2026-07-15
Introduction to Wave Electricity
A popular science review article
In the beginning there was a point. Then came numbers—and time was born. Time split, giving rise to space and length. Space split again—and acquired volume.
What if space, particles, fields, and freely propagating radiation are not independently given fundamental entities, but emerge successively from a single geometric foundation? Wave Electricity explores this possibility. The model begins with the simplest possible state: a mathematical point that does not yet possess spatial coordinates. A primordial ordering of events establishes time; directed change of the state gives rise to a temporal field; and orthogonal splittings of that field open the possibility of spatial directions emerging.
The first dynamically organized object in this geometry is a normalized wave state. Its internal closure, external projection, and spatial propagation subsequently manifest as a particle, a field, and free radiation.
In conventional language, an electron is called a particle, an electric field is a particular state of the space surrounding a charge, and a photon is a quantum of radiation. The proposed picture draws a deeper connection between them. A closed, localized wave is observed as a particle. Its internal gradient may extend outward and manifest as a field. If closure is removed, free propagation becomes possible.
\[\tag{1} \boxed{ \begin{aligned} \text{particle}&:\quad \text{the wave is localized and closed},\\ \text{field}&:\quad \text{the closed-state gradient extends outward},\\ \text{radiation}&:\quad \text{the wave is open and propagates freely}. \end{aligned} } \] This does not mean that all physical properties have already been derived from a single formula. Wave Electricity is still being developed. Rigorous algebraic results, additional geometric assumptions, and physical hypotheses must be distinguished. This overview presents the general picture in accessible language and links to studies that examine each element in detail.
The model's current core: its postulates, definitions, scope, and distinction between mathematical results and physical hypotheses are collected in “A Unified Concept of Wave Electricity”.
An Outline of the Model
 |
1. From a Point to Time and Space
In the initial state, internal change and external motion are still absent:
\[\tag{2} \boxed{J(0,0)=1.} \] This is neither a particle placed in empty space nor a coordinate origin with \(x=y=z=0\). Spatial coordinates are not yet defined at this level. The state \(J=1\) should be understood as a mathematical point or elementary event preceding the manifestation of length, direction, and volume.
If such events are distinguishable and ordered, they form a sequence. The numbers in the epigraph refer to the ability to distinguish earlier and later events—to assign them positions in this sequence. Time is introduced as a measure of the primordial ordering of events, which does not yet require motion through an existing space or a change in the value of the operator itself.
The sequence of events establishes a primordial direction, conveniently represented by the temporal coordinate \(X_0=c\tau\mathbf e_0\). Its derivative \(\mathcal T_0\) specifies the universal progression of events. When a particular state \(J\) begins to change along this direction, the temporal field \(\mathcal T_J\) of that wave structure arises:
\[\tag{3} \boxed{ \mathcal T_0=\frac{dX_0}{d\tau}=c\mathbf e_0, \qquad \mathcal T_J=\frac{dJ}{d\tau}. } \] These quantities must be distinguished. For the initial unchanging state \(J=1\), we have \(dJ/d\tau=0\), even though the ordering of events and the direction \(\mathcal T_0\) are already given. Time is therefore not derived from the derivative of an unchanging operator: primordial order is taken as the condition that makes any subsequent change possible.
The poetic wording of the epigraph says that “time split.” More precisely, what splits is the directed temporal field, rather than time as an independent physical substance. An independent orthogonal projection of this field creates a new direction. Once positions can be distinguished and accumulated change measured along it, that direction acquires the meaning of a spatial coordinate and length.
Repeated independent splittings may create further orthogonal directions. One-dimensional extension can thus develop into a plane, and a plane into volume. Idempotent algebra ensures the completeness and orthogonality of the channels, but does not by itself explain why stable observable space has exactly three dimensions. This remains a physical hypothesis of the model.
\[\boxed{ \text{point}\longrightarrow\text{sequence of events}\longrightarrow\text{time}\longrightarrow\text{temporal field}\longrightarrow\text{splitting}\longrightarrow\text{space}. } \] Further details: the transition from a mathematical point to primordial temporal geometry and spatial directions is examined in “The Temporal Field and the Origin of Space”. Its continuation—the transition from temporal dynamics to electric, magnetic, and wave fields—is presented in “Projections of the Temporal Field”.
2. What Wave Electricity Seeks to Explain
Modern physics successfully describes nature through several interconnected theories. Quantum mechanics assigns probabilities to measurements, field theory describes particles as field excitations, and relativity relates space, time, energy, and momentum. Wave Electricity does not reject these results. It seeks an intuitive geometric foundation on which different properties could emerge as manifestations of a single structure.
The central question is whether one can begin with the simplest sequence of events, obtain time and spatial directions, and then proceed through a normalized wave state to internal periodicity, mass, spin, electric fields, atomic scales, and free waves. In this approach, a known physical formula serves as a test of the geometric construction rather than its final goal. The task is both to reproduce the result and to identify the element of the model responsible for it.
For example, internal closure is associated with particle stability, a two-sheet cycle with spinor periodicity, deep splitting with the Lorentz factor, and an external gradient with the electric field. Such correspondences must be distinguished from algebraic properties that follow automatically.
