2026-08-30
Splitting Space and Motion as a Single Process
Vacuum, Spatial Extension, and the Euclidean Metric
Space is typically considered a pre-existing extended medium within which bodies can be at rest or in motion. This paper considers the opposite possibility: space does not precede motion, but emerges as a metric structure of distinguishable states created by motion. Motion should then be understood not as the transfer of a pre-existing object between pre-existing points, but as a physically manifested idempotent splitting of a state.
This approach allows us to connect several previously obtained results. Geometric splitting of coordinates shows that relative motion separates descriptions of a single event between independent idempotent components. Multilevel splitting creates an infinite sequence of orthogonal channels, the norm of whose natural self-similar state coincides with the Lorentz factor. Finally, the symmetric compensation of the two branches allows us to isolate the ordinary Euclidean metric as an unsplit projection of a more general \(i\)-geometry.
The main hypothesis of the work is as follows: motion and the physically manifested idempotent splitting are two descriptions of a single geometric process. Spatial extension arises as a metric relation between the states of this splitting.
It can be assumed that the vacuum is not an empty and motionless space, as it appears at first glance, but a hidden state of continuous motion. Individual movements within it may be directed oppositely and therefore not create a noticeable overall flow, but this does not mean that motion is truly absent. Perhaps it is precisely this internal dynamic that gives the vacuum extension and creates what we perceive as space. Next, we will consider successively whether this hypothesis can be expressed through idempotent splitting and derived from it motion, spatial metrics, and Euclidean geometry.
In this paper, it is necessary to strictly distinguish three levels of assertions. The algebraic properties of idempotents and state norms are proved mathematically. The identification of the splitting coefficient with velocity is introduced as a physical postulate. The interpretation of the vacuum and the origin of space is a consequence of this postulate, not algebra alone.
1. Idempotent splitting of unity
Let's start with the decomposition of unity into two mutually orthogonal idempotents:
\[\tag{1} \boxed{1=\ep+\em}. \] They satisfy the conditions
\[\tag{2} \ep^2=\ep, \qquad \em^{\,2}=\em, \qquad \ep\em=0. \] The difference of idempotents forms a hyperbolic unit:
\[\tag{3} \j=\ep-\em, \qquad \j^2=1. \] Conversely, idempotents are expressed in terms of \(1\) and \(\j\):
\[\tag{4} \ep=\frac{1+\j}{2}, \qquad \em=\frac{1-\j}{2}. \] Any two values \(A_+\) and \(A_-\) can be stored within a single split object:
\[\tag{5} \mathcal A=\ep A_+ +\em A_-. \] If the values are the same, the splitting ceases to be evident in the observable result:
\[\tag{6} A_+=A_-=A \quad\Longrightarrow\quad \mathcal A=(\ep+\em)A=A. \] The formal equality \(1=\ep+\em\) expresses the potential for a state to split. Splitting becomes physically manifest only when its independent branches acquire different states or amplitudes.
2. Split Coordinate
Let one coordinate correspond to two idempotent values \(dx_+^i\) and \(dx_-^i\). Let us introduce the complete split coordinate.
\[\tag{7} \boxed{ d\mathcal X^i = \ep\,dx_+^i+\em\,dx_-^i }. \] Using formulas (4), we obtain
\[\tag{8} \boxed{ d\mathcal X^i = \frac{dx_+^i+dx_-^i}{2} + \j\frac{dx_+^i-dx_-^i}{2} }. \] Let's denote the mean and difference parts:
\[\tag{9} d\bar x^i = \frac{dx_+^i+dx_-^i}{2}, \qquad d\widetilde x^i = \frac{dx_+^i-dx_-^i}{2}. \] Then the coordinate takes a compact form.
\[\tag{10} \boxed{ d\mathcal X^i=d\bar x^i+\j\,d\widetilde x^i }. \] The quantity \(d\bar x^i\) contains the common part of the two branches, while \(d\widetilde x^i\) measures their difference. The ordinary coordinate is obtained when \(d\widetilde x^i=0\), but this is not the only case where the measured metric becomes unsplit.
