2026-09-01
The hypothesis of the expansion of the Universe as an upward splitting of space
In the Wave Electricity concept, multi-level idempotent splitting is used to describe internal state channels. This article considers a scaled-down extension of this idea: successive ascending splitting, in which each new level is orthogonal to the already observed space. It is hypothesized that such splitting increases the metric norm of spatial intervals and is therefore observed as an expansion of the Universe.
The presented construction is a working cosmological hypothesis. The geometric increase in the norm follows from the orthogonality of successive levels. However, the law determining the rate of ascending splitting and its absolute time scale at this stage are additional assumptions of the model and are subject to verification based on observed data.
1. Two Directions of Splitting
We will distinguish between downward and upward splitting. Downward splitting reveals the internal structure of a localized state: internal branches, additional channels, and deeper levels. Upward splitting, on the other hand, connects already formed states with subsequent spatial scales:
\[\tag{1} \boxed{ \begin{aligned} \text{downward splitting}&:\quad \text{internal structure of the state},\\ \text{upward splitting}&:\quad \text{increase in the scale of spatial projection}. \end{aligned}} \] An upward transition should not be understood as the ordinary movement of a body in an additional physical direction. The new level is orthogonal to the currently observed level and is not directly registered within it. Only the changed total metric length becomes observable.
2. Operator of a Single Ascending Step
Consider two comoving points and denote the distance vector between them at level \(n\) by \(\mathbf D_n\). Let the operator \(S_n\) transfer this interval to a new orthogonal channel:
\[\tag{2} S_n\mathbf D_n\perp\mathbf D_n, \qquad \lVert S_n\mathbf D_n\rVert=\lVert\mathbf D_n\rVert. \] We define one step of the ascending splitting by the operator
\[\tag{3} \boxed{ \mathbf D_{n+1} =U_{\varepsilon_n}\mathbf D_n =\mathbf D_n+\varepsilon_n S_n\mathbf D_n, } \] where \(\varepsilon_n\) is the dimensionless depth of a given step. Due to orthogonality, the cross product disappears:
\[\tag{4} \begin{aligned} D_{n+1}^{2} &=\lVert\mathbf D_n+\varepsilon_nS_n\mathbf D_n\rVert^2\\ &=D_n^2+\varepsilon_n^2D_n^2. \end{aligned} \] Consequently, the physical length of the interval increases according to the law
\[\tag{5} \boxed{ D_{n+1}=D_n\sqrt{1+\varepsilon_n^2}. } \] The hidden component is not visible in the old projection because
\[\tag{6} \mathcal P_n[\varepsilon_nS_n\mathbf D_n]=0, \] but enters the full norm quadratically. Therefore, the increase in distance can be observed without the usual radial motion of an object through local space.
3. Absence of a Distinguished Center
The cosmological expansion should act equally on any free interval. Therefore, it is assumed that within a single cosmological level, the parameter \(\varepsilon_n\) is independent of the position of a pair of points and the direction of the vector connecting them. For any pairs \(A,B\) and \(C,D\), then
\[\tag{7} \frac{D_{AB,n+1}}{D_{AB,n}} =\frac{D_{CD,n+1}}{D_{CD,n}} =\sqrt{1+\varepsilon_n^2}. \] Thus, the operator scales intervals, not absolute coordinates, relative to some center. Each observer sees the same picture of the removal of other free objects.
Homogeneity and isotropy do not follow automatically from orthogonality. They are included in the hypothesis as a requirement: the ascending operator must act equally on all free spatial intervals of a given level.
4. Scale factor and accumulated splitting
After \(N\) successive steps, the scale factor is
\[\tag{8} A_N =\prod_{n=0}^{N-1}\sqrt{1+\varepsilon_n^2}. \] Introduce the accumulated upward splitting variable
\[\tag{9} \boxed{ \Sigma_N =\sum_{n=0}^{N-1}\ln(1+\varepsilon_n^2). } \] Then
\[\tag{10} \boxed{ A_N=e^{\Sigma_N/2}. } \] In the continuous cosmological description, we will use
\[\tag{11} \boxed{ A(t)=e^{\Sigma(t)/2}, \qquad H(t)=\frac{\dot A}{A}=\frac12\dot\Sigma(t). } \] For the comoving distance \(R(t)=A(t)\chi\), where \(\chi\) is constant, the law follows directly from this
\[\tag{12} \boxed{ \dot R=H(t)R. } \] Thus, Hubble's law is a kinematic consequence of general self-similar scaling. It does not yet define the function \(H(t)\), but relates it to the rate of accumulation of upward splitting.
