The observed motion of stars in galaxies indicates the presence of an additional gravitational contribution that cannot be derived from visible matter alone. This contribution is usually explained by invisible matter—dark matter. This paper considers a different, currently hypothetical, possibility: the additional gravity is generated not by a collection of new particles, but by a coherent scalar mode of the internal wave geometry.

This formulation is close to the scalar and wave dark matter models, but here the possible field is associated with the idempotent structure of the operator \(J\). It is important to immediately distinguish this mode from the electromagnetic scalar potential. Henceforth, we use the notation \(\chi\) for it, and \(\Phi_g\) for the ordinary gravitational potential.

The proposed construction is not yet a complete theory of dark matter. Its goal is more modest: to show what geometric channel could produce the observed gravity profile and why the fine structure constant unexpectedly defines the characteristic scale of galactic velocities.

The algebraic model is based on two mutually orthogonal idempotents: \[ \ep^2=\ep, \qquad \em^2=\em, \qquad \ep\em=0, \qquad \ep+\em=1. \tag{1} \] They define two independent internal complex planes. However, these planes should not be literally identified with a single time and a single spatial coordinate. They describe the internal phase structure of the wave, while the observed time and three-dimensional direction arise after its projection into external space.

The overall internal state can be represented as \[ \Psi=\ep\psi_+ + \em\psi_-. \tag{2} \] The phase difference and orientation of these components can manifest themselves in the external projection as charge, magnetic, and polarization properties. However, algebra also allows for quantities that relate equally to both planes and therefore do not specify a distinct external direction. We denote a macroscopic coherent mode of this type by \(\chi\).

Here \(\chi\) is not the time component of the electromagnetic four-potential nor a new notation for the electric potential. It is an independent, effective scalar characteristic of the internal state. Its exact expression in terms of \(\psi_+\) and \(\psi_-\) must be obtained at the next stage of the model's development.

An electromagnetic wave has a transverse external projection: changes in the electric and magnetic channels propagate in space and directly interact with the charge. A scalar mode is not required to have such a transverse projection. Therefore, it may not directly emit or absorb ordinary photons.

However, the absence of direct electromagnetic visibility does not mean the absence of energy. If the \(\chi\) field has a gradient, temporal dynamics, or internal energy, it can contribute to the overall energy-momentum tensor and thereby change the geometry of spacetime. In this case, light interacts with it gravitationally—through the curvature of its trajectory, that is, through gravitational lensing.

At the level of effective description, the scalar mode can be associated with the action \[ S_\chi=\int d^4x\,\sqrt{-g} \left[ -\frac12 g^{\mu\nu} \partial_\mu\chi\,\partial_\nu\chi -V(\chi) \right]. \tag{3} \] This notation does not derive the field from the operator \(J\), but only shows the minimal form of its macroscopic dynamics. The function \(V(\chi)\), the field normalization, and the possible mass are not yet specified here.

First, let's separate the observed result from its possible microscopic cause. Let \(\Phi_g(r)\) be the total gravitational potential in the spherically symmetric approximation. For circular motion, we have \[ \frac{v_c^2(r)}{r}=\frac{d\Phi_g}{dr}. \tag{4} \] If, over a certain range of distances, the velocity is practically independent of the radius, \[ v_c(r)\simeq v_{\rm flat}, \tag{5} \] then integration (4) yields the logarithmic potential: \[ \boxed{ \Phi_g(r)=v_{\rm flat}^2 \ln\left(\frac{r}{r_s}\right)+\operatorname{const} }. \tag{6} \] Here \(r_s\) defines the origin of the potential. Formula (6) is not a special property of the field \(\chi\): it is a direct consequence of the flat rotation curve.