3. Split prime \(i\)-metric
In the extended basis
\[\tag{11} \left\{\ep,\,i\ep,\,\em,\,i\em\right\} \] each idempotent component forms its own complex plane. Let \(dS_+^2\) and \(dS_-^2\) denote positive-definite metric formsof these two planes. Then we call the idempotent-valued quantity a complete \(i\)-metric.
\[\tag{12} \boxed{ d\mathcal S_i^2 = \ep\,dS_+^2+\em\,dS_-^2 }. \] Substituting expressions (4), we find.
\[\tag{13} \boxed{ d\mathcal S_i^2 = \frac{dS_+^2+dS_-^2}{2} + \j\frac{dS_+^2-dS_-^2}{2} }. \] We introduce two parts of the metric:
\[\tag{14} dS_{\mathrm{sym}}^2 = \frac{dS_+^2+dS_-^2}{2}, \qquad dS_{\mathrm{split}}^2 = \frac{dS_+^2-dS_-^2}{2}. \] Then
\[\tag{15} \boxed{ d\mathcal S_i^2 = dS_{\mathrm{sym}}^2 +\j\,dS_{\mathrm{split}}^2 }. \] The first part is common to both branches, the second preserves their metric distinction. Thus, \(i\)-geometry contains the usual scalar metric and an additional hyperbolic component in a single object.
4. Euclidean Geometry as a Symmetric Sector
If the square of the length in both branches is the same,
\[\tag{16} dS_+^2=dS_-^2=dS_E^2, \] then the difference part vanishes:
\[\tag{17} dS_{\mathrm{split}}^2=0. \] The full metric collapses:
\[\tag{18} \boxed{ d\mathcal S_i^2 =(\ep+\em)dS_E^2 =dS_E^2 }. \] Euclidean geometry is a symmetric scalar sector of the more general \(i\)-geometry, in which the coefficient of \(\j\) in the metric form is zero.
It is important that the disappearance of the \(\j\)-component of the metric does not require the disappearance of the \(\j\)-component of the coordinate. Consider opposite displacements.
\[\tag{19} dx_-^i=-dx_+^i=-dx^i. \] Then the mean coordinate is zero, and the difference coordinate is preserved:
\[\tag{20} d\bar x^i=0, \qquad d\widetilde x^i=dx^i, \qquad d\mathcal X^i=\j\,dx^i. \] However, the quadratic form becomes ordinary:
\[\tag{21} \begin{aligned} d\mathcal S_i^2 &=\delta_{ij}d\mathcal X^i d\mathcal X^j \\[1mm] &=\j^2\delta_{ij}dx^i dx^j \\[1mm] &=\delta_{ij}dx^i dx^j. \end{aligned} \] Therefore, the Euclidean metric does not prove the absence of hidden splitting. It shows that the squares of the lengths of the two branches are equal.
5. General criterion for the vanishing of the \(\j\)-part
Use representation (10) and calculate the quadratic form:
\[\tag{22} \begin{aligned} d\mathcal S_i^2 &=\delta_{ij} \left(d\bar x^i+\j d\widetilde x^i\right) \left(d\bar x^j+\j d\widetilde x^j\right) \\[1mm] &=\delta_{ij} \left( d\bar x^i d\bar x^j +d\widetilde x^i d\widetilde x^j \right) \\[1mm] &\quad +2\j\,\delta_{ij}d\bar x^i d\widetilde x^j. \end{aligned} \] Therefore, the precise condition for a Euclidean sector is
\[\tag{23} \boxed{ \delta_{ij}d\bar x^i d\widetilde x^j=0 }. \] In one dimension, this condition is usually realized either by the branches coinciding or by their opposites. In the multidimensional case, the mean and difference parts can also be mutually orthogonal.
6. Multilevel Idempotent Splitting
A single binary splitting is insufficient to describe the self-similar structure of motion. We introduce an independent hyperbolic unit \(\j_n\) at each level:
\[\tag{24} \j_n^2=1, \qquad \j_n\j_m=\j_m\j_n. \] It corresponds to a pair of idempotents
\[\tag{25} \ep_n=\frac{1+\j_n}{2}, \qquad \em_n=\frac{1-\j_n}{2}. \] At each level, the \(\em_n\) branch commits to a new state, and the \(\ep_n\) branch continues to split. Define
\[\tag{26} \boxed{ E_n = \left(\prod_{k=1}^{n}\ep_k\right)\em_{n+1}, \qquad n=0,1,2,\ldots }. \] The first states are of the form
\[\tag{27} E_0=\em_1, \qquad E_1=\ep_1\em_2, \qquad E_2=\ep_1\ep_2\em_3, \quad\ldots \] They are idempotent and mutually orthogonal:
\[\tag{28} \boxed{ E_nE_m=\delta_{nm}E_n }. \] Indeed, for \(m>n\), the state \(E_n\) contains \(\em_{n+1}\), while \(E_m\) contains at the same level \(\ep_{n+1}\). Their product vanishes.
