In standard quantum mechanics, the electron shell filling rule is explained using four quantum numbers (), the Pauli exclusion principle, and spin as a fundamental property of the electron. This is not an explanation – this is a description. The question remains: why exactly ? In the TVM framework, the rule ceases to be an empirical rule or a consequence of quantum postulates. It becomes a geometric necessity – a direct consequence of: The radius of the -th level: (derived in the main work) The wavelength at the fundamental level: (derived in the First supplementary work) The condition of wave stationarity – so that destructive interference does not occur Starting from these TVM results, the work derives the rule as follows: The circumference of the orbit at level is The minimum distance between two waves on the same orbit is The maximum number of waves that can fit on the circumference is The work also derives the geometric origin of the subshells, showing that they are not spatial orientations, but places where extrema (maxima and minima) of the stationary wave on a circular orbit are realized. The number of possible configurations (arrangements of extremum points on the circle, not spatial orientations) relative to zero points gives the capacities 2, 6, 10, 14..., and their sum yields . This demonstrates that: Spin does not exist as a phenomenon – the factor 2 originates from geometry, not from spin The Pauli principle is not a separate law – it is a description of a geometric constraint Orbital quantum numbers are not necessary – at the fundamental level, comes directly The periodic system is not empirical – it is geometrically necessary The rule is not a hypothesis – it is recognized in the TVM structure. It has always been there, in the geometry of the circle, the wave nature of the electron, and the condition of wave stationarity to avoid destructive interference. Keywords: TVM, shell filling, , geometry, wavelength, interference, Pauli principle, periodic system, s, p, d, f, stationary state, standing wave. NOTE: This is the second in a series of supplementary scientific papers that elaborate the application of the TVM formalism to specific physical phenomena. Starting from the results of the First supplementary work from the series (de Broglie relation), this work derives the electron shell filling rule from pure geometry, without introducing spin, the Pauli principle, or orbital quantum numbers. It is recommended that the reader first study the First supplementary work [2], as well as Chapters 3 and 4 of the main work [1].
ABSTRACT In standard quantum mechanics, the spectrum of the harmonic oscillator: is derived by solving the Schrödinger equation for a parabolic potential. This is a mathematical formalism that gives correct numerical results but does not explain why the levels are as they are – why there is a ground state , why the levels are equally spaced, why the difference between them is . In the TVM framework, the spectrum of the harmonic oscillator is recognized through two independent physical processes that are already embedded in the mechanism of the Universe's evolution: Evolution of the first level – how the energy of the original, primary level decreases through successive transformations. Hierarchy of new levels – how, at each transformation, part of the energy separates from the previous level and forms a new, independent level. These two processes are simultaneous and correlated: the first level cannot evolve without creating new levels, and new levels cannot arise without the evolution of the first level. They are two sides of the same process – the impossibility of perfect division upon reaching the transformation limit . In TVM, the ground state of the harmonic oscillator is not a "zero-point energy of the vacuum" – it is a consequence of the first transformation. The formula: comes directly from the TVM mechanism and fractal recursion, where each new level has its own frequency , and the product of amplitude and momentum grows as . Thus, the harmonic oscillator ceases to be a "special" quantum system – it becomes a general pattern that appears wherever a system undergoes a series of transformations, from the initial state of the Universe to local quantum phenomena. Keywords: TVM, harmonic oscillator, spectrum, fractal recursion, transformation, stationarity, energy defect. NOTE: This is the third in a series of supplementary scientific papers that elaborate the application of the TVM formalism to specific physical phenomena. Starting from the foundations laid in the zeroth paper [1], this work derives the harmonic oscillator spectrum from the transformation mechanism and the hierarchy of levels. It is recommended that the reader first study the First supplementary work (de Broglie relation) [2] and the Second supplementary work (Electron shell filling) [3], as well as Chapters 3 and 4 of the main work [1].
This paper presents a geometric and wave interpretation of energy quantization in the hydrogen atom, based on the de Broglie closure condition of the electron wave in a circular orbit. In this concept, energy quantization is a secondary phenomenon resulting from the fact that the electron wave in each orbit consists of exactly n full periods, and the transition to level n+1 corresponds to the addition of one full period. Combining the wave condition with the classical equilibrium of the Coulomb and centripetal forces leads to values of the orbital radii and a discrete energy spectrum consistent with solutions of the Schrödinger equation for the hydrogen atom. It is shown that energy quantization can be interpreted as a consequence of the resonant nature of the electron's wave nature and the condition of phase uniqueness after a complete orbit around the nucleus.1 The model under consideration is semiclassical in nature and serves as an intuitive representation of known results from quantum mechanics. It is assumed that the allowable states correspond to configurations in which the de Broglie wave forms a standing wave containing an integer number of full periods around the orbital circumference. This condition leads directly to the quantization of angular momentum according to the Bohr model. The waveband model provides a one-dimensional analogy of the full quantum description and can serve as a teaching tool to facilitate understanding the geometric aspects of energy quantization in the hydrogen atom. It demonstrates that energy quantization is a natural consequence of the standing wave geometry and the addition of successive full periods along the orbit as the system transitions to the next energy eigenstate after activation. The electron is a stable eigenstate of a quantum field whose behavior in bound systems can be geometrically interpreted as the self-resonance of a de Broglie wave satisfying the condition of single-valent phase, rather than as a local particle with a classical trajectory. Self-resonance of a wave means that the condition of single-valent phase of the wave function after a complete orbit around the nucleus is satisfied, i.e., the requirement that the phase change be an integer multiple of 2π; this is equivalent to the Bohr–de Broglie condition for the closure of the de Broglie wave.