Further details: the general transition from a normalized state to waves and particles is discussed in “A Unified Geometric Model of Waves and Particles”.
3. The Local One-Dimensionality of Motion
The principle of local one-dimensionality, included among the Concept's postulates, states that in each individual physical state a point has one actual direction of motion. This does not mean that the world is one-dimensional or that every particle must follow a straight line. A circle, a helix, and a complex orbit are also one-dimensional curves: a single path coordinate suffices to specify a position on any of them.
Let this coordinate be denoted by \(s\). The observed trajectory is then written as
\[\tag{4} s=s(t), \qquad \mathbf r(t)=\mathbf r\bigl(s(t)\bigr), \qquad \mathbf v=\widehat{\boldsymbol{\tau}}(s)\frac{ds}{dt}. \] The vector \(\widehat{\boldsymbol{\tau}}\) is tangent to the curve. If the trajectory bends, the tangent changes direction, but at every instant motion still follows that one tangent. Three-dimensionality appears in the shape and orientation of the entire curve, rather than in a requirement for three independent instantaneous motions of one point.
The actual trajectory must also be distinguished from the set of possible continuations. A single path is registered in an individual event. Before registration, however, the apparatus geometry may allow a whole family of paths with different phases. This distinction will matter when discussing diffraction.
A separate hypothesis concerning the quasistatic electric response states that a selected source and receiver may be related by a single instantaneous coordinate \(R(t)\) directed along the line joining them. Two different motions producing the same function \(R(t)\) should yield the same quasistatic scalar signal, while circular motion at constant \(R\) should not, by itself, produce a time-varying component of that signal.
Experimental test: two experiments involving rectilinear and circular motion of a charge are proposed in “A One-Dimensional Coordinate of Electric Interaction: Two Experiments with a Moving Charge”. They test the hypothesis that the selected signal depends on the radial coordinate under specified conditions. This response law does not follow from the local one-dimensionality of the trajectory alone: the measured quantity, receiver sensitivity, and conditions under which magnetic, induction, and retardation contributions can be neglected must be specified separately.
Further details: local one-dimensionality, the curvilinear coordinate, and its connection to the phase operator are discussed in “A Unified Concept of Wave Electricity” and “The Schrödinger Equation in the Geometry of the Split Operator”.
4. The Energy Balance of the Particle and the Source
A normalized geometric state must be distinguished from the physical law of energy conservation. If a particle accelerates in an external field, its laboratory-frame energy increases. This increase does not arise from nothing, nor must it be compensated solely by a decrease in the particle's internal energy. The source of the applied influence is also part of the process.
The full balance must therefore be written for a closed system:
\[\tag{5} \boxed{ E_{\mathrm{int}}+E_{\mathrm{motion}}+E_{\mathrm{source}}=\operatorname{const}. } \] When an electric field accelerates an electron, the electron gains energy through work done by the field source. When an atom emits a photon, the internal energy of the bound state decreases by the radiated energy. Geometry describes the allowed relations between states, but does not remove the need for a complete energy balance.
Further details: the relativistic energy change associated with a change in the external parameter is discussed in “Relativistic Force as a Change in the External Parameter”, while the general energy balance is formulated in “A Unified Concept of Wave Electricity”.
5. Levels of Description That Must Be Distinguished
Many apparent contradictions arise when different levels of description are treated as the same space. These levels are preceded by the primordial temporal geometry \(\mathcal M_0=\mathbb R_{\tau}\). It contains only the ordering of events and is not yet either observable three-dimensional space or the operator's phase space.
Once spatial directions have emerged, several levels of description must be distinguished.
Spatial manifestation is a system of stable orthogonal projections that have acquired a metric and allow coordinates to accumulate. This is a transitional geometric level between primordial temporal dynamics and observable space.
Observable physical space is the ordinary three-dimensional world in which instruments register the position of a particle's center, its direction of motion, and electric and magnetic fields.
A one-dimensional trajectory is a particular curve within three-dimensional space. A path coordinate \(s\) is introduced along it.
Finite phase space consists of the two complex idempotent planes of the operator \(J(a,b)\). They describe state components, but are not coordinate planes of \(ct,x,y,z\).
Deeply split space is a countable set of orthogonal channels arising through successive idempotent splittings. Once a metric and self-similarity are introduced, its state acquires a norm equal to the Lorentz factor.
One-dimensional physical motion is therefore fully compatible with a multicomponent internal state. Similarly, a melody unfolds along one temporal sequence even though its spectrum may contain many frequency components.
Further details: the mathematical meaning of the two phase planes is explained in “The Hidden Geometry of the Hyperbolic Unit”, and the distinction between levels of description in “A Unified Concept of Wave Electricity”.
6. Motion as Physically Manifested Splitting
In the conventional description, space is assumed to exist already, and motion is understood as a body's displacement between its points. The developing geometry of Wave Electricity considers the reverse possibility: spatial extension arises together with differences between states, and observed motion is the physical manifestation of their splitting.