The residual branch after \(N\) continuations is equal to
\[\tag{29} R_N=\prod_{k=1}^{N}\ep_k, \qquad R_0=1. \] At each level,
\[\tag{30} R_n=E_n+R_{n+1}, \] therefore, after a finite number of splittings
\[\tag{31} \boxed{ 1=\sum_{n=0}^{N}E_n+R_{N+1} }. \] 7. Self-similar splitting state
To avoid defining motion in terms of pre-existing coordinates, we first introduce a purely algebraic transition coefficient
\[\tag{32} \lambda\in(-1,1). \] We require that the ratio of the amplitudes of adjacent levels be constant:
\[\tag{33} \boxed{ \frac{a_{n+1}}{a_n}=\lambda }. \] Otsyuda
\[\tag{34} a_n=a_0\lambda^n. \] We define the squared norm of the state \(\mathcal A=\sum a_nE_n\) as the sum of the squares of the coefficients of mutually orthogonal channels:
\[\tag{35} \|\mathcal A\|^2 = \sum_{n=0}^{\infty}a_n^2. \] The unit norm condition yields
\[\tag{36} 1 =a_0^2\sum_{n=0}^{\infty}\lambda^{2n} =\frac{a_0^2}{1-\lambda^2}. \] Therefore, when choosing a positive initial amplitude
\[\tag{37} a_0=\sqrt{1-\lambda^2}. \] The only normalized self-similar state is of the form
\[\tag{38} \boxed{ \mathcal\Phi_{\lambda} = \sqrt{1-\lambda^2} \sum_{n=0}^{\infty}\lambda^nE_n }. \] In fully expanded idempotent form
\[\tag{39} \begin{aligned} \mathcal\Phi_{\lambda} =\sqrt{1-\lambda^2}\big(& \em_1 +\lambda\ep_1\em_2 \\[1mm] &+\lambda^2\ep_1\ep_2\em_3 \\[1mm] &+\lambda^3\ep_1\ep_2\ep_3\em_4 +\cdots\big). \end{aligned} \] The quantity \(\mathcal\Phi_{\lambda}\) itself is not idempotent. It represents the normalized state in a basis of mutually orthogonal idempotent channels \(E_n\).
8. Norm Preservation
Due to the orthogonality of the channels, the square of the state is equal to
\[\tag{40} \mathcal\Phi_{\lambda}^{\,2} = (1-\lambda^2) \sum_{n=0}^{\infty}\lambda^{2n}E_n. \] To neatly transition from an idempotent result to a scalar, we introduce a functional on absolutely summable coefficients.
\[\tag{41} \tau\left(\sum_{n=0}^{\infty}c_nE_n\right) = \sum_{n=0}^{\infty}c_n. \] Then
\[\tag{42} \|\mathcal A\|^2 \equiv \tau\left(\mathcal A^2\right), \] and for state (38)
\[\tag{43} \begin{aligned} \|\mathcal\Phi_{\lambda}\|^2 &=(1-\lambda^2) \sum_{n=0}^{\infty}\lambda^{2n} \\[1mm] &=1. \end{aligned} \] The geometric fraction of the norm of the \(n\)-th channel is
\[\tag{44} \boxed{ P_n=(1-\lambda^2)\lambda^{2n} }, \qquad \sum_{n=0}^{\infty}P_n=1. \] In this paper, \(P_n\) is called the geometric fraction of the norm. Its probabilistic interpretation is not required and is not separately postulated.