5. Relationship with the Second Moment of Vacuum Motion
The resulting scaling factor can be related to the more general hypothesis of the vacuum as a balanced state of motion. Let the local components of this motion be characterized by dimensionless velocities \(\beta_i\), with their first moment equal to zero and their second moment conserved:
\[\tag{13} \left\langle\beta_i\right\rangle=0, \qquad C_{ij}(t)=\left\langle\beta_i\beta_j\right\rangle\ne0. \] The tensor \(C_{ij}\) describes the distribution of the latent vacuum motion by directions. For a homogeneous and isotropic state
\[\tag{14} \boxed{ C_{ij}(t) =\frac{\left\langle\beta^2(t)\right\rangle}{3}\delta_{ij}. } \] The normalized tensor \(3C_{ij}/\langle\beta^2\rangle\) defines the shape of the spatial metric but eliminates its overall scale. To connect the second moment with the cosmological expansion, we introduce an additional postulate: the square of the scale factor is equal to the ratio of the traces \(C_{ij}\) at the considered and reference moments of time:
\[\tag{15} \boxed{ A^2(t) =\frac{\operatorname{Tr}C(t)}{\operatorname{Tr}C(t_{\mathrm{ref}})} =\frac{\left\langle\beta^2(t)\right\rangle} {\left\langle\beta^2(t_{\mathrm{ref}})\right\rangle}. } \] Comparison with the formula \(A=e^{\Sigma/2}\) gives a physical interpretation of the accumulated variable:
\[\tag{16} \boxed{ \Sigma(t) =\ln\frac{\left\langle\beta^2(t)\right\rangle} {\left\langle\beta^2(t_{\mathrm{ref}})\right\rangle}. } \] Therefore, the Hubble parameter is expressed in terms of the rate of change of the second moment of the vacuum motion:
\[\tag{17} \boxed{ H(t) =\frac12\dot\Sigma(t) =\frac12\frac{d}{dt}\ln\left\langle\beta^2(t)\right\rangle. } \] In this interpretation, expansion does not mean the resulting vacuum flux, but rather the growth of the metric scale of its split state. The first moment remains zero, while changing the second moment increases the distances between free comoving points.
The relation \(A^2\propto\operatorname{Tr}C\) does not follow automatically from the orthogonality of the branches alone. It is introduced as an additional cosmological postulate linking the upward splitting with the metric structure of the vacuum motion.
6. Discreteness and the Continuous Limit
If the steps occur at a fundamental interval \(\tau_n\), then on each segment
\[\tag{18} H_n =\frac{1}{2\tau_n}\ln(1+\varepsilon_n^2). \] For small splitting
\[\tag{19} H_n\approx\frac{\varepsilon_n^2}{2\tau_n}. \] For a finite continuous limit, it is necessary that
\[\tag{20} \boxed{ \varepsilon^2(t)=2H(t)\,dt. } \] Therefore, the elementary orthogonal addition has scale \(\varepsilon\sim\sqrt{dt}\), while the observed norm change is proportional to \(dt\). This indicates that the continuous pattern is the limit of a large number of microscopic steps, rather than a simple smooth motion in a single hidden direction.