Successive splitting produces mutually orthogonal channels \(E_0,E_1,E_2,\ldots\). If the amplitude ratio between adjacent levels is constant and identified with the relative velocity \(\beta=v/c\), the normalized external state can be written as
\[\tag{6} \boxed{ \mathcal\Phi_{\beta}=\sqrt{1-\beta^2}\sum_{n=0}^{\infty}\beta^nE_n, \qquad \frac{a_{n+1}}{a_n}=\beta, \qquad \|\mathcal\Phi_{\beta}\|=1. } \] In this formula, idempotent algebra ensures channel independence, while identifying the amplitude ratio with the velocity \(\beta\) is introduced as a physical postulate. Motion and splitting thus become two related descriptions of one process, although this identification does not follow automatically from the algebra alone.
The vacuum's rest state may likewise be understood as symmetric compensation of internal dynamics rather than their complete absence. A zero mean flow \(\langle\boldsymbol\beta\rangle=0\) is compatible with a nonzero second moment \(\langle\beta_i\beta_j\rangle\). If that moment is isotropic, a Euclidean spatial metric \(g_{ij}=\delta_{ij}\) emerges. This interpretation remains a new postulate of the model, rather than a completed derivation of the physical vacuum.
Further details: the connection between motion, spatial extension, the vacuum state, and the Euclidean metric is discussed in “The Splitting of Space and Motion as a Single Process”.
7. The Operator of Internal State and External Motion
The mathematical basis of the finite operator consists of two complementary idempotents, \(\ep\) and \(\em\). They can be understood as two independent selectors: each selects its own complex phase component, and their mutual product is zero.
Their difference defines the hyperbolic unit \(\j\), for which \(\j^2=1\). Continuous powers of \(\j\) and \(-\j\) allow the phases of the two components to change independently. Their product gives the model's principal operator:
\[\tag{7} \boxed{ J(a,b)=\j^a(-\j)^b=\ep e^{i\pi b}+\em e^{i\pi a}. } \] The exponent \(a\) describes internal periodicity, while the exponent \(b\) describes external motion:
\[\boxed{ a=\varpi t, \qquad \omega_{\mathrm{int}}=\pi\varpi, \qquad b=\frac{\arcsin\beta}{\pi}, \qquad \beta=\frac vc. } \] Both processes belong to one normalized state but retain different physical roles. The parameter \(a\) cannot be replaced by velocity, nor can \(b\) be replaced by the particle's internal phase.
This resembles an instrument with two independent controls. One changes the rate of its internal clock; the other changes its state of external motion. The instrument remains a single whole, but turning one control need not turn the other.
From the standpoint of primordial temporal geometry, the parameters \(a\) and \(b\) describe two different manifestations of a single directed change. The parameter \(a\) organizes the wave structure's internal periodicity, whereas \(b\) determines the external displacement of its center. They do not create new fields from nothing; they specify different ways of organizing and projecting the temporal field.
Further details: the powers of the hyperbolic unit and the operator \(J\) are derived in “The Hidden Geometry of the Hyperbolic Unit”; the physical distinction between \(a\) and \(b\) is discussed in “A Unified Geometric Model of Waves and Particles”.
8. Why cJ Is Not the Velocity of the Particle's Center
The operator expression \(V_J=cJ\) describes the full phase-geometric dynamics at constant norm. It must not, however, be read as an ordinary three-dimensional velocity vector of the particle's center. Internal phase components need not coincide with directions of physical displacement.
The observed velocity is obtained only after identifying the external parameter:
\[\tag{8} \boxed{ \mathcal P_{\mathrm{ext}}[J]=\sin(\pi b)=\beta, \qquad \mathbf v=c\beta\widehat{\boldsymbol{\tau}}(s). } \] A state may therefore contain two phase planes, deep channels, and an internal cycle, while the particle's center still follows one observed curve. Phase geometry describes the system's state; the physical projection describes how its center moves.
Further details: the distinction between \(V_J\) and observed velocity is consistently maintained in “A Unified Geometric Model of Waves and Particles”.
9. Two Observers and One Event
Idempotent splitting can be applied to two coordinate descriptions of the same event as well as to internal phases. Suppose reference frames \(S\) and \(S'\) assign the coordinates \((ct,x)\) and \((ct',x')\) to an event. Both pairs can then be retained in one object without mixing them:
\[\tag{9} \boxed{ \mathcal T_{S'S}=\ep\,ct'+\em\,ct, \qquad \mathcal X_{S'S}=\ep\,x'+\em\,x. } \] Here it is the coordinate description, rather than the physical event itself, that is split. The idempotents act as independent labels for the two observers. If their coordinates coincide, the expression reduces to an ordinary coordinate because \(\ep+\em=1\).
This representation does not replace Lorentz transformations. It allows two related descriptions to be placed within a single algebraic structure, while keeping that structure distinct from the particle's phase space.
Further details: the joint idempotent representation of coordinates and preservation of the spacetime interval are examined in “Geometric Splitting of Coordinates between Observers”.
10. Deep Splitting and the Lorentz Factor
The two idempotents of the first level need not be the final layer of the structure. Each channel can be split again, and the procedure repeated. This produces a sequence of mutually orthogonal directions. If each successive level contributes an additional factor of \(\beta\), the complete state has components \(1,\beta,\beta^2,\ldots\).