9. Physical Identification with Motion
Until now, \(\lambda\) was only an algebraic self-similarity coefficient. Now let's introduce a physical postulate:
\[\tag{45} \boxed{ \lambda=\beta=\frac{v}{c} }. \] Then the state of external motion takes the form
\[\tag{46} \boxed{ \mathcal\Phi_{\beta} = \sqrt{1-\beta^2} \sum_{n=0}^{\infty}\beta^nE_n }. \] When \(\beta=0\), only the initial channel is manifested:
\[\tag{47} \mathcal\Phi_0=E_0. \] For any \(\beta\ne0\), all levels acquire nonzero amplitudes:
\[\tag{48} a_n=\sqrt{1-\beta^2}\,\beta^n\ne0. \] Therefore, any nonzero external motion corresponds to an infinite sequence of manifest orthogonal states, although at low speeds, the amplitudes of deep levels rapidly decrease.
External motion does not create idempotents out of nothing. They belong to the potential structure of space. Motion manifests this structure by distributing the unit norm among its levels.
10. Inferring Motion from Splitting
Equivalence requires not only constructing a splitting based on a known velocity, but also reconstructing the velocity from the state itself. Let
\[\tag{49} \mathcal\Phi=\sum_{n=0}^{\infty}a_nE_n \] be a normalized self-similar state. Then
\[\tag{50} \boxed{ \beta=\frac{a_{n+1}}{a_n} }. \] In particular,
\[\tag{51} \boxed{ v=c\frac{a_1}{a_0} }. \] The initial amplitude determines the Lorentz factor:
\[\tag{52} a_0=\sqrt{1-\beta^2}=\frac{1}{\gamma}, \qquad \boxed{\gamma=\frac{1}{a_0}}. \] Thus, within the class of normalized self-similar states, the mapping is one-to-one:
\[\tag{53} \boxed{ \text{relative motion} \quad\Longleftrightarrow\quad \text{self-similar idempotent splitting} }. \] This statement does not apply to an arbitrary formal decomposition of identity. Orthogonality of the channels, unit norm, and a constant ratio of adjacent amplitudes are required.
11. The Lorentz Factor as the Norm of a Split State
Consider the Unnormalized State
\[\tag{54} \boxed{ \mathcal\Gamma_{\beta} = \sum_{n=0}^{\infty}\beta^nE_n }. \] Its square and norm are equal
\[\tag{55} \mathcal\Gamma_{\beta}^{\,2} = \sum_{n=0}^{\infty}\beta^{2n}E_n, \] \[\tag{56} \begin{aligned} \|\mathcal\Gamma_{\beta}\|^2 &=\tau\left(\mathcal\Gamma_{\beta}^{\,2}\right) \\[1mm] &=\sum_{n=0}^{\infty}\beta^{2n} \\[1mm] &=\frac{1}{1-\beta^2}. \end{aligned} \] Hence,
\[\tag{57} \boxed{ \|\mathcal\Gamma_{\beta}\| = \frac{1}{\sqrt{1-\beta^2}} =\gamma(\beta) }. \] The normalized and metric states are related by the formula
\[\tag{58} \mathcal\Phi_{\beta} = \frac{\mathcal\Gamma_{\beta}} {\|\mathcal\Gamma_{\beta}\|} = \sqrt{1-\beta^2}\,\mathcal\Gamma_{\beta}. \] A detailed construction of the infinite chain and two proofs of the origin of the Lorentz factor are given in the paper "Origin of the Lorentz Factor from Infinite Idempotent Splitting". Here, this result is included in the general connection between motion, space, and metrics.
12. Composition of Motions
In "Sum and Difference of Velocities as a Composition of Doppler Projections", the relativistic law of composition was obtained through the external Doppler scale.
