2026-08-28
The temporal field and the origin of space
Physical fields are usually introduced as quantities already distributed in space and changing over time. With this approach, space itself is considered a pre-existing basis: coordinates, distances, and directions are first specified, and then particles and fields are placed within them. However, this is not sufficient for a geometric model. If the observed properties must derive from a single construct, we must ask: what exists first—space or a more primordial directional structure, the splittings of which alone form spatial directions?
This paper proposes to begin with an extremely simple state: a mathematical point that has neither internal rotation, nor external motion, nor spatial coordinates. The only difference arises between the successive events of its existence. From this sequence, we obtain the primary temporal direction, its gradient, and then the temporal field of a specific wave structure.
The main hypothesis of the paper is that space need not precede the field. On the contrary, spatial directions can arise as independent orthogonal splittings of the primary temporal field, and the observed fields can arise as its projections onto the resulting directions.
\[\tag{1} \boxed{\text{change}\;\longrightarrow\;\text{time}\;\longrightarrow\;\text{temporal gradient}\;\longrightarrow\;\text{temporal field}\;\longrightarrow\;\text{splitting}\;\longrightarrow\;\text{space}} \] In the first part, only the general foundation of this scheme will be constructed. Electric, magnetic, and other known fields are not yet derived here. Their classification and derivation from different projections of the temporal field will constitute the task of the second part.
1. General Principle of Gradient Generation
Let a certain quantity \(Q\) depend on a parameter \(s\). Two close states differ by the value
\[\tag{2}\Delta Q=Q(s+\Delta s)-Q(s).\] The ratio \(\Delta Q/\Delta s\) shows how much the state changes when moving over a small parameter step. In the limit, we obtain a local gradient along \(s\):
\[\tag{3}\boxed{G_s[Q]=\lim_{\Delta s\to0}\frac{Q(s+\Delta s)-Q(s)}{\Delta s}=\frac{dQ}{ds}}\] Thus, for a gradient to appear, two conditions are necessary: adjacent states must be distinguishable, and the transition between them must be ordered. If a state does not change, then its gradient is zero. If, however, change occurs at every point in the sequence under consideration, the gradient becomes a gradient field.
A gradient arises where adjacent states are distinguishable. Its direction is determined by the order of transition from one state to another.
Here, it is important not to confuse the three levels. A finite difference compares two states, a derivative specifies the local rate of their change, and a field associates such a local gradient with each state of the structure under consideration.
2. A Mathematical Point Without Space
Consider the initial state of the global operator of Wave Electricity
\[\tag{4}J(a,b)=\j^a(-\j)^b.\] Parameter \(a\) describes the internal state, and parameter \(b\) describes the external motion. Let
\[\tag{5}a=0,\qquad b=0.\] Then
\[\tag{6}\boxed{J(0,0)=1}.\] Such a mathematical point does not yet have internal rotation, external motion, sheet structure, or spatial position. However, one should not write \(x=y=z=0\), because such a notation already presupposes the existence of three-dimensional space and a chosen origin within it. At the initial stage, the coordinates \(x\), \(y\), and \(z\) are not equal to zero—they are not yet defined.
This state should be understood not as a particle placed in empty space, but as an extremely simple event before the manifestation of spatial directions.
3. Sequence of Events and the Emergence of Time
Let a mathematical point be represented by a sequence of events
\[\tag{7}P(\tau_1),\quad P(\tau_2),\quad P(\tau_3),\quad\ldots,\qquad \tau_1<\tau_2<\tau_3.\] The events are not yet separated by spatial distance, but they differ in their position in the sequence. The parameter \(\tau\) measures this order. Within the proposed construction, time is introduced as a measure of the sequence of distinguishable events:
\[\tag{8}\boxed{\text{time}=\text{measure of an ordered sequence of changes}}\] Transition
\[\tag{9}P(\tau)\longrightarrow P(\tau+\Delta\tau)\] creates the first directional relationship. It is not yet movement in space, but already distinguishes the subsequent event from the previous one. This is how the primary temporal axis arises.