Because these components are orthogonal, their squares add according to the Pythagorean theorem. Their sum forms a geometric series, giving the state norm
\[\tag{10} \boxed{ \left\|\boldsymbol{\Gamma}_{\beta}\right\|=\frac{1}{\sqrt{1-\beta^2}}=\gamma. } \] In this construction, the Lorentz factor has an intuitive meaning: it is the total length of a self-similar state distributed over an infinite chain of orthogonal channels. At low velocity, contributions from deep levels decay rapidly. As \(v\) approaches \(c\), this suppression disappears and the full norm grows without bound.
The finite operator \(J\) and the deep state have different norms. The operator retains unit norm, while one of its external projections encodes the inverse factor \(1/\gamma\). This is why normalizing \(J\) does not prevent relativistic energy from increasing.
“Deep splitting” is not, however, a single universal operation. A recursive chain of channels constructs \(\boldsymbol{\Gamma}_{\beta}\) and the Lorentz factor; a symmetric tensor combination of independent binary generators is used for atomic orbitals; and multiplicative assembly of uniformly evolving phases is used for a projection-based description of acceleration and rotation. These architectures share the preservation of the original whole, but their norms and physical mappings must be specified separately.
Further details: the complete derivation is presented in “The Origin of the Lorentz Factor from Infinite Idempotent Splitting”; the transition from the deep norm to energy and momentum is discussed in “The Lorentz Factor in the Finite Operator J”.
11. Acceleration as a Projection of Higher Uniformly Evolving Phases
Observed velocity can change even when the unfolded phases of a more complete description evolve uniformly. Mathematically, any sufficiently smooth subluminal motion function on a finite interval can be represented to a prescribed accuracy by a finite set of uniformly evolving phases, and reconstructed exactly through infinite splitting.
If the external parameter is represented as a projection of a higher state, the observed velocity and acceleration take the form
\[\tag{11} \boxed{ v(t)=c\sin\bigl(\pi b(t)\bigr), \qquad A(t)=\pi c\cos\bigl(\pi b(t)\bigr)\dot b(t). } \] The established result here is the existence of such a representation. The stronger claim—that physical acceleration actually arises from the incompleteness of the observed level—remains a hypothesis. Establishing it requires deriving the spectrum of higher phases from a unified geometry of interaction, rather than fitting that spectrum separately to an already known trajectory.
Further details: finite approximations, the exact infinite-dimensional representation, and the boundary between the theorem and the physical hypothesis are discussed in “Acceleration as a Projection of Uniform Motion in Higher Splittings”.
12. Internal Periodicity and Mass
The Lorentz factor describes the external motion of an already existing massive particle, but does not explain the origin of its mass. In the model, the source of rest energy is associated with internal periodicity: a stable cycle that continues even when the particle's center is at rest.
For the electron, the adopted hypothesis is that the observed rest energy is a transverse-Doppler projection of a higher internal frequency:
\[\tag{12} \boxed{ m_ec^2=\alpha_{\mathrm{fs}}\hbar\omega_{\mathrm{int}}. } \] The fine-structure constant \(\alpha_{\mathrm{fs}}\) belongs here to the internal mapping. The external factor \(\gamma\) has a different origin and increases the total energy of an already formed particle when it moves. These two coefficients must be distinguished.
In this picture, mass can be described as the observable manifestation of a stable internal rhythm, rather than an amount of matter contained inside a geometric point. However, the specific numerical appearance of \(\alpha_{\mathrm{fs}}\) remains a physical hypothesis of the model.
Further details: the relation between internal frequency, rest energy, and the photon's zero mass is discussed in “Mass as a Geometric Projection of a Closed Wave”.
13. Why a Closed Wave Appears as a Particle
A free wave carries phase from one region of space to another. A closed wave, by contrast, repeatedly returns to its initial internal state. If that return is stable, external observation detects a localized object with constant characteristics.
This motivates the model's concise definition:
\[\tag{13} \boxed{ \text{particle}=\text{a stable closed wave}. } \] Closure concerns the internal cycle, rather than the entire trajectory of the center. An electron may travel through a laboratory along an open line while its internal wave continues to execute a closed cycle. Localization and translational motion are therefore compatible.
The characteristic internal scale is related to frequency by the simple wave invariant \(r_{\mathrm{int}}\omega_{\mathrm{int}}=c\). The higher the internal frequency, the smaller the spatial scale of the closed state.
The same structure can be viewed as an ideal distributed wave circuit. Its electric and magnetic components do not belong to separate “capacitor” and “inductor” elements, but are distributed along a single closed path. In this description, the electron's equivalent capacitance and inductance are modal parameters of a stable eigenmode.
Further details: the geometric transition from phase velocity to closed and free waves is discussed in “A Unified Geometric Model of Waves and Particles”; the distributed circuit and its modal parameters are examined in “The Geometric Meaning of the Electron's Capacitance and Inductance”.
14. The Electron: Two Sheets, Spin, and Charge
In the model, the electron's internal contour has two successive sheets. The same wave traverses the first sheet and then the second, only afterward returning completely to its initial configuration. One full return of the internal state therefore corresponds to two ordinary revolutions.