\[\tag{59} D(\beta) = \sqrt{\frac{1+\beta}{1-\beta}}. \] Its logarithm forms the additive Doppler coordinate
\[\tag{60} \boxed{ q=\ln D=\operatorname{artanh}\beta }, \qquad \beta=\tanh q. \] For two collinear transformations
\[\tag{61} q_{12}^{(\pm)}=q_2\pm q_1, \] therefore
\[\tag{62} \boxed{ \beta_{12}^{(\pm)} = \frac{\beta_2\pm\beta_1} {1\pm\beta_1\beta_2} }. \] This Doppler derivation is not repeated in the present work. It is only necessary to require that the composition of the motions be compatible with the composition of the split states. Since
\[\tag{63} \sqrt{1-\beta^2}=\operatorname{sech}q, \] get
\[\tag{64} \boxed{ \mathcal\Phi_q = \operatorname{sech}q \sum_{n=0}^{\infty}\tanh^n q\,E_n }. \] The desired composition operation must satisfy
\[\tag{65} \boxed{ \mathcal\Phi_{q_1}\star\mathcal\Phi_{q_2} = \mathcal\Phi_{q_1+q_2} }. \] For relative motion:
\[\tag{66} \boxed{ \mathcal\Phi_{q_2}\star\mathcal\Phi_{-q_1} = \mathcal\Phi_{q_2-q_1} }. \] The operation \(\star\) currently specifies the required physical composition of states. It cannot be replaced by an ordinary algebraic product, which would multiply the coefficients \(\beta_1\beta_2\). Deriving \(\star\) from nested splitting operators remains a separate problem.
13. Relationship with the external parameter \(b\)
In the external part of the operator
\[\tag{67} J(t)=\j^a(-\j)^b \] parameter \(b\) is related to the velocity by the formula
\[\tag{68} b=\frac{\arcsin\beta}{\pi}, \qquad \beta=\sin(\pi b). \] Therefore, the normalized state of the external splitting can be written as
\[\tag{69} \boxed{ \mathcal\Phi_b = \cos(\pi b) \sum_{n=0}^{\infty} \sin^n(\pi b)\,E_n }. \] Its recursive structure is as follows
\[\tag{70} \mathcal\Phi_b = \cos(\pi b)E_0 +\sin(\pi b)\,\mathcal S\mathcal\Phi_b, \] where the transition operator satisfies
\[\tag{71} \mathcal S E_n=E_{n+1}. \] Thus, the parameter \(b\) can be interpreted not only as the angle of external motion, but also as a compact parameter of the depth of self-similar splitting. In this case, equality between the initial factor \((-\j)^b\) and the complete infinite state \(\mathcal\Phi_b\) is not yet postulated. It remains to be determined whether \((-\j)^b\) is its first projection or a compact algebraic representation.
14. Vacuum as Balanced Motion
Now let us move from the motion of an individual state to space as a whole. Suppose that the vacuum contains opposite components of motion that do not produce a resulting directed flow:
\[\tag{72} \boxed{ \left\langle\boldsymbol\beta\right\rangle_{\mathrm{vac}}=0 }. \] However, compensating for the first moment does not eliminate the quadratic value:
\[\tag{73} \boxed{ \left\langle\beta^2\right\rangle_{\mathrm{vac}}>0 }. \] For the pair \(+\boldsymbol\beta\) and \(-\boldsymbol\beta\)
\[\tag{74} \frac{\boldsymbol\beta+(-\boldsymbol\beta)}{2}=0, \qquad \frac{\beta^2+(-\beta)^2}{2}=\beta^2. \] The same can be seen from the norm fractions of the split states:
\[\tag{75} P_n(+\beta) =P_n(-\beta) =(1-\beta^2)\beta^{2n}. \] The sign of the direction changes the odd amplitudes, but does not destroy the square of the norm. Therefore, the vacuum can appear motionless, preserving the non-zero internal structure of the twomotion.
A vacuum in this hypothesis is not the absence of motion, but a state of motion without net displacement: the first moment is zero, and the second moment remains nonzero.
15. The Emergence of Spatial Extension
If only one indistinguishable state exists, then it is impossible to define a relationship "between" states. There is no spatial interval in such a description. Splitting creates a sequence of independent channels.
\[\tag{76} E_0,E_1,E_2,\ldots, \] between which a metric relationship can then be introduced. Therefore, the proposed sequence has the form
\[\tag{77} \boxed{ \text{change of state} \longrightarrow \text{splitting} \longrightarrow \text{distinguishable states} \longrightarrow \text{metric} \longrightarrow \text{extension} }. \] This definition avoids a logical circle. The primary motion here is not yet defined by the ratio \(dx/dt\), since the coordinate \(x\) must first emerge. Motion at the pre-geometric level is understood as a transition between distinguishable algebraic states: \[\tag{78} \frac{d\mathcal V}{d\tau}\ne0, \]
where \(\tau\) is the primary parameter of the sequence of states. After the emergence of a metric structure, coordinate and conventional velocity are introduced: \[\tag{79} \mathcal V(\tau) \longrightarrow \{E_n\} \longrightarrow x \longrightarrow v=\frac{dx}{dt}. \]
Space in this interpretation is not a receptacle of motion, but a metric manifestation of motion. Its extension is maintained by the continuous splitting of the vacuum state.
