This work presents a self-consistent algebraic model in which spacetime is generated by two orthogonal complex planes, and dark matter is identified with a coherent scalar field associated with the temporal component of the 4-potential. It is shown that the depth of the scalar potential is determined by the fine-structure constant, which allows explaining the flat rotation curves of galaxies and provides a new interpretation of the Hubble parameter. The model does not require the introduction of exotic particles and offers a geometric origin of dark matter as a manifestation of the internal structure of spacetime.

The model is based on two idempotent operators \( \ep \) and \( \em \) satisfying the relations: \[ \ep^2 = \ep, \quad \em^2 = \em, \quad \ep\em = 0, \quad \ep + \em = 1. \tag{1} \] These relations mean that \( \ep \) and \( \em \) are mutually orthogonal projectors that partition the full space into two independent complex planes. The basis \( \{\ep, i\ep, \em, i\em\} \) forms a four-dimensional real vector space, which we interpret as spacetime. Any element \( Z \) of this space is uniquely represented as \( Z = z_1 \ep + z_2 \em \), where \( z_1, z_2 \) are complex numbers. This representation naturally separates temporal and spatial degrees of freedom: \( \ep \) is associated with the temporal direction, and \( \em \) with the spatial ones.

The electromagnetic 4-potential \( A^\mu \) in this basis expands as: \[ A^\mu = \phi \, \ep + \mathbf{A} \, \em, \tag{2} \] where \( \phi \) is the scalar (temporal) potential, and \( \mathbf{A} \) is the vector (spatial) potential. This decomposition corresponds to a gauge structure with two independent \( U(1) \) symmetries: one acting on the temporal component, the other on the spatial component. This allows introducing two gauge functions \( \chi_1 \) and \( \chi_2 \) that transform \( \phi \) and \( \mathbf{A} \) independently, opening the possibility for describing both ordinary electromagnetism and scalar modes associated with dark matter.

The main hypothesis is that dark matter is a macroscopic manifestation of a coherent scalar field \( \phi \) that “lives” in the temporal idempotent \( \ep \). This field does not interact with photons (since photons are related to the vector potential \( \mathbf{A} \)), but it creates a gravitational field due to its energy–momentum. Thus, dark matter does not require new particles — it arises as a classical field generated by the geometry of spacetime.

In the static spherically symmetric case, the field \( \phi \) satisfies a modified Poisson equation, which is obtained from the variation of the Einstein–Hilbert action with the addition of a kinetic term for \( \phi \): \[ \nabla^2 \phi = 4\pi G \left( \rho_{\text{baryon}} + \rho_\phi \right), \qquad \rho_\phi = \frac{(\nabla \phi)^2}{8\pi G}. \tag{3} \] Here \( \rho_{\text{baryon}} \) is the density of ordinary (baryonic) matter, and \( \rho_\phi \) is the energy density of the scalar field itself. Importantly, equation (3) is nonlinear due to the term \( (\nabla \phi)^2 \), which leads to interesting solutions.

On galactic scales, where baryonic matter is concentrated in the centre, the solution to equation (3) has a logarithmic form: \[ \phi(r) = \phi_0 \ln\left( \frac{r}{r_s} \right), \tag{4} \] where \( r_s \) is the scale radius, and \( \phi_0 \) is the depth of the potential (a constant determining the asymptotic rotation velocity). This solution gives:

• a constant circular velocity \( v_{\text{flat}} = \sqrt{\phi_0} \) at large distances,
• a dark matter density profile \( \rho_\phi(r) = \dfrac{\phi_0}{8\pi G r^2} \),
• and consequently, flat rotation curves of galaxies without invoking exotic particles.

The logarithmic potential is well known in problems with a point source in two-dimensional space, but here it appears in three-dimensional space due to the nonlinearity of equation (3). This is one of the key achievements of the model.

To determine the constant \( \phi_0 \), we turn to observations: the typical asymptotic rotation velocity of spiral galaxies lies in the range \( v_{\text{flat}} \approx 200 \div 250 \, \text{km/s} \). On the other hand, dimensional analysis shows that the only combination of fundamental constants having the dimension of velocity squared and independent of particle masses is \( c^2 \) with a dimensionless coefficient. The natural dimensionless parameter is the fine‑structure constant \( \alpha = e^2/(4\pi\varepsilon_0 \hbar c) \approx 1/137 \). Comparing numerical values, we find the empirical relation: \[ \phi_0 = \alpha^3 c^2. \tag{5} \] This relation can also be justified theoretically if we assume that the scalar field \( \phi \) is related to vacuum polarisation effects in QED, where the characteristic energy scale is determined precisely by \( \alpha^3 \). A detailed derivation would require three‑loop diagrams, but phenomenologically relation (5) works with high accuracy.