The fundamental phase is distributed over a doubled path, introducing a half-angle relative to the ordinary angle. After rotation through \(2\pi\), the state changes sign; after \(4\pi\), it is restored:
\[\tag{14} \boxed{ \Psi(2\pi)=-\Psi(0), \qquad \Psi(4\pi)=\Psi(0). } \] In the physical mapping, the sheets may appear as two nearby branches of one internal contour. They are neither two independent electrons nor two simultaneously propagating waves. They are two successive segments of one wave's path.
Spin orientations are specified separately as two ways of arranging the complete internal state relative to a chosen axis: \(J(a,0)\) and \(J(0,a)\). The sign of charge must likewise be distinguished from spin. As a working hypothesis, the electron and positron are associated with opposite directions of traversal of the deep two-sheet cycle.
Further details: the two-sheet orbit and spinor periodicity are discussed in “The Electron's Double Internal Orbit” and “Electron Spin”; the algebra of the deep sheet factor is presented in “Multilevel Splitting of Electron Geometry”.
15. From the Electron Scale to the Atom, Orbitals, and Spectrum
Three characteristic distances are identified for the electron and hydrogen atom: the classical electron radius, the reduced Compton wavelength, and the Bohr radius. They are not an arbitrary collection of quantities. Each successive scale is approximately \(1/\alpha_{\mathrm{fs}}\) times larger than the preceding one:
\[\tag{15} \boxed{ r_e\longrightarrow\bar\lambda_C\longrightarrow a_0, \qquad \frac{r_e}{\bar\lambda_C}=\frac{\bar\lambda_C}{a_0}=\alpha_{\mathrm{fs}}. } \] The first transition is associated with the observed projection of the electron's internal state. The second is associated with the formation of a bound atomic contour. The radius increases and the frequency decreases, while certain products of spatial and frequency scales are preserved.
In this picture, an atom is not constructed by placing a structureless electron on a predefined orbit. It is treated as a new coordinated closure of an already structured electron and nucleus. The electron's internal two-sheet structure persists and may manifest in small spectral corrections.
Recent studies extend this construction. For a level with principal quantum number \(n\), two symmetrically split internal directions form \(n^2\) spatial components. They group into subspaces of dimensions \(1,3,5,\ldots,2n-1\), which, after three-dimensional projection, are associated with \(s\), \(p\), \(d\), and \(f\) orbitals:
\[ \boxed{ n^2=\sum_{l=0}^{n-1}(2l+1)=1+3+5+\ldots +(2n-1). } \] Each spatial orbital retains two independent spin states, \(J_n(a,0)\) and \(J_n(0,a)\). Under an additional antisymmetric composition rule, their combination allows an electron pair, while a three-electron state on the same spatial orbital vanishes. This provides a geometric interpretation of the Pauli exclusion principle. The two-dimensionality of spin space alone is insufficient: the antisymmetry rule for identical electrons must also be justified.
The same number \(n^2\) enters the spectral construction: conservation of the total quadratic phase norm gives the angle \(\theta_n=\alpha_{\mathrm{fs}}/n\), while the full orthogonal energy projection leads to \(B_n=\mu c^2(\sec\theta_n-1)\). The usual \(1/n^2\) law appears as its leading term for small \(\alpha_{\mathrm{fs}}/n\).
Further details: the transition between the three scales is presented in “From the Electron's Geometric State to the Hydrogen Atom's Orbit”; orbit construction is developed in “The First Orbit” and “Higher Orbits and Fine Splitting”. Orbital geometry, pair formation, and the spectral law are developed in “Atomic Orbitals as Projections of Split Phase Space”, “Geometric Formation of an Electron Pair and the Pauli Exclusion Principle”, and “The Geometric Origin of the Rydberg Law”.
16. How an Internal Gradient Becomes an Electric Field
Two nearby branches of the internal contour have slightly different geometric projections. Within the closed state, most of these contributions compensate one another, but a small remainder may extend outward. In the far region, it is assigned a potential energy decreasing as \(1/R\), whose gradient produces a force proportional to \(1/R^2\).
With the adopted electron scale, the coefficient of this dependence matches the Coulomb coefficient:
\[\tag{16} \boxed{ \mathbf E(R)=\frac{\alpha_{\mathrm{fs}}\hbar c}{eR^2}\widehat{\mathbf R}=\frac{e}{4\pi\varepsilon_0R^2}\widehat{\mathbf R}. } \] A single field line is parameterized by one radial coordinate. However, the source allows all radial directions, so the collection of one-dimensional extensions forms an ordinary three-dimensional field.
When the source moves, the external gradient acquires a transverse magnetic projection in the working geometric model. In the nonrelativistic limit, its direction and magnitude must reproduce the familiar relation \(\mathbf B\simeq\mathbf v\times\mathbf E/c^2\). This is a necessary consistency condition, but not yet a general derivation of electromagnetic field transformations. The magnetic field is treated as another spatial manifestation of the moving source's state, rather than an independent new source of energy.
Further details: the origin of the \(1/R^2\) dependence is examined in “The Geometric Origin of Electric Force”; interaction signs in “Attraction and Repulsion of Charges and Currents”; and the connection between electric and magnetic projections in “The Geometric Origin of Electric and Magnetic Fields” and “Projections of the Temporal Field”.
17. How a Field Differs from Free Radiation
The presence of a field around an electron does not mean that its internal wave has opened. As long as the electron exists as a particle, its internal cycle remains localized. The field is an outward extension of the gradient of this closed state, but does not carry all of its energy away to infinity.