4. Movement Preceding Space
In the most general sense, movement can be understood not only as a change in spatial coordinates, but also as a transition between distinguishable states. The spatial displacement \(d\mathbf r\) is not yet defined, but progression along the sequence of events already exists.
To associate the time coordinate with the dimension of length, we introduce
\[\tag{10}x^0=c\tau.\] We write the position of a mathematical point on the primary time axis as
\[\tag{11}X_0(\tau)=c\tau\,\mathbf e_0,\] where \(\mathbf e_0\) is the unit temporal direction. Then
\[\tag{12}\boxed{\frac{dX_0}{d\tau}=c\mathbf e_0}.\] Formula (12) does not describe the velocity of a particle in three-dimensional space. It characterizes the movement of an event along the time coordinate. Spatial velocity will only appear after the emergence of an independent spatial direction.
5. The Primary Temporal Manifold
The original geometry can be represented as a one-dimensional temporal manifold.
\[\tag{13}\boxed{\mathcal M_0=\mathbb R_\tau}.\] Only the coordinate \(x^0=c\tau\) is defined in it. The spatial part of \(\mathbb R^3\) has not yet been introduced, so \(\mathcal M_0\) cannot yet be identified with ordinary four-dimensional spacetime.
It is more accurate to speak of a primary temporal geometry, from which, with further splitting, additional independent directions may emerge. It is the initial state of future spacetime, but for now contains only the axis of the sequence of events.
6. Temporal gradient
For two adjacent events, the difference in their temporal positions is
\[\tag{14}\Delta X_0=X_0(\tau+\Delta\tau)-X_0(\tau)=c\Delta\tau\,\mathbf e_0.\] Dividing it by the step \(c\Delta\tau\) and taking the limit, we obtain the unit temporal gradient:
\[\tag{15}\boxed{\widehat{\mathcal T}_0=\lim_{\Delta\tau\to0}\frac{\Delta X_0}{c\Delta\tau}=\mathbf e_0}.\] Its dimensional form coincides with the derivative of the position with respect to the parameter \(\tau\):
\[\tag{16}\boxed{\mathcal T_0=\frac{dX_0}{d\tau}=c\mathbf e_0}.\] The temporal gradient defines the first direction of geometry. It exists due to the difference between successive events and does not require the prior existence of space.
The temporal gradient is the directionality of the course of events themselves. It is not yet an individual physical field of the particle.
7. Two Different Gradients
At this stage, it is especially important to distinguish between the gradient of the event's progress and the gradient of the global state \(J\). For \(a=b=0\), we have
\[\tag{17}J=1,\qquad \frac{dJ}{d\tau}=0,\] but at the same time
\[\tag{18}\frac{dX_0}{d\tau}=c\mathbf e_0\ne0.\] Therefore, the universal temporal gradient
\[\tag{19}\mathcal T_0=\frac{dX_0}{d\tau}\] exists already thanks to the sequence of events. Individual state field
\[\tag{20}\mathcal T_J=\frac{dJ}{d\tau}\] will only appear when the state itself begins to change. This distinction does not allow us to assign frequency, energy, mass, or charge to the state \(J=1\).