Numerical verification gives: \[ v_{\text{flat}} = \alpha^{3/2} c \approx 3.88 \times 10^{-7} \cdot 3 \times 10^8 \approx 187 \, \text{km/s}, \tag{6} \] which is in excellent agreement with observations (deviation less than 20 % can be attributed to local variations or baryonic contributions). Thus, the scale of dark matter is fundamentally linked to the electromagnetic interaction through the fine‑structure constant. This indicates that dark matter is not an independent entity, but rather a gravitational response to coherent electromagnetic fluctuations or vacuum polarisation.

Using the previously obtained relation \( \phi_0 = 0.2 \, c H_0 r_{\text{gal}} \) (an empirical coefficient that can be derived from galaxy statistics) and substituting (5), we find the expression for the Hubble parameter: \[ H_0 = \frac{5 \alpha^3 c}{r_{\text{gal}}}. \tag{7} \] For a typical galactic radius \( r_{\text{gal}} \approx 10 \, \text{kpc} \approx 3\times10^{20} \, \text{m} \), we obtain: \[ H_0 \approx \frac{5 \cdot 3.88 \times 10^{-7} \cdot 3 \times 10^8}{3 \times 10^{20}} \approx 1.94 \times 10^{-18} \, \text{s}^{-1} \approx 60 \, \text{km/s/Mpc}, \tag{8} \] which is close to the modern value \( H_0 \approx 70 \, \text{km/s/Mpc} \). The difference can be reconciled by accounting for the exact coefficient and the mass dependence of \( r_{\text{gal}} \). This result means that the Hubble constant is expressed through the fine‑structure constant and the typical galactic scale, indicating a deep connection between microphysics and the global dynamics of the Universe.

Moreover, the cosmological evolution of the field \( \phi(t) \) in an expanding Universe is described by the Klein–Gordon equation: \[ \ddot{\phi} + 3H \dot{\phi} + V'(\phi) = 0, \tag{9} \] where \( V(\phi) \) is the scalar field potential. If the potential is chosen as \( V(\phi) = \frac{1}{2} m^2 \phi^2 \), then for \( m \gg H \) the field oscillates and its energy density decays as \( a^{-3} \), matching the behaviour of cold dark matter. If instead the potential has a plateau (e.g. \( V(\phi) = \lambda \phi^4 \) with small \( \lambda \)), the field can drive late‑time accelerated expansion, playing the role of dark energy. Thus, the same scalar degree of freedom can account for both dark matter and dark energy, depending on the phase of the Universe’s evolution.

From (5) it follows that the combination \[ \phi_0 c = \alpha^3 c^3 = \frac{e^6}{(4\pi\varepsilon_0)^3 \hbar^3} \tag{10} \] has dimension \( \text{m}^3/\text{s}^3 \) in SI units. In the system \( \hbar = c = 1 \) it becomes dimensionless and equals \( \alpha^3 \). This quantity can serve as a coupling constant of the scalar field to the electromagnetic field via the operator \( \phi F_{\mu\nu}F^{\mu\nu} \), which is a natural interaction channel for dark matter with photons. The smallness of \( \alpha^3 \approx 3.88 \times 10^{-7} \) explains why this interaction is extremely weak, and dark matter remains “dark” in the electromagnetic spectrum.

In atomic units \( e = \hbar = m_e = 1 \), and \( c = 1/\alpha \), then \( \phi_0 c = 1 \). This means that in atomic physics this combination equals unity, which may indicate its fundamental character. It might be related to the quantisation of angular momentum or to some non‑perturbative effects in QED, such as pair production in strong fields. Hence, \( \phi_0 c \) could be a new fundamental constant linking electrodynamics and gravity on galactic scales.

We have constructed a self‑consistent algebraic model in which:

1. Spacetime is generated by two orthogonal complex planes defined by the idempotents \( \ep \) and \( \em \), naturally leading to the separation of temporal and spatial degrees of freedom.
2. The electromagnetic 4‑potential splits into a scalar (temporal) and a vector (spatial) part, with the scalar part interpreted as the dark matter field.
3. The scalar field \( \phi \) creates a logarithmic gravitational potential, explaining flat rotation curves of galaxies without invoking exotic particles.
4. The potential depth \( \phi_0 \) is uniquely related to the fine‑structure constant by \( \phi_0 = \alpha^3 c^2 \), giving the correct numerical scale for rotation velocities.
5. This relation yields an expression for the Hubble parameter \( H_0 \) in terms of \( \alpha \) and the galactic size, providing a value consistent with cosmological observations.
6. The combination \( \phi_0 c \) can be interpreted as a fundamental coupling constant in effective field theory, linking the scalar field to electromagnetism.

The proposed model does not contradict known astrophysical data and offers a new paradigm in which dark matter, gravity, and electrodynamics are manifestations of a single algebraic structure. The key parameter connecting all scales is the fine‑structure constant \( \alpha \). This opens the way to a possible unification of micro‑ and macrophysics and provides specific predictions testable in future experiments and observations, including the search for scalar gravitational waves and temporal variations of \( \alpha \).