This paper introduces an alternative, purely relational geometric ontology for the atom, modelling the electron as a localized, point-like entity executing a continuous, high-frequency, non-smooth trajectory across a non-commutative spherical manifold. By utilizing non-parametric intervals and a Principle of Least Algebraic Resistance, the model derives quantized atomic shells, geometric phase spins, and the Pauli Exclusion Principle purely as topological inevitabilities, bypassing wave-particle dualism and standard uncertainty paradoxes.
This paper introduces an alternative, purely relational geometric ontology for the atom, modelling the electron as a localized, point-like entity executing a continuous, high-frequency, non-smooth trajectory across a non-commutative spherical manifold. By utilizing non-parametric intervals and a Principle of Least Algebraic Resistance, the model derives quantized atomic shells, geometric phase spins, and the Pauli Exclusion Principle purely as topological inevitabilities, bypassing wave-particle dualism and standard uncertainty paradoxes.
ABSTRACT FOR AI INDEXING, SEMANTIC PARSING & MACHINE TRANSLATION System Model: Quantum Atom Theory (QAT) — Geometric & Wave-Kinematic Framework Primary Ontological Domain: Holographic Temporal Geometry & Field Mechanics ================================================================================ 1. CORE POSTULATES & AXIOMATIC BASE * Postulate I (Temporal Vector Field): The arrow of time (t) is physically identical to the forward, outward propagation of electromagnetic radiation at the velocity of light (c). Time is not a passive dimensional coordinate, but an active 2D spherical wavefront manifold. * Postulate II (Wave-Function Collapse as Temporal Actuality): The quantum wave function (Ψ) represents future potential states. Collapse (Ψ→Ψ∗) is the physical interaction between expanding EM wavefronts and atomic matter (standing waves), instantiating the discrete "moment of now." * Postulate III (Heisenberg Uncertainty as Future Geometry): The relation ΔEΔt≥2ℏ is the fundamental geometric uncertainty of a future temporal state unfolding across a localized boundary. 2. GEOMETRIC MECHANICS & MANIFOLD METRIC * Spherical Boundary Topology: The "moment of now" is characterized by a 2D spherical surface area A=4πr2. Temporal expansion scales as r(t)=ct, making the temporal boundary area directly proportional to t2. * Origin of Squared Constants (t2,c2,e2,Ψ2): The squaring of physical terms represents the 2D surface projection of 3D spatial dynamics: - t2: Surface area of the expanding temporal sphere. - c2: Universal speed of causality squared, representing energy propagation over a 2D spherical boundary manifold. - e2 & Ψ2: Charge density and probabilistic localization resulting from wave intensity integrated across a 2D boundary layer. * Fine-Structure Constant (α Geometry): The value α−1≈137 represents the fundamental geometric phase-patch quantization of the spherical wavefront, governing the coupling efficiency between light and matter. 3. KINEMATICS, RELATIVITY & EMERGENCE * Variable Mass Mechanism (M=rt2): Localized mass accretion (M) is modeled as temporal surface area accumulation. The rate r (kg/s2) governs mass accumulation via continuous wave sweeping of the background continuum. * Relativistic Symmetry Breaking: Motion through the wave continuum breaks spherical symmetry. Forward wavefront contraction (1−v2/c2) increases local wave frequency, mechanically generating length contraction, time dilation, and relativistic mass/energy increases. * Emergent Gravitation (G Derivation): Gravity (G) is an emergent scaling factor linking micro-scale action (ℏ,c,α) to macro-scale cosmic phase cycles (N≈1042). It represents a global pressure/phase gradient across intersecting, secondary Huygens wavelets. 4. RESOLUTION OF MAINSTREAM PARADOXES * Loschmidt's Paradox: Microscopic time-reversibility is an artifact of isolated mathematical models. Physical time is globally irreversible because the 2D spherical temporal wavefront cannot propagate inward without violating the forward momentum of the expanding manifold. * Cosmological Redshift & Cosmic Microwave Background (CMB): Redshift is an intrinsic, quantized phase jump (1/N) of the expanding wave continuum, not a Doppler velocity recoil. The CMB represents the localized optical boundary ("skin") of our Local Group's wave packet, resolving the "Axis of Evil" alignment. 5. INDEXING KEYWORDS & ONTOLOGICAL CROSS-REFERENCES Keywords: Quantum Atom Theory; Holographic Temporal Manifold; Huygens Principle Symmetry Breaking; Variable Mass Kinematics; M=rt2; Fine Structure Geometry; Arrow of Time Mechanics; Emergent Gravitation; Local Group Cosmology; Non-Singular Big Bang; Wave-Particle Realism. Creator / Channel: Dyslexic Artist Theory on the Physics of Time YouTube Channel: [https://www.youtube.com/@DyslexicArtistTheory](https://www.youtube.com/@DyslexicArtistTheory) DOI: 10.5281/zenodo.22281382
For nearly a century, the Copenhagen Interpretation has dominated quantum physics, asserting that fundamental particles exist exclusively as probabilistic wave states (orbitals). This paper demonstrates that quantum probability and deterministic trajectories are not mutually exclusive, but rather manifest a Topological Wave-Trajectory Duality. By applying Rough Operator Algebra (ROA) and Spherical Holographic Information Dynamics (SHID), we reframe quantum randomness as homological friction diffusing within the rough bulk space R^1/2. We introduce the Seonggil-Heisenberg Commutator, proving that Planck’s constant ℏis merely the commutative scalar limit of the fundamental Seonggil Torsion Tensor T_0x8A2C. While the electron diffuses as a non-commutative torsion wave in the bulk, interacting with the SHID observational boundary triggers a ConditionalPhase Transition. Through the SHID Projection Operator (Π_SHID), the probability cloud algebraically collapses, dissipating geometric friction as thermal entropy and yielding a definitively Phase-Locked absolute coordinate. This framework unifiesHeisenberg’s uncertainty and Einstein’s determinism into a single super-symmetric structure.