7. Minimal Dynamical Hypothesis
Geometry determines the ratio \(H=\dot\Sigma/2\), but does not specify the absolute rate of splitting. To close the model, we adopt the following working principle: the square of the total rate of upward splitting is the sum of the contribution of the matter density and the free contribution of space itself.
\[\tag{21} \boxed{ \dot\Sigma^2 =\frac{32\pi G}{3}\rho+4\kappa^2. } \] Here \(\rho\) is the density of matter and radiation, and \(\kappa\) is the natural frequency of free ascending splitting. The positive root corresponds to the expanding branch:
\[\tag{22} \boxed{ \dot\Sigma =2\sqrt{\frac{8\pi G}{3}\rho+\kappa^2}. } \] The density coefficient is consistent with the energy balance of a homogeneous spherical region. For radius \(R\), density \(\rho\), and mass
\[\tag{23} M(R)=\frac{4\pi}{3}\rho R^3 \] the specific balance of radial motion has the form
\[\tag{24} \frac{\dot R^2}{2}-\frac{GM(R)}{R}=\mathcal E. \] When \(\dot R=HR\), we obtain
\[\tag{25} H^2 =\frac{8\pi G}{3}\rho+\frac{2\mathcal E}{R^2}. \] For a spatially flat branch \(\mathcal E=0\). The independent orthogonal contribution of the free splitting adds quadratically and yields
\[\tag{26} \boxed{ H^2=\frac{8\pi G}{3}\rho+\kappa^2. } \] Equation (26) is not a derivation of cosmological dynamics from a single idempotent algebra. It is the minimal dynamical law of the hypothesis, which reconciles the orthogonal addition of disintegration rates with a uniform gravitational energy balance.
8. Matter and Radiation
Since the volume of the free comoving region changes as
\[\tag{27} V\propto A^3=e^{3\Sigma/2}, \] for non-interacting non-relativistic matter, we obtain
\[\tag{28} \boxed{ \rho_m(\Sigma)=\rho_{m0}e^{-3\Sigma/2}. } \] For radiation, the energy of each quantum further decreases, therefore
\[\tag{29} \boxed{ \rho_r(\Sigma)=\rho_{r0}e^{-2\Sigma}. } \] The closed equation of evolution takes the form
\[\tag{30} \boxed{ \dot\Sigma =2\sqrt{ \kappa^2 +\frac{8\pi G}{3} \left( \rho_{m0}e^{-3\Sigma/2} +\rho_{r0}e^{-2\Sigma} \right)}. } \] If today \(A(t_0)=1\), then \(\Sigma(t_0)=0\). Let's introduce modern fractions
\[\tag{31} \Omega_{m0}=\frac{8\pi G\rho_{m0}}{3H_0^2}, \qquad \Omega_{r0}=\frac{8\pi G\rho_{r0}}{3H_0^2}, \qquad \Omega_{s0}=\frac{\kappa^2}{H_0^2}. \] For a flat model
\[\tag{32} \Omega_{m0}+\Omega_{r0}+\Omega_{s0}=1. \] Then
\[\tag{33} \boxed{ \dot\Sigma =2H_0\sqrt{ \Omega_{r0}e^{-2\Sigma} +\Omega_{m0}e^{-3\Sigma/2} +\Omega_{s0}}. } \] 9. Latent Acceleration
Differentiating equation (26) and using the laws of density variation, we obtain
\[\tag{34} \dot H =-4\pi G \left( \rho_m+\frac{4}{3}\rho_r \right). \] Therefore, the acceleration of the scale factor is
\[\tag{35} \boxed{ \frac{\ddot A}{A} =\kappa^2 -\frac{4\pi G}{3} \left( \rho_m+2\rho_r \right). } \] Matter and radiation slow the expansion, while free upward spallation creates positive acceleration. It becomes observable when
\[\tag{36} \boxed{ \kappa^2> \frac{4\pi G}{3} \left( \rho_m+2\rho_r \right). } \] In this interpretation, acceleration is not an additional force pushing galaxies apart. It arises from the multiplicative growth of the total norm: each new orthogonal contribution is proportional to the scale already achieved.