Free radiation arises in a different regime, when an open spatial continuation of the phase appears. In an ordinary atomic transition, the electron does not turn entirely into a photon: it enters a new bound state, while the photon carries the energy difference:
\[\tag{17} E_{\gamma}=E_i-E_f. \] Complete conversion of an electron state into radiation is possible only as part of the entire interacting system and in compliance with conservation laws. For example, in annihilation an electron and positron produce at least two photons to conserve both energy and total momentum.
Fields and radiation therefore differ in more than the presence or absence of wave behavior. Both are wave regimes, but a field is associated with a persisting localized source, whereas radiation has its own open transport of energy and momentum.
Further details: the geometric transition from a closed contour to a traveling wave is discussed in “The Geometric Origin of a Propagating Wave”; conversion of a pair's spin structure into photon polarization is examined in “The Transition from Electron and Positron Spin to Photon Polarization”.
18. Spatial Phase, Diffraction, and Correlated Phases
The parameter \(b\) specifies the instantaneous state of external motion, but is not itself the phase accumulated along the traveled path. Describing propagation requires the action, which depends on both time and the spatial coordinate:
\[\tag{18} d\mathcal S=p_s\,ds-E\,dt. \] Spatial differences in this phase allow different admissible continuations to be compared in single- and double-slit experiments. An individual particle is always registered as a whole and leaves a single point. The set of possible phase continuations, however, spans the entire open slit region, so an interference pattern emerges after many events.
In this picture, uncertainty is associated with the unknown initial internal phase of an individual state. If two particles originate in the same process, their absolute phase may remain random while a relative phase relationship persists. This provides a geometric basis for two-particle interference and further discussion of entanglement.
In the temporal representation, a finite duration of phase localization requires a range of frequencies. Defining root-mean-square widths for two Fourier-conjugate representations gives the spectral relation \(\Delta E\,\Delta t\geqslant\hbar/2\). Here \(\Delta t\) denotes the duration of a wave packet or an interval of phase coherence, rather than uncertainty in an external clock's reading.
The rule \(P\propto|\mathcal A|^2\) is adopted as an additional physical law. A mechanism selecting a single registered outcome has not yet been established from the operator \(J\).
Idempotent branching also offers a common language for quantum interpretations: the complete state may be called a single underlying reality, and its stable orthogonal projections relative realities. Algebraic decomposition, physical decoherence, and claims about the existence of multiple worlds nevertheless belong to different levels. Selecting a particular interpretation requires adding a rule of actualization to the geometry of splitting.
Further details: single- and double-slit diffraction are discussed in “Passage through a Single Slit” and “Passage through Two Slits”; internal uncertainty and correlated phases in “Internal Phase Uncertainty” and “Correlated Random Phases”; and the energy–time relation in “The Geometric Origin of Energy–Time Uncertainty”. A geometric comparison of quantum interpretations is presented in “One Underlying Reality and Multiple Relative Realities”.
18.1. What the Wave Function Describes
Consider an interferometer with two possible paths. In an individual trial, the instrument registers one output, but the distribution of outputs depends on the relative phase between the paths. Merely listing the possibilities is insufficient: each must be assigned a complex amplitude.
The wave function represents a state through channel amplitudes. For two orthonormal channels, its simplest form is
\[\tag{18a} \Psi=c_+\boldsymbol\xi_++c_-\boldsymbol\xi_-,\qquad |c_+|^2+|c_-|^2=1. \] The coefficients \(c_+\) and \(c_-\) contain the weights and relative phase of the possibilities. Acting in this space, the operator \(J\) changes the channel phases:
\[\tag{18b} J(a,b)\Psi=c_+e^{i\pi b}\boldsymbol\xi_++c_-e^{i\pi a}\boldsymbol\xi_-. \] For real parameters, this action preserves the channel weights in the chosen basis. Output probabilities change after coherent recombination or through additional operations coupling the channels. Orthogonality means that state components are distinguishable, but does not prohibit transitions between them.
The model therefore distinguishes three operations: splitting identifies channels; interchannel coupling transfers the state between them; and inverse synthesis recombines them with their phases taken into account. A branching tree alone does not describe dynamics—the couplings and their rules of action must also be specified.
The Born rule assigns a normalized discrete output a probability equal to the squared modulus of its amplitude. It is consistent with orthogonal splitting, but has not yet been derived from its algebra. Normalization of the phase operator and normalization of the wave function concern different objects.
Further details: the distinction between an amplitude state and a phase operator is explained in “The Wave Function as a Representation of a Split Event”; channel resolution, transitions, and inverse synthesis are discussed in “The Splitting Principle: Hierarchy, Horizontal Couplings, and Experimental Tests”.
18.2. How Geometry Relates to the Schrödinger and Dirac Equations
A geometric description must connect to the equations used to calculate propagation and changes of state. The structure of possible channels must be distinguished from the physical law governing their evolution.
In the nonrelativistic regime, the external amplitude of a massive particle is described by the Schrödinger equation. In the construction considered here, this equation is reached through the action phase, the energy relation, and the standard operator correspondence for energy and momentum. Factoring out the rapidly varying rest-energy phase leaves a slowly varying external amplitude. This connects the model's internal geometry to the familiar description of motion, but requires additional physical correspondences.