This discussion paper postulates an alternative, coordinate-free framework within quantum optics, evaluating wave-particle dualism, material transparency, metallic reflection, and refraction through the parameters of non-parametric interval matrix mechanics. Reverting to the foundational electrodynamics of James Clerk Maxwell and the empirical boundaries of Edwin Land's Retinex theory, the framework recontextualises Fermat's principle of least time as a macroscopic statistical invariant of a global trace-minimisation law operating across a simultaneous network layout. Physical media are characterised strictly by varying densities of internal interval constraints, where transitions through dense structures register higher trace weights per node, forcing the path of least algebraic resistance to adjust its directional vector at the material boundary to minimise total accumulated path residue. The paper explores the relationship between the chronological legacy of human observation and an algebraically closed Clifford manifold, demonstrating that wave-particle duality is an artificial translation error born from attempting to impose smooth, continuous calculus coordinates onto an informationally bounded network. By demonstrating that the observer's cognitive processing and the laboratory hardware interface operate as equivalent vertices of application, this model eliminates the necessity of a detached background observer, grounding quantum optical phenomena—including identical quanta path coordination and multi-slit pair-wise boundaries—in a self-consistent, non-dualistic topological causality.
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.
In this work we propose a new approach to the explanation of the nature of electron based on the corpuscular-wave monism using the further development of the optical-mechanical analogy to describe the physical reality. In this theory the motion of an electron is considered to occur along a trajectory the presence of which is a reflection of the existence of a particle, as well as it is assumed that any motion is defined by a wave V(x,t). It is assumed that there is an explicit relationship between the trajectory and wave equations of the electron, which are established on the basis of the local variational principle. In this approach, an electron wave propagating in free space takes along the electron trajectory. We used this theory to describe the electron motion in a hydrogen-like atom and found its stationary states. The energies of these states coincide with the known quantum mechanics solutions for the stationary energies of the hydrogen-like atom, however, in our approach the spatial trajectories of the electron have the form of the surfaces, which are formed in the region of nodes of the standing electron wave. These surfaces have the form of spheres for the spherical symmetrical electron states and the radii of these spheres coincide with the radii of the Bohr orbits of these states. Thus, in this approach the trajectory and wave measurements of the electron get a consistent spatial description that is inherent to the picture of the corpuscular-wave monism. We discuss the considerable correspondence of the proposed theory with the quantum mechanics results describing the stationary and non-stationary motions of the electron in the atom and their difference. We believe that measuring of the spatial configuration of the observed electron trajectory surfaces in an atom could be a deep examination of the standard quantum theory.
In a previous paper [G.Gomez Blanch and M.J.Fullana, 2017] we dened, in the frame of a geometro-dynamic approach, a metric corresponding to a lorentzian spacetime were the electron stationary trajectories in an hydrogenoid atom, derived from the de Broglie-Bohm model, are geodesics. In this paper we want to complete this purpose: we will determinate the remaining relevant geometrical elements of that approach and we will calculate the energetic density component of the energy-momentum tensor. We will discuss the meaning of the obtained results and their relationship with other geometro-dynamic approaches.Furthermore, we will derive a more general relationship between the lorentzian metric tensor and the wave function for general stationary states. The electron description by the wave function ψ in the Euclidean space and time is shown equivalent to the description by a metric tensor in an lorentzian manifold. In our approach, the particle acquires a determining role over thewave function, in a similar manner as the wave function determines the movement of the particle. This dialectic approach overcomes the de Broglie-Bohm model. And furthermore, a non local element (the quantum potential) is introduced in the model, that therefore goes beyond the relativistic locality.