8. Internal dynamics along time
Now we include internal change, preserving the absence of external motion:
\[\tag{21}a=a(\tau),\qquad \dot a\ne0,\qquad b=0.\] For uniform internal motion
\[\tag{22}a=\varpi\tau,\qquad \omega=\pi\varpi.\] The global state takes the form
\[\tag{23}J(\tau)=\j^{a(\tau)}.\] In an idempotent basis, the hyperbolic power is decomposed into a fixed and a rotating components:
\[\tag{24}\j^a=\ep+\em e^{i\pi a}.\] Substituting \(a=\varpi\tau\) and \(\omega=\pi\varpi\), we obtain an explicit dependence of the state on the proper evolution parameter:
\[\tag{25}\boxed{J(\tau)=\ep+\em e^{i\omega\tau}}.\] Now we apply the general gradient principle introduced in the first section to the operator \(J\). For two close events, the difference in states is equal to
\[\tag{26}\Delta J=J(\tau+\Delta\tau)-J(\tau)=\em e^{i\omega\tau}\left(e^{i\omega\Delta\tau}-1\right).\] Dividing by \(\Delta\tau\), we get
\[\tag{27}\frac{\Delta J}{\Delta\tau}=\em e^{i\omega\tau}\frac{e^{i\omega\Delta\tau}-1}{\Delta\tau}.\] As \(\Delta\tau\to0\), the expression in the last factor tends to \(i\omega\). Therefore, the local gradient of the global state has the form
\[\tag{28}\boxed{\mathcal T_J=\frac{dJ}{d\tau}=i\omega\em e^{i\omega\tau}}.\] The same result is obtained using the chain rule:
\[\tag{29}\mathcal T_J=\dot a\frac{\partial J}{\partial a}=\varpi\frac{\partial\j^a}{\partial a}.\] Now, not only the position of the event on the time axis changes, but also the internal state associated with this event. The primary gradient \(\mathcal T_0\) defines the direction of evolution, and \(\mathcal T_J\) shows how exactly a particular wave structure changes along this direction:
\[\tag{30}\boxed{\mathcal T_J=D_{\mathcal T_0}J}.\] 9. Temporal Field
Let us call the temporal field of a wave structure the directed change of its global state \(J\) along the primary time coordinate.
\[\tag{31}\boxed{\text{temporal field:}\qquad \mathcal T_J=\frac{dJ}{d\tau}}\] This definition contains two levels. The universal temporal gradient creates the possibility of directional change, and the specific dependence \(J(\tau)\) imbues this directionality with internal dynamics.
With a unit normalization of the rotating component, the modulus of the temporal field is determined by the internal frequency:
\[\tag{32}\boxed{|\mathcal T_J|=\omega}.\] The field is periodic with the same period as the internal state:
\[\tag{33}T=\frac{2\pi}{\omega},\qquad \mathcal T_J(\tau+T)=\mathcal T_J(\tau).\] Over the entire period, the state returns to the original, so the integral change is zero:
\[\tag{34}\int_{\tau}^{\tau+T}\mathcal T_J(\tau')\,d\tau'=J(\tau+T)-J(\tau)=0.\] This does not mean the local field disappears. Like the velocity in a circle, \(\mathcal T_J\) is nonzero at every point in the cycle, although the total increase over the closed period is zero.
Since the norm \(J\) is conserved, the derivative is tangential to the internal rotation and orthogonal to the state itself:
\[\tag{35}\boxed{J\cdot\mathcal T_J=J\cdot\frac{dJ}{d\tau}=0}.\] After introducing the quantum of action, the internal frequency corresponds to an energy scale.
\[\tag{36}E=\hbar\omega.\] The detailed transition from internal frequency to observable energy and mass is discussed in other works of Wave Electricity. The only significant point here is that energy appears as a characteristic of change over time even before the appearance of observable spatial fields.
The temporal field is primary not because it is another field within the existing spacetime, but because it is defined before the appearance of spatial directions.
10. The Mathematical Principle of Splitting
The mathematical possibility of dividing a single state into independent components is based on idempotent splitting. Its properties are discussed in detail in this article. Only three basic principles are needed here.