This work extends the Methane Metauniverse (MMU) framework by introducing a projection principle that connects the internal elastic geometry of the dual-tetrahedral MMU cell to the observable quantum numbers of atomic physics. The MMU model describes spacetime using three internal modes: an electric stretching mode 𝑤2, a torsional mode 𝑤3, and a volumetric mode 𝑤4. In this paper, these internal modes are shown to project deterministically onto the radial, angular, and spin degrees of freedom of the Schrödinger equation. A central result of this study is that the three coupled torsional motions of the tetrahedral faces naturally generate the familiar angular-momentum spectrum 𝐿2=𝑙(𝑙+1)ℏ. This spectrum emerges mechanically from the geometry rather than from group-theoretic postulates. By equating torsional kinetic and elastic energies, the MMU further yields a new identity 𝐿2=𝑚𝑐𝑎, linking angular momentum to the internal scale of the tetrahedral cell. The dual nature of the torsional mode provides a geometric origin for spin-½ and enforces the Pauli exclusion principle as an intrinsic occupancy rule. Summing all angular sublevels reproduces the shell capacities 2,8,18,32,… , showing that orbital structure follows from internal symmetry rather than quantum statistics. This projection principle offers a unified geometric explanation for quantum orbitals, spin, and Pauli exclusion within the elastic architecture of the MMU spacetime model.
Abstract The aim of this work is to determine the dependence of the total energy of the atomic system on the mutual orientation of the orbital angular moments of electrons. It is shown that the minimum value of the energy is attained for those configurations that are an eigenfunction of the orbital angular momentum operator L ̂ 2 . Thus, at least for individual atoms of the periodic table it is shown that Dirac’s postulate on the necessity of a wave function to be an eigenfunction of L ̂ 2 is a consequence that follows from the properties of the Hamiltonian, and it is not a complementary condition. Additionally, it is shown that the second Hund’s rule is incorrect with respect to the mutual orientation of the electron orbital angular momentum projections.
This paper fully recognizes repeatable experimental facts related to atomic spectral observations electron shell arrangement and energy level transitions and respects all observed phenomena. It does not deny the statistical laws corresponding to energy level effects and the Pauli exclusion effect. The core purpose of this paper is to correct the long standing model inversion misunderstanding in quantum theory that is treating artificial definitions and mathematical approximation diagrams used by humans to simplify calculation and facilitate academic communication directly as physical reality in the microscopic world. This paper clearly distinguishes practical calculation rules from underlying physical mechanisms. Induced laws for human deduction and analysis cannot replace the true interaction nature of particles. This paper constructs a microscopic atomic picture based purely on mechanical balance with the core conclusions as follows. Atomic energy levels are not discrete geometric thin lines but continuous spherical potential energy intervals centered on atomic nuclei. The steady state of electron orbits arises from the combined balance of Coulomb repulsion and nuclear attraction instead of being preset in advance. The textbook diagrams showing intersecting orbits and dynamic avoidance of electrons are only effects of planar simplified drawing and do not represent the true microscopic state. Electrons in the same layer maintain stable equal spacing by Coulomb repulsion but spherical surfaces of inner and outer layers have different potential energy gradients leading to obvious differences in energy level gaps and finally differentiated transition spectra. Segmentation of visible light spectra originates from the difference in photoreception of cone cells in human eyes. Man is the measure of all things. Classification of spectra and energy levels by discrete criteria comes from human observational induction while natural ontology remains continuous. The Pauli exclusion effect is not an innate axiom of the universe but a natural mechanical result formed jointly by three dimensional space Coulomb repulsion and potential energy balance. This paper takes mechanical causality as the only main line to uniformly explain atomic energy levels spectral features and electron arrangement rules. All inferences strictly rely on experimental observations.