10. Cosmological Redshift
If the free wavelength scales with free spatial intervals, then
\[\tag{37} 1+z =\frac{A(t_0)}{A(t_{\mathrm{em}})}. \] Using \(A=e^{\Sigma/2}\), we get
\[\tag{38} \boxed{ 1+z =\exp\left[ \frac{\Sigma(t_0)-\Sigma(t_{\mathrm{em}})}{2} \right]. } \] When normalized, \(\Sigma(t_0)=0\)
\[\tag{39} \boxed{ \Sigma(z)=-2\ln(1+z), \qquad \Delta\Sigma(z)=2\ln(1+z). } \] Here, it is necessary to distinguish between the absolute accumulated splitting and the value measured relative to the present moment. If \(\Sigma_{\mathrm{abs}}(t)\) increases monotonically, then the variable used in cosmological formulas is defined as
\[\tag{40} \boxed{ \Sigma(t) =\Sigma_{\mathrm{abs}}(t)-\Sigma_{\mathrm{abs}}(t_0). } \] Therefore, a positive absolute accumulation does not contradict a negative value of \(\Sigma(z)\) in the past: it only means that fewer levels of upward splitting had accumulated by the time of emission than by the present.
Hence, equation (33) yields the observed dependence.
\[\tag{41} \boxed{ H(z) =H_0\sqrt{ \Omega_{r0}(1+z)^4 +\Omega_{m0}(1+z)^3 +\Omega_{s0}}. } \] Equality (39) itself is not yet an independent confirmation of the model: it is equivalent to determining the scale factor. Law (41) becomes verifiable if the splitting parameters are determined independently.
11. The Age of the Universe and the Definition of \(\kappa\)
In the "matter plus free splitting" approximation, when the contribution of radiation in the late epoch is small, the solution of the equation is
\[\tag{42} \boxed{ A(t) =\left( \frac{\Omega_{m0}}{\Omega_{s0}} \right)^{1/3} \operatorname{sinh}^{2/3} \left( \frac{3\kappa t}{2} \right). } \] From the condition \(A(t_0)=1\) it follows
\[\tag{43} \boxed{ \kappa =\frac{2}{3t_0} \operatorname{arsinh} \sqrt{ \frac{1-\Omega_{m0}}{\Omega_{m0}} }. } \] With approximate values
\[\tag{44} t_0=13.797\, \text{billion years}, \qquad \Omega_{m0}=0.315 \] we get
\[\tag{45} \boxed{ \kappa \approx 0.05705\, \text{billion years}^{-1} \approx 1.81\cdot10^{-18}\, \text{s}^{-1}. } \] The characteristic free decay time is
\[\tag{46} \boxed{ \tau_{\kappa}=\frac{1}{\kappa} \approx 17.53\, \text{billion years}. } \] From \(H_0=\kappa/\sqrt{1-\Omega_{m0}}\) we get
\[\tag{47} H_0\approx67.4\ \frac{\text{km/s}}{\text{pc}}. \] The quantity equivalent to the cosmological constant is
\[\tag{48} \boxed{ \Lambda_{\mathrm{eff}} =\frac{3\kappa^2}{c^2} \approx1.09\cdot 10^{-52}\, \text{m}^{-2}. } \] In the proposed hypothesis, \(\Lambda_{\mathrm{eff}}\) is interpreted not as an independent substance, but as the square of the natural frequency of free upward splitting.
The number \(\kappa t_0\approx0.787\) is not considered a new fundamental constant. It depends on the present-day ratio \(\Omega_{s0}/\Omega_{m0}\). Therefore, \(\kappa\), \(t_0\), or an equivalent pair of cosmological parameters must remain observable input data for now.
12. What is derived and what is assumed in the hypothesis
From the orthogonal geometry of the upward splitting it follows:
1. increase in the norm of one interval according to the law \(D_{n+1}=D_n\sqrt{1+\varepsilon_n^2}\);
2. multiplicative accumulation of scale \(A=e^{\Sigma/2}\);
3. Kinematic relation \(H=\dot\Sigma/2\);
4. Law of removal of free accompanying objects \(\dot R=HR\);
5. Possibility of hidden acceleration due to the growth of the full norm of orthogonal components.
Additionally accepted:
1. Universality of the ascending operator for all free intervals;
2. Relationship of the cosmological scale with the second moment of vacuum motion according to the law \(A^2=\operatorname{Tr}C/\operatorname{Tr}C_{\mathrm{ref}}\);
3. Dynamical law \(H^2=8\pi G\rho/3+\kappa^2\);
4. Constancy of \(\kappa\) in its simplest form;
5. Scaling of the free wavelength along with the spatial scale;
6. Absence of cosmological scaling within stably coupled systems.
13. Testability of the hypothesis
In its simplest form, the \(H(z)\) dependence mathematically coincides with a flat model containing matter, radiation, and a constant vacuum contribution. Therefore, simply replacing the notation \(\Lambda\) with \(\kappa\) does not create a new observable prediction.