16. Closed Wave and Internal Geometry
The absence of external motion does not mean the absence of motion at all. For a particle stationary as a whole,
\[\tag{80} \beta=0, \qquad b=0, \] but the internal parameter can continue to change:
\[\tag{81} a=\varpi t, \qquad \dot a=\varpi\ne0. \] Then
\[\tag{82} J(t)=\j^a. \] If the internal state is periodically closed,
\[\tag{83} J(t+T)=J(t), \] it creates a compact internal geometry without causing open external motion. Therefore, a distinction must be made.
\[\tag{84} \boxed{ \begin{aligned} \text{closed motion} &\longrightarrow \text{internal geometry of the particle}, \\[1mm] \text{open motion} &\longrightarrow \text{external spatial extent}. \end{aligned} }. \] In this representation, the particle can be viewed as a stable, locally closed state of the general motion of the vacuum, rather than as an external object placed in an initially empty space.
17. The First and Second Moments of Vacuum Motion
Let the external state, after the emergence of directions, be characterized by components \(\beta_i\). The first moment
\[\tag{85} M_i^{(1)}=\left\langle\beta_i\right\rangle \] determines the observed flow. For a vacuum at rest
\[\tag{86} M_i^{(1)}=0. \] The metric structure is associated with the second moment.
\[\tag{87} \boxed{ C_{ij} = \left\langle\beta_i\beta_j\right\rangle }. \] Therefore, a zero first moment does not mean the disappearance of the metric:
\[\tag{88} \left\langle\beta_i\right\rangle=0, \qquad \left\langle\beta_i\beta_j\right\rangle\ne0. \] This difference is the basis for the transition from latent compensated motion to observable spatial extent.
18. Isotropy and the Euclidean Metric
The equality \(\langle\beta_i\rangle=0\) is not sufficient for Euclidean geometry. The compensated motion can remain anisotropic. An additional condition for the isotropy of the second moment is necessary:
\[\tag{89} \boxed{ \left\langle\beta_i\beta_j\right\rangle = \frac{\left\langle\beta^2\right\rangle}{3} \delta_{ij} }. \] Let's define a normalized spatial metric
\[\tag{90} \boxed{ g_{ij} = \frac{3}{\left\langle\beta^2\right\rangle} \left\langle\beta_i\beta_j\right\rangle }. \] For an isotropic vacuum
\[\tag{91} g_{ij}=\delta_{ij}. \] Therefore, the spatial interval takes on Euclidean form
\[\tag{92} \boxed{ d\ell^2 = \delta_{ij}dx^i dx^j = dx^2+dy^2+dz^2 }. \] Thus, two different conditions perform two different functions:
\[\tag{93} \boxed{ \begin{aligned} \left\langle\beta_i\right\rangle=0 &\quad\Longrightarrow\quad \text{no resulting flow}, \\[1mm] \left\langle\beta_i\beta_j\right\rangle \propto\delta_{ij} &\quad\Longrightarrow\quad \text{the metric is isotropic and Euclidean}. \end{aligned} }. \] Euclidean geometry arises not from the absence of motion, but from the symmetry and isotropic compensation of motion hidden in the split structure of the vacuum.
19. Symmetry Breaking and the Common Metric
If two idempotent branches have different squares of length,
\[\tag{94} dS_+^2\ne dS_-^2, \] then the \(\j\)-component of the metric is preserved:
\[\tag{95} d\mathcal S_i^2 = dS_{\mathrm{sym}}^2 +\j dS_{\mathrm{split}}^2, \qquad dS_{\mathrm{split}}^2\ne0. \] If the second moment of motion is not isotropic, then a general positive-definite spatial metric arises.
\[\tag{96} g_{ij} \propto \left\langle\beta_i\beta_j\right\rangle, \qquad g_{ij}\ne\delta_{ij}. \] This opens the possibility of further linking the anisotropy or curvature of the observed space with the symmetry breaking of the vacuum splitting. However, such a conclusion would require constructing the field \(g_{ij}(x,t)\), determining its dynamics, and comparing it with experiment. A gravitational interpretation is not introduced in this paper.