Abstract Where does the integrality of quantization conditions come from? Standard quantum mechanics treats it as a fundamental postulate, or attributes it to the single-valuedness of the wave function, without asking whether a deeper geometric reason exists. This paper argues, starting from the winding structure of closed standing waves, that integrality is a consequence of geometry. The fundamental object of the paper is the rotating standing wave, which possesses three simultaneous degrees of freedom: breathing (radial oscillation), spin, and orbital motion. First layer: the core proof of circulation quantization. The total length of the closed path is
2π. The even-number theorem shows that the antisymmetry condition forces zeros to appear in pairs, so an antisymmetric structure necessarily exists. The phase-locking condition states that one full rotation advances one full wave; the closed path accommodates
Q full waves, so the total effective number of half-wave steps is
2LQ. The step length is \Delta\phi = \frac{\pi}{LQ}. The path integral gives \oint \mathbf{P} \cdot d\mathbf{l} = 2LQ \cdot \Delta, and the quantum number is n = 2LQ, where
L∈{1,1/2} and
Q∈Z+, so that
n is always an integer. Comparison with the Bohr-Sommerfeld condition \oint \mathbf{p} \cdot d\mathbf{l} = 2\pi n \hbar = n h yields the correspondence
n=2LQ,
Δ=h=2πℏ. Within the present framework, the integral part of the quantization condition is a theorem rather than a postulate; the proportionality constant
Δ=h is calibrated by experiment. Second layer: the geometric origin of the spin dichotomy. As the rotating standing wave completes one orbital revolution, it simultaneously spins. A spin-to-orbit ratio of
1:1 sweeps out a cylindrical band whose surface normal is restored after one circuit; a ratio of
1:2 sweeps out a Möbius band whose surface normal is restored only after two circuits. Other ratios fail to satisfy the planar gluing condition and do not close. Hence only two topologies of closed band exist, and spin admits only integer and half-integer values. Third layer: how the standing wave realizes
1:2. In a standing wave,
E and
B are spatially orthogonal, and a phase difference of exactly
90∘ corresponds to the dead point of a synchronous motor--an unstable equilibrium. Any infinitesimal perturbation introduces a phase offset, and the system, once displaced from the dead point, rotates at the synchronous speed. Each half-wave contains one pair of magnetic poles: a pole-pair number
p=1 corresponds to the ratio
1:1, and
p=2 to the ratio
1:2. Energy minimization and frequency locking jointly determine the critical state. Fourth layer: the geometric essence of parity non-conservation. The Möbius band is non-orientable and possesses no global left-right chirality. Topological inertia is unidirectional: two Möbius bands combine into a Klein bottle automatically and at low energy, whereas a cylinder is cut, flipped, and reglued into a Möbius band only at high energy and unidirectionally. The weak interaction is precisely this unidirectional flip operation. Parity non-conservation is therefore not a dynamical symmetry breaking but a manifestation of topological inertia. Fifth layer: the geometric origin of magnetism. The pole-pair number
p∈Z+, and magnetic poles must appear in pairs. Within the closed-standing-wave framework,
∇⋅B=0 is demoted from a postulate to a corollary. Verification: at the particle scale, 328 massive particle modes yield a hit rate of
99.4% (
326/328) in the
L=1/2 spectrum, with
p=1.18×10−71. For PDB biological macromolecules, 40 independent configurations yield a hit rate of
95.0% (
38/40) under the prior-
L strategy, with
p=1.21×10−6; the counterexamples A-RNA and A-DNA share the same deviation value of
32.7∘, rooted in the disruption of phase locking by the C3'-endo sugar-ring conformation. At the macroscopic scale, 55 Galactic cool stars (Bowler 2023) yield a hit rate of
90.9% (
50/55) under the prior-
L strategy, with
p=8.47×10−3. For orbital resonances, 152 pairs show a small-integer-ratio (
≤10) fraction of
92.8% (
141/152), with a direct-resonance hit rate of
100%. The structural distribution of both hits and counterexamples jointly constitutes the verification: the theory predicts not only the phase-locking spectrum but also the mechanism by which phase locking is disrupted. This paper introduces no wave function, no Hilbert space, and no formal postulates of quantum mechanics. Its core claims are that the circulation quantization condition arises from geometry; the integrality of quantization arises from geometry; the spin dichotomy arises from the spin-to-orbit ratio; parity non-conservation arises from topological inertia; magnetism arises from the standing-wave transposition sequence; and the proportionality constant is calibrated by experiment. The paper further argues that fermions and bosons are two configurations of the same topological entity (the Möbius band and the cylinder), that the direct topological transformation between them is geometrically permitted, that quantum entanglement corresponds to topological winding, and that self-replication corresponds to the topological splitting of a twisted surface. Statement of epistemic status: this paper makes no numerical predictions and does not replace the computational power of standard quantum mechanics. It addresses the question of "why"--why the quantization condition is integral, why spin admits only two types, why parity is not conserved. The Standard Model addresses "what"--it predicts all observed values precisely. The two are complementary and do not compete. The mode of verification in this paper is structural: if the theory is correct, the phase-locking spectrum should appear in three-scale data, and counterexamples should cluster around the theoretically predicted disruption mechanisms. This paper derives the geometric origin of the integer
n in the quantization condition and does not claim to derive the numerical value of Planck's constant, which is calibrated by experiment.