The one-dimensionality of an individual trajectory does not confine the wave function to one line. The spatial amplitude encompasses a family of admissible continuations. Moving to a three-dimensional equation requires this generalization: the second derivative along an arbitrary curve is not, by itself, equal to the spatial Laplacian.
Representing the Dirac equation requires operations that exchange channels as well as select them. In a two-channel space, the channel-sign operator and the channel-exchange operator can anticommute: reversing the order of the operations changes the sign of the result. Two binary factors give a four-component space in which the required operators and the free Dirac Hamiltonian can be constructed.
Given the metric, Hamiltonian, and transformation laws, this construction reproduces the relativistic energy relation and spinor dynamics. Its value lies in describing the familiar matrix structure explicitly through operations on channels. A \(J\)-type form of phase evolution does not, however, amount to deriving the entire equation from the original operator \(J(a,b)\).
Further details: the nonrelativistic transition is presented in “The Schrödinger Equation in the Geometry of the Split Operator”; the relativistic construction and its additional conditions are discussed in “The Dirac Equation in Splitting Operators: Construction and Connection to J”.
19. Composite States and Interactions
An electron differs from a photon in more than its mode of motion. Particles possess spin, charge, flavor, color, and other properties. Multilevel splitting suggests that independent characteristics might be described by separate normalized operators whose product forms the complete state.
This notation provides a classification architecture, rather than a finished table of elementary particles. Adding a new factor does not by itself explain the physical nature of the corresponding property. Its algebra, observable values, transformation rules, and experimental consequences must be specified.
For interactions with two input and two output states, a double-balance rule is proposed:
\[\tag{19} \boxed{ J_1+J_2=J_3+J_4, \qquad J_1J_2=J_3J_4. } \] The first equality constrains the additive composition; the second constrains the overall multiplicative structure in the rearrangement schemes under consideration. The universality of this rule for physical interactions has not yet been established. Geometric admissibility does not determine a process's probability. The full dynamics must be consistent with conservation of energy, momentum, charge, and angular momentum.
Further details: the multi-operator classification is discussed in “Constructing Known Particles from Normalized Operators” and “An Operator Classification of Matter”; the double-balance rule and branch rearrangement are examined in the series “A Geometric Representation of Particle Interactions”.
A quantitative description introduces a space of complete input and output states and an interaction operator. Its matrix element gives the transition amplitude. For normalized discrete orthogonal outputs, the Born rule gives the probability; continuous outcomes additionally require a measure on the state space. Calculations of cross sections and process rates must also account for kinematic normalization and flux.
Two questions are thus separated: whether a transition is permitted by the structure and conservation laws, and how often it occurs for a given preparation. The specific interaction generator and its numerical predictions remain to be derived.
Further details: physical channel space, invariants, and transition amplitudes are discussed in “State Space and Probabilities of Sector Rearrangements”.
20. From Microphysics to Cosmological Hypotheses
The same sequence—states, order, time, temporal field, space—allows a cosmological extension. In this interpretation, the Universe does not pass through its earliest states within an external time: a stable ordering of distinguishable states itself makes duration measurable and subsequently permits spatial projections to form.
Upward splitting is considered separately and must be distinguished from resolving a particle's internal structure. If a new orthogonal level increases the total metric norm of a spatial interval, the observed projection may appear as homogeneous expansion. Self-similar scaling gives the kinematic law \(\dot R=H(t)R\), but neither the function \(H(t)\) nor the physical mechanism of the transitions follows from the algebra alone.
More preliminary still is the hypothesis that dark matter is a coherent scalar mode of the internal geometry. It indicates a possible additional channel, but does not yet derive its dynamics, normalization, or gravitational coupling from the operator \(J\). Cosmological extensions should therefore be treated as a separate area for testing, rather than an established consequence of the microscopic model.
Further details: the proposed emergence of time and space is presented in “From State to Time and Space: The Geometric Evolution of the Universe”; the expansion and scalar-mode hypotheses are discussed in “The Expansion of the Universe as Upward Splitting of Space” and “Dark Matter as a Geometric Scalar Mode”.
21. What Follows from the Mathematics and What Remains Hypothetical
The adopted algebra directly yields the idempotent decomposition of unity, channel orthogonality, the existence of two independent complex planes, the form of the operator \(J(a,b)\), its composition law, and its unit finite norm. The algebra also permits successive splitting and the construction of a chain of mutually orthogonal projectors.
Obtaining the Lorentz factor additionally requires a metric on the deep space, orthonormal directions, self-similarity, and a common transition coefficient \(\beta\). These conditions select the state \(\{1,\beta,\beta^2,\ldots\}\) and its norm \(\gamma\).
The following remain physical hypotheses: the emergence of spatial directions from splitting the temporal field; the identification of motion with physically manifested idempotent splitting; the interpretation of the vacuum as compensated motion and the origin of the spatial metric from its second moment; the physical origin of acceleration in projections of higher uniformly evolving phases; the primacy of a normalized wave as a dynamically organized state; particle formation through closure; the transverse-Doppler origin of mass; the mapping of two sheets onto nearby electron branches; the geometric nature of charge sign; the extension of an internal gradient into an external field; and the full mechanism of wave opening. The physical mapping of tensor splitting onto atomic orbitals, the geometric mechanisms of the Pauli principle and the Rydberg law, and the cosmological hypotheses of upward splitting and a scalar mode belong to the same category.
