What if space, particles, fields, and free radiation are not independently defined primary entities, but instead emerge sequentially from a single geometric foundation? This is precisely the possibility explored by Wave Electricity. The model begins with an extremely simple state a mathematical point that does not yet possess spatial coordinates. A sequence of distinguishable states creates time; a directed change creates a temporal field; and the orthogonal splittings of this field open up the possibility for spatial directions to emerge.

The first dynamically organized object within this geometry is a normalized wave state. The ways in which it undergoes internal closure, external projection, and spatial propagation manifest subsequently as a particle, a field, and free radiation.

In conventional language, an electron is called a particle, an electric field is described as a special state of the space surrounding a charge, and a photon is termed a quantum of radiation. The proposed framework establishes a deeper connection between them. A closed and localized wave is observed as a particle. Its internal gradient can extend outward and manifest as a field. If the closure is released, free propagation occurs.

This does not mean that all physical properties have already been derived from a single formula. Wave Electricity is currently under development. Within this framework, it is necessary to distinguish between rigorous algebraic results, additional geometric assumptions, and physical hypotheses. The aim of this overview article is to present the big picture in simple terms and provide references to works where each element is examined in detail.

In the initial state, internal change and external motion are not yet present:

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 the simplest event preceding the manifestation of length, direction, and volume.

If such events are distinguishable and ordered, a sequence emerges. The numbers in the epigraph signify precisely the ability to distinguish between preceding and succeeding events that is, to number their positions within this sequence. Time is introduced as a measure of ordered change that does not yet require displacement within a pre-existing space.

A sequence of events creates a primary directionality, conveniently represented by the temporal coordinate \(X_0=c\tau\mathbf e_0\). Its derivative, \(\mathcal T_0\), defines the universal flow of events. When a specific state \(J\) begins to change along this flow, the temporal field of this wave structure, \(\mathcal T_J\), emerges:

These quantities cannot be equated. 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 established. Therefore, time is not derived from the derivative of an invariant operator; rather, the primary order is posited as a precondition for any subsequent change.

The poetic formula in the epigraph states that "time split." More precisely, it is not time as an independent physical substance that splits, but the directed temporal field. Its independent orthogonal projection creates a new direction. When positions can be distinguished and accumulated change measured along this direction, it acquires the meaning of a spatial coordinate and length.

Repeated independent splittings can create new orthogonal directions. Thus, a one-dimensional extent can transform into a plane, and a plane into a volume. Idempotent algebra ensures the completeness and orthogonality of the channels; however, on its own, it does not yet explain why the stable observable space possesses precisely three dimensions. This remains a physical hypothesis of the model.

Modern physics successfully describes nature through several interconnected theories. Quantum mechanics determines measurement probabilities, field theory describes particles as field excitations, and relativity links space, time, energy, and momentum. Wave Electricity does not dispute these results. Instead, it seeks an intuitive geometric foundation upon which various properties might emerge as manifestations of a unified structure.

At the heart of the model lies the question: is it possible to start with the simplest sequence of events, derive time and spatial directions, and then via a normalized wave state sequentially arrive at intrinsic periodicity, mass, spin, the electric field, atomic scales, and the free wave? In this approach, a known physical formula is viewed not as the ultimate goal, but as a means of verifying the geometric construction. It is important not only to reproduce the result but also to demonstrate which element of the model is responsible for it.

For instance, internal closure is linked to particle stability, a double-sheeted cycle to spinor periodicity, deep splitting to the Lorentz factor, and an external gradient to the electric field. At the same time, such correspondences must be distinguished from algebraic properties that arise automatically.

The model's first postulate states that, in any given physical state, a point has a single actual direction of motion. This does not imply that the world is one-dimensional or that all particles must move in straight lines. 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 \(s\). Then the observed trajectory is expressed as

The vector \(\widehat{\boldsymbol{\tau}}\) is tangent to the curve. If the trajectory curves, the tangent changes direction, yet at any given moment, motion still proceeds along that single tangent line. Three-dimensionality manifests itself in the shape and orientation of the entire curve, rather than in a requirement for three independent instantaneous movements of a single point.

It is important here to distinguish between the actual trajectory and the set of possible continuations. A single path is recorded in any given event. However, prior to recording, the geometry of the setup may allow for an entire family of paths with different phases. This distinction will be relevant when discussing diffraction.

