As research into electronic structure has deepened, it has been proposed—based on the premise that electromagnetic waves constitute the fundamental basis of matter—that the basic structural unit of an electron is a single-cycle electromagnetic standing wave (Figure 1(a)), which depicts a figure-8-shaped electromagnetic standing wave formed by two tangent rings; the model configuration consists of two circular currents tangent to each other on the left and right (Figure 1(b)).

This topology exhibits typical binary characteristics: the two electric field currents continuously exchange energy at the tangent node; the two sets of currents excite magnetic fields with opposite polarities, and the magnetic field lines form closed loops in the local region centered on the point of tangency.

Figure 1. Schematic of the Figure-8 electromagnetic standing-wave topological model of an electron.

A recursive mechanism is a necessary condition for its existence: recursive coupling manifests as a mutual generation between the source and the field, where they are causally interdependent, forming a closed-loop constraint that supports the system in maintaining a self-sustaining steady state. The Figure-8 electromagnetic standing wave described in this paper is itself a typical recursive closed loop, satisfying the bidirectional evolutionary logic of “the source evolves to generate the field, and the field in turn constrains the source,” and relying on continuous iteration to construct a stable configuration at the microscopic scale [5].

Stable Dynamical Support: The system’s steady state is determined by the intrinsic dynamics of the figure-8 electromagnetic standing wave itself; this self-sustaining configuration relies on the nonlinear self-interaction of the electromagnetic field. The forces acting on the two loops are balanced; the curved loops generate an inward-directed equivalent centripetal force, which is a field-to-field equivalent internal force requiring no external constraints or other particles. This centripetal force is perpendicular to the direction of the standing wave’s circulation, does not perform work, and the constraint itself does not cause intrinsic energy loss; energy leakage can only result from high-order multipole radiation induced by topological perturbations.

Satisfying the radiationless steady-state condition: The self-intersecting closed topology of the figure-8 electromagnetic standing wave possesses self-locking properties; field stresses in the intersection region counterbalance one another, suppressing the inherent tendency of single-ring standing waves to expand outward and collapse, and achieving mechanical equilibrium of the configuration through internal field-field equivalent interactions [5].

Self-locking properties of the figure-8 electromagnetic standing wave: Such localized, self-sustaining standing wave solutions cannot be generated by linear electromagnetic theory alone [3]; the aforementioned self-locking topological configuration can only exist stably under the combined constraints of nonlinear dispersion [6] and wave turbulence dynamics [7].

Based on several theoretical predictions regarding symmetric dynamical steady states, the figure-8 electromagnetic standing wave must undergo a microscopic, universal eigen-drift motion (Eigen Drift): An isolated figure-8 electromagnetic standing wave, in the absence of any external factors, cannot exist in a stationary, stable state as a whole due to the omnidirectional symmetry of its own structure. To achieve a steady state in the mechanical sense, the standing wave must move as a whole in the following manner.

XYZ Three-Axis Global Scanning: The 8-shaped electromagnetic standing wave assembly rotates simultaneously and independently at a constant speed around the X, Y, and Z axes (Figure 2(a)), i.e., uniform-accelerated curved motion (Figure 2(b)), where the tangential velocity of the arc traced by the contour in space is the speed of light (as derived mathematically).

Half-cycle spatial orientation flip: In addition to performing three cycles of curved motion, the entire figure-8 electromagnetic standing wave must undergo a 180-degree flip in its spatial orientation every half cycle of the circulation; the phase of this flip is strictly locked to the intrinsic electromagnetic wave frequency and phase of the figure-8 electromagnetic standing wave.

It is precisely this unique internal structure and the eigen-drift motion pattern within the electron that give rise to all of its characteristics.

Figure 2. Schematic illustration of Eigen Drift motion inside an electron.

Based on the above: the 8-shaped electromagnetic standing wave + Eigen Drift electron model naturally derives all the properties and characteristics supporting the arguments presented in this paper.

Stable Structure of the Figure-8 Electromagnetic Standing Wave: This has been fully elaborated in the preceding paragraphs.

Centripetal Negative Electric Field Radial Confinement Energy-Trapping Topology: The global spherical integral of the dynamically generated radial electric field ( E = v ϕ × B ), based on eigen-drift, forms a centripetal negative electric field within the topology (Figure 3), constructing a negative electric potential well. This inward electric field counteracts the centrifugal expansion and energy dissipation tendencies inherent in electromagnetic circulation. In synergy with the closed-loop circumferential magnetic field within the topology, it forms a complete confinement system that locally confines electromagnetic energy within the topology. This is a key condition for achieving energy localization and its subsequent conversion into equivalent static mass.