An independent verification will be possible if at least one of the following results can be obtained from the splitting structure:
1. An independent value of \(\kappa\);
2. A weak evolution of \(\kappa(z)\) that differs from constant;
3. A connection between \(\kappa\) and the density or internal frequencies of matter;
4. A difference in the growth of cosmological inhomogeneities;
5. The limit at which upward splitting ceases to act on bound systems.
The evolution of free splitting can be directly related to the observed equation of state of dark energy. We allow the quantity \(\kappa\) to depend on the scale factor:
\[\tag{49} \boxed{ H^2 =\frac{8\pi G}{3}\rho+\kappa^2(A). } \] The corresponding effective density of free splitting is
\[\tag{50} \boxed{ \rho_s(A)=\frac{3\kappa^2(A)}{8\pi G}. } \] If we imagine it as a cosmological component with state parameter \(w_s\), then from the law of conservation of energy it follows
\[\tag{51} \boxed{ \begin{aligned} w_s(A) &=-1-\frac{2}{3}\frac{d\ln\kappa}{d\ln A},\\ w_s(z) &=-1+\frac{2(1+z)}{3}\frac{d\ln\kappa}{dz}. \end{aligned} } \] Thus, the constant free decay rate corresponds to the cosmological constant, and its change creates a testable deviation:
\[\tag{52} \boxed{ \begin{aligned} \kappa=\mathrm{const} &\quad\Longrightarrow\quad w_s=-1,\\ \frac{d\kappa}{dA}<0 &\quad\Longrightarrow\quad w_s>-1,\\ \frac{d\kappa}{dA}>0 &\quad\Longrightarrow\quad w_s<-1. \end{aligned} } \] Therefore, the observational recovery of \(w_s(z)\) is simultaneously a recovery of the function \(\kappa(z)\). If \(w_s=-1\) at all redshifts, the data are consistent with a constant frequency of free upward splitting. If a stable evolution of \(w_s\) is detected, the model should explain the corresponding change in \(\kappa\) from the internal splitting dynamics.
The calculated functions \(H(z)\), luminosity distance, and angular separation can then be compared with observations:
\[\tag{53} D_C(z)=c\int_0^z\frac{dz'}{H(z')}, \qquad D_L(z)=(1+z)D_C(z), \qquad D_A(z)=\frac{D_C(z)}{1+z}. \] Conclusion
ProposedThe second hypothesis associates the expansion of the Universe with a successive ascending splitting of space. Each new level is orthogonal to the previous one and is directly hidden, but increases the overall metric norm of the intervals. In the representation of the vacuum as a balanced motion, this growth is expressed through a change in the second moment \(C_{ij}=\langle\beta_i\beta_j\rangle\). Self-similarity makes the scaling multiplicative, giving rise to the Hubble law, and the free component of the splitting \(\kappa\) is capable of producing late acceleration.
\[\tag{54} \boxed{ \begin{aligned} C_{ij}=\left\langle\beta_i\beta_j\right\rangle &\quad\longrightarrow\quad A^2=\frac{\operatorname{Tr}C}{\operatorname{Tr}C_{\mathrm{ref}}}=e^{\Sigma},\\ H=\frac12\dot\Sigma &\quad\longrightarrow\quad \dot R=HR. \end{aligned} } \] At this stage, the model should be considered a hypothesis for a geometric expansion mechanism. The value of \(\kappa\) is determined from observed cosmological data and has not yet been obtained from the geometry of the upward splitting. The main further task is to construct a fundamental law for this process that would allow for an independent calculation of \(\kappa\) or a prediction of its evolution over time.