An experimentally testable proposition regarding electric interaction follows from this postulate: a chosen source and receiver can be linked by a single instantaneous coordinate \(R(t)\) directed along the line connecting them. Two different motions that generate the same function \(R(t)\) should yield the same quasistatic scalar signal, whereas motion along a circle at constant \(R\) should not, in itself, generate a variable component of such a signal.

A normalized geometric state should not be confused with the physical law of conservation of energy. If a particle accelerates in an external field, its laboratory energy increases. This increase does not arise from nothing, nor is it necessarily compensated solely by a decrease in the particle's internal energy. The source of the influence also participates in the process.

Therefore, the total balance is written for a closed system:

When an electric field accelerates an electron, the electron's energy increases at the expense of work done by the field source. When an atom emits a photon, the internal energy of the bound state decreases by the energy of the radiation. Geometry describes the permissible relationships between states but does not, in itself, negate the necessity of a total energy balance.

Most apparent contradictions arise when different levels of description are mistaken for the same space. These levels are preceded by a primordial temporal geometry, \(\mathcal M_0=\mathbb R_{\tau}\). It contains only the ordering of events and is neither the observable three-dimensional space nor the operator's phase space.

Once spatial directions emerge, it is necessary to distinguish between several levels of description.

Spatial manifestation a system of stable orthogonal projections that have acquired a metric and allow for the accumulation of coordinates. This is a transitional geometric level between primordial temporal dynamics and observable space.

Observable physical space the ordinary three-dimensional world where instruments record the position of a particle's center, the direction of motion, and electric and magnetic fields.

One-dimensional trajectory a specific curve within three-dimensional space. A path coordinate \(s\) is defined along it.

Finite phase space the two complex idempotent planes of the operator \(J(a,b)\). They describe state components but are not the coordinate planes \(ct,x,y,z\).

Deeply split space a countable set of orthogonal channels arising from successive idempotent splitting. Upon the introduction of a metric and self-similarity, its state acquires a norm equal to the Lorentz factor.

Thus, one-dimensional physical motion is fully compatible with a multi-component internal state. Similarly, a melody unfolds along a single temporal sequence, even though its spectrum may contain multiple frequency components. \[\tag{6} \boxed{ \mathcal\Phi_{\beta}=\sqrt{1-\beta^2}\sum_{n=0}^{\infty}\beta^nE_n, \qquad \frac{a_{n+1}}{a_n}=\beta, \qquad \|\mathcal\Phi_{\beta}\|=1. } \] \[\tag{7} \boxed{ J(a,b)=\j^a(-\j)^b=\ep e^{i\pi b}+\em e^{i\pi a}. } \] \[\boxed{ a=\varpi t, \qquad \omega_{\mathrm{int}}=\pi\varpi, \qquad b=\frac{\arcsin\beta}{\pi}, \qquad \beta=\frac vc. } \] \[\tag{8} \boxed{ \mathcal P_{\mathrm{ext}}[J]=\sin(\pi b)=\beta, \qquad \mathbf v=c\beta\widehat{\boldsymbol{\tau}}(s). } \] \[\tag{10} \boxed{ \left\|\boldsymbol{\Gamma}_{\beta}\right\|=\frac{1}{\sqrt{1-\beta^2}}=\gamma. } \] \[\tag{11} \boxed{ v(t)=c\sin\bigl(\pi b(t)\bigr), \qquad A(t)=\pi c\cos\bigl(\pi b(t)\bigr)\dot b(t). } \] \[\tag{12} \boxed{ m_ec^2=\alpha_{\mathrm{fs}}\hbar\omega_{\mathrm{int}}. } \] \[\tag{13} \boxed{ \text{particle}=\text{stable closed wave}. } \] Its electric and magnetic components do not belong to separate "capacitor" and "coil" 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 natural wave. \[\tag{14} \boxed{ \Psi(2pi)=-\Psi(0), \qquad \Psi(4pi)=\Psi(0). } \] \[\tag{15} \boxed{ r_e\longrightarrow\bar\lambda_C\longrightarrow a_0, \qquad \frac{r_e}{\bar\lambda_C}=\frac{\bar\lambda_C}{a_0}=\alpha_{\mathrm{fs}}. } \] \[ \boxed{ n^2=\sum_{l=0}^{n-1}(2l+1)=1+3+5+\ldots +(2n-1). } \]