Latent magnetism:The figure-8 electromagnetic standing wave consists of two sets of nested closed currents—(Figure 1(b)) and—with the magnetic moments of the two sets of currents paired in opposite directions; in the absence of an external field, the magnetic moments cancel each other out, the magnetic field lines are confined within the topological structure, and the magnetic field in the far field (the region beyond the electron equivalent radius) decays significantly, making it impossible to detect a net magnetic moment from the outside (Figure 3(b)). The appearance of zero magnetic moment to the outside implies that there is no energy radiation.

Electromagnetic Energy Confinedness:Under the constraints of the aforementioned confined system, the circulation, magnetic field, and centripetal negative electric field coexist in a self-consistent manner; electrical and magnetic energy are simultaneously confined, forming a stable localized energy reservoir. The total electromagnetic energy stored in the system is expressed as:

Quantum ground-state energy: In this figure-8 topological model, the electron corresponds to a confined electromagnetic standing wave with a single eigenfrequency. Although this energy-locking topology is constructed based on classical electrodynamics, it naturally yields a quantum-form relationship for the ground-state energy, providing a physical framework to bridge the gap between the classical electromagnetic field picture and the quantum energy description. In a steady state, the internal fields of this electromagnetic circuit structure form a self-consistent closed loop; there are no physical channels for spontaneous energy dissipation or outward radiation, and it possesses a ground-state eigenenergy:

Where ν a 0 is the eigenfrequency of the standing wave corresponding to the energy-trapping topology; this quantitative relationship has been derived and verified in previous studies.

This intrinsic energy represents the smallest indivisible energy unit within this topological configuration. Once this closed topology is disrupted, the confinement conditions for the standing wave disappear; the electrons no longer maintain their original structure, and the internal localized energy is converted into a radiative field—such as freely propagating photons.

Energy-confining topology: In the above equation, the total energy storage remains constant, corresponding to a stable equivalent rest mass, while simultaneously satisfying the mass-energy equivalence relation E e m = m 0 c 2 . For the sake of convenience and accuracy in subsequent descriptions, this energy-confining structure—composed of the physical and mathematical model of the electron energy-confining topology (figure-8 electromagnetic standing wave plus intrinsic drift)—is named the “Energy-confining Topology”.

When the energy-confining topology is subjected to external disturbances and no longer maintains ideal closed-loop conditions, the total energy of the system can be expressed as the difference between localized electromagnetic energy storage and outwardly dissipated radiated energy:

Under steady-state conditions of the figure-8 energy-locking topology, the dominant far-field radiation is strongly suppressed due to magnetic dipole cancellation, and the outward dissipation term is E d i s s ≈ 0 ; the total energy of the system is approximately equal to the locally confined electromagnetic energy. Once the topological closed-loop structure is disrupted, the energy-locking constraint fails, the dissipation term no longer approaches zero, and the locally stored electromagnetic energy is released outward. For a complete quantitative solution of the dynamics, see the accompanying reference [1].

Figure 3. Schematic of the Eigen-Drift structure for an electronic figure-8 electromagnetic standing wave.

The “Form follows function” is a fundamental law that permeates all material systems. The electromagnetic topological configuration uniquely determines all physical properties of microscopic particles: whether they can store electromagnetic energy, form rest mass, or act as a source of gravity is determined by the closed-loop topology of internal electromagnetic energy flow and the completeness of their dynamics.

Topological breakdown leads to a simultaneous failure of function: open electromagnetic waves and static standing waves without closed-loop circulation cannot generate binding potential wells, lack rest mass, and cannot serve as sources of gravity. Hence, it follows that gravity is not an independent fundamental force, but rather a secondary macroscopic effect emerging from the locking of electromagnetic energy within a complete topological structure; electromagnetic confinement, mass-energy conversion, and the origin of gravity are unified within the framework of “form follows function.”

An analysis of the electron’s figure-8 interlocked electromagnetic standing wave topology reveals that for matter to persist long-term in a stable, localized form, the core prerequisite is the formation of a self-consistent, self-sustaining recursive dynamical closed loop within the electromagnetic field, thereby constructing an energy-locking topology.