First, one is represented as the sum of two idempotents:
\[\tag{37}\ep+\em=1.\] Second, each component preserves itself when reapplied, and different components are mutually orthogonal:
\[\tag{38}\ep^2=\ep,\qquad \em^2=\em,\qquad \ep\em=0.\] Therefore, an arbitrary state \(Q\) can be represented as the sum of two independent projections without losing the original whole:
\[\tag{39}\boxed{Q=\ep Q+\em Q}.\] Each of the resulting planes can be split again using a new set of projections \(\prp\) and \(\prm\). This is how a multi-level structure emerges. For this article, the general idea is important: a single direction can generate several distinct but related projections, without the original quantity disappearing or doubling.
Idempotent splitting preserves the whole, divides it into orthogonal components, and allows this action to be repeated at deeper levels.
11. Splitting the Temporal Field
\[\tag{40}\prp+\prm=1,\qquad \prp^2=\prp,\qquad \prm^2=\prm,\qquad \prp\prm=0.\]
Now the single temporal field is represented by two independent projections:
\[\tag{41}\boxed{\mathcal T_J=\prp\mathcal T_J+\prm\mathcal T_J}.\] We denote the first projection as preserving the original temporal directionality, and the second as the new orthogonal Channel:
\[\tag{42}\mathcal T_{\mathrm t}=\prp\mathcal T_J,\qquad \mathcal T_{\mathrm s}=\prm\mathcal T_J.\] Then
\[\tag{43}\boxed{\mathcal T_J=\mathcal T_{\mathrm t}+\mathcal T_{\mathrm s}},\qquad \mathcal T_{\mathrm t}\mathcal T_{\mathrm s}=0.\] Idempotent mathematics guarantees the completeness and orthogonality of the two channels, but by itself does not yet determine the magnitude of the observed projection. For the normalized implementation of splitting, we introduce the parameter \(\theta\) and the operator \(\mathcal R\), which transforms the field into the orthogonal direction:
\[\tag{44}\mathcal T_{\mathrm t}(\theta)=\mathcal T_J\Cos\theta,\qquad \mathcal T_{\mathrm s}(\theta)=\mathcal R\mathcal T_J\Sin\theta.\] When two components are orthogonal, the full norm is preserved:
\[\tag{45}\boxed{|\mathcal T_{\mathrm t}|^2+|\mathcal T_{\mathrm s}|^2=|\mathcal T_J|^2}.\] At \(\theta=0\), there is no new projection. At \(\theta>0\), part of the original directionality manifests itself in the orthogonal channel. The field does not double or gain additional energy: only the distribution of its projections changes.
This does not mean that part of time literally transforms into a ready-made spatial coordinate. First, an independent projection of the temporal field arises. It becomes a spatial direction when it can distinguish positions, specify changes, and compare projections of different states.
12. Space as a System of Splitting Directions
Let a nonzero orthogonal component define a unit direction
\[\tag{46}\mathbf e_1=\frac{\mathcal T_{\mathrm s}}{|\mathcal T_{\mathrm s}|}.\] However, a single direction is not sufficient for the emergence of space: we must be able to distinguish positions along it. We normalize the temporal field and assign its full directionality scale \(c\). Then the spatial fraction of the directed motion is equal to
\[\tag{47}\mathcal V_{\mathrm s}=c\,\mathbf e_1\Sin\theta.\] Its modulus determines the rate of accumulation of the new coordinate:
\[\tag{48}\boxed{\frac{dx_1}{dt}=c\Sin\theta}.\] Consequently, the spatial position arises as the accumulated projection of the temporal motion:
\[\tag{49}\boxed{x_1(t)-x_1(t_0)=\int_{t_0}^{t}c\Sin\theta(t')\,dt'}.\] For a constant splitting angle:
\[\tag{50}x_1(t)=x_1(t_0)+c\Sin\theta\,(t-t_0).\] In the special case of external motion \(\theta=\pi b\). Since \(b=\arcsin\beta/\pi\), we obtain
\[\tag{51}\Sin(\pi b)=\beta=\frac vc,\qquad \frac{dx_1}{dt}=c\Sin(\pi b)=v.\] Therefore
\[\tag{52}\boxed{x_1(t)-x_1(t_0)=\int_{t_0}^{t}v(t')\,dt'}.\] Now the new direction has not only an orientation, but also a coordinate, allowing us to distinguish positions. Only after this is the original temporal manifold expanded by the first spatial direction:
\[\tag{53}\boxed{\mathcal M_0=\mathbb R_\tau\quad\longrightarrow\quad \mathcal M_1=\mathbb R_\tau\oplus\mathbb R_{x_1}}.\] In this construction, space is not added to time from the outside. It appears along with the orthogonal projection of the temporal field. The new direction and its field component arise as two sides of a single splitting.