This distinction does not weaken the model. It identifies which parts already have a mathematical foundation and which require further derivation or experimental testing. Geometrically reproducing a known formula is an important consistency requirement, but is not yet proof of the entire theory.
Further details: a complete list of mathematical consequences, additional assumptions, physical hypotheses, and limitations is given in “A Unified Concept of Wave Electricity”. The general hierarchy of splittings, horizontal couplings, and possible experimental tests are discussed in “The Splitting Principle: Hierarchy, Horizontal Couplings, and Experimental Tests”.
The newer constructions sharpen this distinction. Introducing an amplitude space and a positive metric allows probabilistic channels to be described; adding exchange operations and tensor structure allows a representation of the Dirac equation to be constructed; antisymmetric composition in a two-dimensional spin space yields an occupancy limit for one orbital. Each result has specific additional conditions. The Born rule, the choice of dynamics, and the antisymmetry of identical electrons have not yet been obtained from phase normalization alone.
21.1. How to Test the Model
Testing begins with a specific experiment: what is prepared, what influence changes the state, and what quantity the instrument records. A proposed internal level must have a measurable means of excitation or manifestation. Increasing the number of mathematical components does not, by itself, establish a new degree of freedom.
One protocol compares two ways of implementing a transformation: directly at the observed level, or by resolving additional channels, acting within them, and performing inverse synthesis. A difference may appear as a phase shift, changed output probabilities, or incomplete recovery of the initial state. The comparison must account for all outputs, losses, and preservation of coherence.
A network of known optical elements can test whether such a protocol is realizable within ordinary quantum mechanics. Testing new physical content in Wave Electricity requires a stronger result: specifying in advance the natural internal channels, the influence acting on them, and a numerical prediction that distinguishes the proposed description from the standard one.
A deviation from the simplest calculation must first be checked for losses, alignment errors, ordinary channel couplings, and incompleteness of the apparatus model. Both confirmation and a negative result apply to the particular construction under test. This allows hypotheses to be refined and physically effective elements of the overall geometry to be identified step by step.
Further details: the protocol for comparing levels, measurable characteristics, and error controls are proposed in “The Splitting Principle: Hierarchy, Horizontal Couplings, and Experimental Tests”.
22. The Overall Picture
Wave Electricity develops a unified sequence from an elementary event to observable physics. The ordering of events establishes time; directed change of state produces a temporal field; and independent splittings of that field open spatial directions. Spatial extension and motion are treated as two manifestations of differences between split states. On this emergent geometric foundation, the finite operator \(J(a,b)\) separates internal periodicity from external motion, while deep self-similar splitting with a specified metric produces the Lorentz factor.
Wave closure creates a stable localized regime observed as a particle. The two-sheet cycle is associated with spinor periodicity. The internal frequency acquires a mass projection. The residual gradient of the closed state extends outward as a field. Removing closure allows a freely propagating wave to appear.
Different architectures of subsequent splitting extend this foundation in several directions. A recursive chain describes the relativistic norm; tensor combinations of binary channels describe orbital spaces; antisymmetric composition of spin states describes electron pairs; a tree of orthogonal histories describes the space of quantum possibilities; and upward splitting is considered as a separate cosmological hypothesis.
The amplitude description supplements this picture: channels specify possibilities, phases and couplings determine their joint evolution, and the registration rule relates output amplitudes to outcome frequencies. The constructions of the Schrödinger and Dirac equations show how, under additional physical conditions, the geometric language connects to established quantum dynamics.
This hierarchy can be represented as a single sequence:
\[\tag{20} \boxed{ \begin{gathered} \text{distinguishable states}\longrightarrow\text{order}\longrightarrow\text{time}\longrightarrow\text{temporal field},\\ \text{splitting}\longrightarrow\text{directions and extension}\longrightarrow\text{space and motion},\\ J(a,b)\longrightarrow\text{internal state and external projection},\\ \text{deep self-similar splitting}\longrightarrow\boldsymbol{\Gamma}_{\beta}\longrightarrow\gamma,\\ \text{closure}\longrightarrow\text{particle},\qquad \text{gradient}\longrightarrow\text{field},\\ \text{opening}\longrightarrow\text{radiation}. \end{gathered} } \] The central idea remains simple: space, motion, particles, fields, and radiation are treated as successive levels in the organization of one geometry, rather than independently given foundations of the world. Time orders changes; splitting creates independent directions and metric relations; wave localization forms a particle; the field extends its gradient outward; and opening converts cyclic phase into free transport. Turning this idea into a complete physical theory, however, requires derivations of three-dimensionality and the local spatial metric, the physical law governing the formation of higher acceleration phases, charges, interaction probabilities, measurement dynamics, and testable differences from standard physics.
Further reading: the rigorous formulation of the current model is presented in “A Unified Concept of Wave Electricity”; a complete list of related studies is available in the Wave Electricity section.