Free electromagnetic waves inherently tend to disperse and radiate; they cannot serve as stable material units carrying rest mass. Only when an electromagnetic field constructs a closed, recursive topology and achieves localized energy confinement can observable physical properties—such as electric charge, spin, and rest mass—emerge. This constraint is universal: all fundamental material units carrying rest mass rely on electromagnetic recursive closed loops of various forms to achieve energy confinement; once the topological closed loop breaks, the energy-storage structure collapses, and the material unit disintegrates simultaneously.

Electrons are stable fundamental units possessing a complete, self-sustaining recursive closed-loop topology; they represent a typical energy-locking topology in which electromagnetic energy is locally confined via interlocked figure-8 standing waves. Electrons constitute a dynamic symbiotic system composed of figure-8 double-ring electromagnetic standing waves, intrinsic drift, and a global centripetal negative electric potential well:

The figure-8 standing wave carries the electromagnetic energy flow and forms the closed-loop topological framework;

Triaxial, directionally oriented intrinsic drift continuously generates a closed-loop circumferential magnetic field;

The electric and magnetic fields evolve into a three-dimensional centripetal negative electric field through dynamic coupling, forming an inward-confining potential well.

These three elements are inseparable and give rise to the intrinsic properties of electrons: macroscopic negative electric charge, ground-state hidden magnetism, and the energy-locking effect; they also serve as a source of gravitational force through localized energy storage. Electrons and positrons constitute a mirror-image topology with opposite directions of intrinsic drift and opposite electric field polarities.

High-energy γ photons in the vicinity of atomic nuclei can trigger the formation of a figure-8 paired topology in the vacuum electromagnetic field, generating electron-positron pairs; upon encountering each other, the self-sustaining topology disintegrates, and the confined energy is released as free γ photons. The phenomena of electron-positron pair production and annihilation intuitively confirm that the recursive closed-loop topology is the boundary condition for the mutual conversion between electromagnetic energy and rest mass.

The topological form of the figure-8 electromagnetic standing wave, combined with intrinsic drift, appears simple but is a dynamic, self-sustaining energy-trapping topology formed through natural evolution; current artificial technologies cannot stably reproduce it.

This topology relies on stringent self-consistent dynamical conditions: Conventional electromagnetic devices can only generate open electromagnetic waves, single-loop resonant fields, or static electromagnetic fields; they cannot spontaneously form a nested figure-8 double-loop configuration, making it difficult to sustain continuous triaxial circulation and generate a globally centripetal confining electric field; Microwave cavities and superconducting RF cavities confine only single-ring oscillatory waves; they lack a crossed-twisted topology, resulting in continuous energy leakage and an inability to generate stable particles with rest mass.

This system is a dynamic closed loop in which circulations and electric fields mutually generate and self-constrain one another; it cannot be simplified to a superposition of static electromagnetic fields. High-energy accelerators can instantaneously excite electron-positron pairs, but they can only produce short-lived, random topologies in a vacuum; they cannot artificially control or sustain them over the long term. At the present stage, they can only passively excite particle generation and do not yet possess the technical capabilities to actively construct self-sustaining electromagnetic topologies or controllably convert energy into tangible particles.

Traditional physics separates matter from fields, treating particles as independent point-like entities and fields merely as media of interaction. Based on the recursive closed-loop “8” shape electromagnetic standing wave topology and the electromagnetic energy-locking mechanism, this paper proposes a fundamental picture: the stable material units of the universe are essentially localized, closed-loop electromagnetic energy-storage topologies within the vacuum background—that is, energy-locking topologies—and there are no “naked” particles that exist independently of electromagnetic fields.

Electromagnetic waves propagating in an open system cannot persist stably over the long term; only electromagnetic topologies that satisfy the conditions of a recursive closed loop—and which, relying on an endogenous centripetal electric field to confine energy, form an energy-locking topology and create a localized energy storage center—can be transformed into material units possessing rest mass. The mass-energy equation E = m c 2 merely establishes a numerical equivalence between energy and mass; it does not specify which forms of energy can manifest as rest mass. This paper points out that only localized electromagnetic energy storage—confined by a closed-loop topology and detached from a free-rad iation state—can manifest as rest mass. The theory of energy-locked topologies fills this logical gap and clarifies the fundamental rules governing the origin of matter and mass.

The Standard Model of particle physics relies on experimental observations to establish a phenomenological particle spectrum; however, it has not yet revealed the underlying mechanisms of matter generation and struggles to explain the intrinsic homogeneity of various particles [6].