Space can be defined as the set of independent directions arising from the orthogonal splitting of the primary temporal field.
This changes the usual sequence. Instead of the "first space, then the field in it" scheme, we get:
\[\tag{54}\boxed{\text{temporal field}\;\longrightarrow\;\text{orthogonal projection}\;\longrightarrow\;\text{direction}\;\longrightarrow\;\text{coordinate}\;\longrightarrow\;\text{space}}\] 13. Internal and External Spatial Manifestations
In the operator \(J(a,b)=\j^a(-\j)^b\), the parameters \(a\) and \(b\) describe differentMethods of manifestation of spatial components.
When \(a>0\) and \(b=0\), internal rotation occurs. The temporal field acquires directions in the internal phase planes, but the center of the wave structure does not yet move in external space. Here, internal orbits, sheets, a direction of rotation, and proper momenta may subsequently appear.
When \(b>0\), external motion of the center occurs. Its parameter is related to the observed velocity:
\[\tag{55}b=\frac{\arcsin\beta}{\pi},\qquad \beta=\frac vc.\] In this case, part of the temporal structure acquires the direction of external translation. Therefore, the parameters perform different functions:
\[\tag{56}\boxed{a>0\;\longrightarrow\;\text{internal spatial manifestation},\qquad b>0\;\longrightarrow\;\text{external spatial manifestation}}\] Neither \(a\) nor \(b\) create a new field out of nothing. They define different ways of organizing and projecting an existing temporal field.
14. Possible Origin of Multidimensionality
One independent splitting creates one new direction. Repeated orthogonal splittings can increase the number of independent directions:
\[\tag{57}\mathbb R_\tau\longrightarrow\mathbb R_\tau\oplus\mathbb R_{x_1}\longrightarrow\mathbb R_\tau\oplus\mathbb R_{x_1}\oplus\mathbb R_{x_2}\longrightarrow\mathbb R_\tau\oplus\mathbb R^3.\] This leads to the hypothesis that the dimension of space can be determined by the number of independent stable splittings of the temporal field. However, the idempotent decomposition alone does not yet indicate why the observable space must have precisely three dimensions.
For a rigorous derivation, it is necessary to further establish the mutual orthogonality of the three directions, a spatial metric, a distance measurement rule, and a general method for reconciling the local projections of different particles. Therefore, formula (38) should be understood as a direction for further research, not as a complete proof of three-dimensionality.
15. Observable Fields as Projections
After the emergence of independent spatial directions, the temporal field can have different observable projections. We denote the rule for such a projection by \(\Pi_k\). Then
\[\tag{58}\boxed{\mathcal F_k=\Pi_k[\mathcal T_J]}.\] The subscript \(k\) denotes a specific way of organizing and observing the projection. It depends not on a new primary source, but on the geometric properties of the wave structure itself:
\[\tag{59}\Pi_k=\Pi_k(a,N,b,\chi,\Omega,\dot b,\ldots),\] where \(N\) is the number of sheets, \(\chi\) is the degree of closure, \(\Omega\) is the orientation of the internal plane, and \(\dot b\) characterizes the change in external motion.
All observed spatial fields are considered as different projections of a single temporal field, and not as independent primary entities.
