interaction, electron correlation, and atomic structure. A key aspect of the model is that the electron is not fully confined to three-dimensional space, but extends along an internal coherence dimension. Observable behavior is interpreted as a projection of this deeper structure, providing a structural basis for delocalization, tunneling, and orbital formation. Within this framework, charge arises from asymmetry in lateral coherence, while spin is understood as an internal phase-topological property. Electron–electron interaction is described in terms of coherence interference, offering a structural explanation for repulsion, pairing, and exclusion behavior. Electromagnetic phenomena are interpreted as coherence dynamics: fields correspond to continuous structure, while photons are treated as localized propagating excitations. This provides a unified but differentiated description of electromagnetic interaction. The framework is consistent with established theoretical descriptions while providing a physically interpretable basis for electron structure and its role in electromagnetism and atomic systems. It aims to clarify conceptual foundations rather than replace existing predictive models, and establishes a basis for further work in atomic structure and chemical behavior.
The current consensus on the atomic structure was formulated about a hundred years ago. In the enlightenment of the newly developed instrumentations and theories the electronic structure of the atoms is reconsidered. It has been concluded that in order to comply with experiments the electron must go through phase transformation from point to surface charge when captured by the nucleus. The surface charge electron shell model of the atoms is consistent with all the known features of the atoms and offers a physical explanation for Schrodinger`s wave equation.