20. A New Postulate of Motion and Space
The resulting mathematical and physical principles allow us to formulate a new postulate of the concept.
Motion and the physically manifested idempotent splitting are a single geometric process. The ratio of the amplitudes of adjacent levels of external self-similar splitting is equal to the relative velocity \(\beta=v/c\). Spatial extension arises as a metric relation between the distinguishable states of this splitting.
The mathematical part of the postulate is written as
\[\tag{97} \boxed{ \mathcal\Phi_{\beta} = \sqrt{1-\beta^2} \sum_{n=0}^{\infty}\beta^nE_n, \qquad \frac{a_{n+1}}{a_n}=\beta, \qquad \|\mathcal\Phi_{\beta}\|=1 }. \] The vacuum state postulate has the form
\[\tag{98} \boxed{ \left\langle\boldsymbol\beta\right\rangle_{\mathrm{vac}}=0, \qquad \left\langle\beta^2\right\rangle_{\mathrm{vac}}>0 }. \] The Euclidean projection postulate:
\[\tag{99} \boxed{ dS_+^2=dS_-^2 \quad\Longrightarrow\quad d\mathcal S_i^2=dS_E^2 }. \] For a three-dimensional isotropic space:
\[\tag{100} \boxed{ \left\langle\beta_i\beta_j\right\rangle = \frac{\left\langle\beta^2\right\rangle}{3}\delta_{ij} \quad\Longrightarrow\quad g_{ij}=\delta_{ij} }. \] 21. Domain of Applicability and Open Questions
The proposed construction proves the properties of a self-similar idempotent state after orthogonal channels and a metric functional are specified. The physical equality \(\lambda=\beta\), the vacuum interpretation, and the origin of extension are postulates of the model that must be evaluated by their further consequences.
Several problems remain to be solved before the theory is complete. It is necessary to derive the operation \(\star\) directly from the algebra of multilevel splitting; establish the precise relationship between \((-\j)^b\) and the state \(\mathcal\Phi_b\); separately determine the boundary case \(|\beta|=1\); demonstrate the origin of precisely the three observed spatial directions; construct a local metric \(g_{ij}(x,t)\) and find verifiable differences from existing theories.
Furthermore, latent vacuum motion cannot be understood as the mechanical displacement of matter in an already existing space. Otherwise, a logical circle and a separate rest frame would arise. The primary process must be a transition between pre-geometric states, and its observed spatial interpretation must preserve the principle of relativity.
Conclusion
This paper constructs a consistent relationship between idempotent splitting, motion, spatial extent, and the Euclidean metric. The initial binary decomposition of unity creates two independent branches. Repeated splitting forms an infinite system of orthogonal idempotents \(E_n\). The only normalized self-similar state of this system is determined by the constant ratio of adjacent amplitudes.
After physically identifying this ratio with the velocity \(\beta=v/c\), motion and splitting become mutually reconcilable descriptions of a single process. The unnormalized length of the corresponding state is equal to the Lorentz factor, and the logarithmic Doppler coordinate \(q\) ensures the correct composition of collinear motions.
The vacuum is interpreted as a balanced motion: its first moment is zero, so there is no directed flow, but its second moment remains nonzero and maintains the metric extension. Atequality of metrics of two idempotent branches, the \(\j\)-component of the quadratic form disappears. With the additional isotropy of the second moment, the usual Euclidean metric arises.
\[\tag{101} \boxed{ \begin{gathered} \text{motion} \quad\Longleftrightarrow\quad \text{splitting}, \\[1mm] \text{splitting} \quad\Longrightarrow\quad \text{metric extension}, \\[1mm] \left\langle\boldsymbol\beta\right\rangle=0, \quad \left\langle\beta^2\right\rangle>0 \quad\Longrightarrow\quad \text{stationary, but extended vacuum}, \\[1mm] dS_+^2=dS_-^2 \quad\Longrightarrow\quad d\mathcal S_i^2=dS_E^2. \end{gathered} }. \] Space in the proposed picture is not an empty stage within which movement occasionally arises. It is a geometric manifestation of the continuous movement of a split vacuum state.