16. Criteria for the Formation of Spatial Fields
Internal Dynamics. The condition \(\dot a\ne0\) defines the frequency, phase, energy, and internal rotation of the wave structure.
Number of Sheets. A single-sheet structure contains a common field. Multi-sheet structure divides it into average and difference components, which can have different external projections.
External Motion. When \(b>0\), part of the internal structure acquires the direction of external spatial transfer.
Internal Circulation. A closed oriented projection can create its own momentum even with a stationary center.
Opening. When the closure is broken, the internal sequence of states unfolds along the direction of propagation. For a two-sheet structure, the two sheets become two consecutive parts of the wave period.
Acceleration. If \(b=b(\tau)\), then an additional component appears, proportional to \(\dot b\), which can be associated with radiation.
Orientation of the internal plane. It determines the direction of spatial projection and can subsequently manifest itself as the orientation of the spin or polarization.
Splitting depth. Repeated levels of idempotent splitting can create additional independent difference components and corresponding fields.
In the second part, these features will be considered not separately, but in combination. It is the combination of internal dynamics, number of sheets, movement, and closure that should determine the physical appearance of the observed field.
17. Result Boundaries
The proposed construction demonstrates a logically consistent mechanism by which spaceTime directions can arise from the splitting of the temporal field. However, at this stage, this is a geometric principle of the model, not a complete derivation of physical space.
We still need to obtain:
— the existence of precisely three stable spatial dimensions;
— their mutual orthogonality and a common metric;
— a rule for measuring distances;
— the commonality of space for different wave structures;
— the consistency of local projections of interacting particles;
— equations and experimentally verifiable coefficients of specific fields.
— their mutual orthogonality and a common metric;
— a rule for measuring distances;
— the commonality of space for different wave structures;
— the consistency of local projections of interacting particles;
— equations and experimentally verifiable coefficients of specific fields.
Therefore, it is necessary to distinguish between the already obtained mathematical principle of splitting and its proposed physical interpretation. Idempotent mathematics ensures the completeness and orthogonality of the components. The identification of these components with spatial directions is a new postulate, the implications of which remain to be tested.
18. Transition to Part Two
Part Two will examine how classification features transform the temporal field into known spatial manifestations. The following sequence is expected:
\[\tag{60} \boxed{\begin{aligned}\text{two-sheet difference projection}&\;\longrightarrow\;\text{electric field},\\ \text{two-sheet nature and }b>0&\;\longrightarrow\;\text{magnetic field},\\ \text{internal circulation}&\;\longrightarrow\;\text{magnetic moment},\\ \text{opening of two sheets}&\;\longrightarrow\;\text{electromagnetic wave period},\\ \dot b\ne0&\;\longrightarrow\;\text{radiation}. \end{aligned}}\] Conclusion
The initial state \(a=b=0\) contains no internal or external motion and is described by the operator \(J=1\). Nevertheless, the sequence of events creates a primary temporal direction. Moving along it defines a temporal gradient.
\[\tag{61}\mathcal T_0=c\mathbf e_0.\] When the global state \(J\) begins to change along this direction, a temporal field of a specific wave structure emerges:
\[\tag{62}\boxed{\mathcal T_J=\frac{dJ}{d\tau}}.\] Idempotent splitting allows one to represent a unified field as a sum of mutually orthogonal components without destroying or doubling the original value. A new independent component can define a direction distinct from time. Therefore, within the framework of the proposed model, space can be viewed as a system of directions of splitting of the primary temporal field.
\[\tag{63}\boxed{\text{temporal field}\;\longrightarrow\;\text{orthogonal splittings}\;\longrightarrow\;\text{space and spatial projections}}\] In this understanding, space is not a pre-existing container for a field. A spatial direction arises simultaneously with the projection of the temporal field onto this direction. And all observed fields represent different ways of organizing, splitting, and spatially mapping a single primary temporal dynamic.
Space does not contain a primary field. Space itself arises as a structure of independent projections of the temporal field.