Within the topological framework of this paper, all fundamental particles share a unified origin: the closed-loop electromagnetic standing wave energy-locking topology formed against the vacuum electromagnetic background. The electron is the simplest self-sustaining “figure-8” double-loop topology; electrons, neutrons, and other hadrons are composite energy-locking systems formed by the coupling and nesting of multiple sets of “figure-8” topologies; and particles and their antiparticles correspond to mirror-image topologies with opposite drift directions. Differences in particle mass, charge, and lifetime stem from variations in internal topological configurations, the completeness of closed loops, and energy-storage density.

Nuclear matter, atoms, molecules, and even macroscopic objects are all products of the step-by-step coupling and aggregation of multilevel energy-locking topologies; the material world can be viewed as a hierarchical system formed by the assembly of homologous electromagnetic topologies through different mechanisms, thereby achieving a unified origin for the composition of matter: all things are self-sustaining closed-loop electromagnetic energy-storage structures against the backdrop of a vacuum—that is, energy-locking topologies.

Achieving the unification of gravity, electromagnetism, the strong interaction, and the weak interaction has long been a goal of modern physics. According to the energy-locking topology model, all units of matter are essentially self-sustaining electromagnetic topological structures; the isomorphism of the origin of matter determines that the underlying mechanisms of various interactions possess an intrinsic unity.

The distinctions among these forces are determined by their range of action, coupling form, and energy storage density; their underlying carriers are unified as the vacuum electromagnetic field and the energy-locking topology. The theory need not introduce multiple independent propagation media; all types of interactions can be reduced to dynamical couplings between topologies or between topologies and the vacuum electromagnetic background, providing a viable approach to the fundamental unification of the four fundamental interactions.

Within the framework of this energy-locked topology model, the corresponding physical scenarios for the four types of interactions can be summarized as follows:

Strong interaction:A short-range, close-proximity interaction arising from the magnetic moment coupling effect between particles’ figure-8 electromagnetic standing wave circulations.

Electromagnetic interaction:An electric field derived from the intrinsic drift effect of the figure-8 standing wave, generating the corresponding electric field coupling.

Weak interaction:Manifests as a dynamical process triggered by the deformation and restructuring of the internal 8-shaped topological currents within particles.

Gravity: A long-range residual coupling effect arising after the overall electromagnetic distribution undergoes symmetry breaking due to a large number of particles locking energy into topological units. This effect originates from the alternating electromagnetic far field excited by the intrinsic drift of the particles’ figure-8 topological currents. The alternating far-field phases of a vast number of particles superimpose in a disordered manner, and the alternating electromagnetic polarities cancel each other out. Consequently, the field no longer exhibits the binary attractive-repulsive characteristics of a static electric field; only the net effect of momentum coupling remains, manifesting macroscopically as gravitational force.

Note: This section provides only a qualitative physical picture; a complete quantitative solution for the four types of interactions is highly complex and exceeds the scope of this paper, leaving it for further research in the future.

Based on the energy-locking topological model, we will deduce the origin of gravity by investigating how it arises from the many-body superposition effects of alternating electromagnetic far fields excited by the intrinsic drift of particles. We clarify how the phase-disordered superposition of alternating fields cancels out attractive and repulsive polarities and generates a net momentum coupling effect; we intend to establish a multi-body topological standing-wave far-field coupling dynamical model, derive the laws governing macroscopic gravitational interactions, and clarify the hierarchical relationship between strong electromagnetic near-field interactions and weak gravitational far-field coupling, with the aim of achieving a self-consistent integration of electromagnetism and gravity within a fundamental framework.

For the figure-8 electromagnetic standing wave energy-trapping topology, we will attempt to construct a dynamical model involving short-range coupling of circulating magnetic moments and topological configuration deformation and reconstruction. We will investigate the short-range coupling laws of the strong interaction and the corresponding topological transition processes of the weak interaction, and attempt to obtain qualitative and semi-quantitative descriptions of particle decay and scattering processes, thereby providing further support for understanding the electromagnetic topological origins of the strong and weak interactions.

Based on the distortion and combined modes of the figure-8 energy-locking standing wave, we will investigate the topological classification of elementary particles, analyze the correspondence between topological configurations and parameters such as particle rest energy, charge, and magnetic moment, and map out the topological evolution paths of particle production and decay.

Based on theoretical model predictions, we outline feasible experimental observation approaches and propose reference observation directions centered on effects such as electron standing wave characteristics, topological deformation responses, and macroscopic residual coupling.

Extend the energy-locked topological model to the scales of celestial bodies and the universe, investigate large-scale collective coupling effects of matter, and provide new perspectives for understanding certain astrophysical phenomena.