In the Newtonian limit, the effective source density is determined by Poisson's equation: \[ \nabla^2\Phi_g=4\pi G\rho_{\rm eff}. \tag{7} \] Substituting (6) yields \[\boxed{ \rho_{\rm eff}(r)= \frac{v_{\rm flat}^2}{4\pi G r^2} }. \tag{8} \] The corresponding mass inside radius \(r\) is \[ M_{\rm eff}(r)= \frac{v_{\rm flat}^2}{G}r. \tag{9} \]

The linear growth (9) shows the limits of applicability of the logarithmic solution. It can describe the galactic halo only over a finite range of radii. At large distances, the field coherence must be broken, the profile must change, or one halo must merge with the general field of the galaxy group. Otherwise, the total mass would be unlimited.

If the additional gravitational contribution is created by the field \(\chi\), then its stationary distribution should lead to the effective density (8). However, one cannot simply set \[ \rho_\chi\propto(\nabla\Phi_g)^2, \] since \(\Phi_g\) is the resulting gravitational potential, not the field \(\chi\) itself. Furthermore, the square of the gravitational gradient has the dimensions of the energy density only after introducing the appropriate normalization.

The correct sequence should be as follows:

1. The field \(\chi\) and its energy are derived from the internal geometry;
2. The gravitational contribution of the field is found from the energy of the field;
3. The joint problem for baryonic matter and \(\chi\) is solved;
4. The rotation curve is calculated from the resulting \(\Phi_g(r)\).

In this paper, we perform the reverse, phenomenological step: using the observed flat curve, we determine which profile the future field equation should reproduce.

The fine structure constant already appears in the model as the coefficient of projection of the internal frequency onto the observed energy and mass. Therefore, it is natural to check whether it can also participate in the macroscopic projection of the scalar mode.

The combination \[ \Phi_*=\afs^3c^2 \tag{10} \] has the dimension of the gravitational potential. It corresponds to the velocity \[ \boxed{ v_*=\sqrt{\Phi_*} =\afs^{3/2}c \approx187\;\text{km/s} }. \tag{11} \] When calculating, it is important to distinguish between \[ \afs^{3/2}\approx6.24\times10^{-4}, \qquad \afs^3\approx3.89\times10^{-7}. \tag{12} \]

Value (11) indeed falls within the characteristic range of velocities of large spiral galaxies. However, it cannot be considered a universal velocity: observed galaxies have different masses, sizes, and rotation velocities. Therefore, at this stage, only the parameterization \[ v_{\rm flat}^2=\eta_g\,\afs^3c^2, \tag{13} \] where the dimensionless quantity \(\eta_g\) should be determined by the distribution of baryonic matter, the boundary conditions, and the state of the scalar mode.

Formula (10) is still a hypothesis about the characteristic scale, not a definitive conclusion. The very fact that the required order of magnitude appears merits attention, but the geometric reason for the power \(\afs^3\) remains to be found. Referring solely to the order of quantum electrodynamic corrections is no substitute for such a conclusion.

A constant velocity alone is insufficient to describe real galaxies. Observations show a close relationship between the distribution of ordinary matter and the total centripetal acceleration. Furthermore, the baryonic mass of a galaxy is related to the asymptotic velocity by the approximate law \[ M_b\propto v_{\rm flat}^4. \tag{14} \]

Therefore, the field \(\chi\) cannot form a completely identical background around all galaxies. It must either be excited by baryonic matter or change its stationary distribution under the influence of the geometry it creates. The mechanism for the emergence of dependence (14) is the main next step in the model.

In terms of wave electricity, it can be assumed that a multitude of closed waves of ordinary matter creates not only local electric and gravitational gradients, but also a common, slowly varying scalar component of the internal phase. With sufficient coherence, this component can extend far beyond the visible disk. For now, this is a qualitative interpretation; it requires deriving a superposition equation and a coherence condition.

It is fundamentally important not to equate the scalar mode \(\chi\) with the electromagnetic scalar potential \(\varphi_{\rm em}\). The electromagnetic potential depends on the choice of gauge, whereas the physical density of dark matter should be observable through its gravitational action and should be independent of this choice.

Therefore, in the proposed picture there are three rdifferent levels:

• \(\varphi_{\rm em}\) and \(\mathbf A\) describe the electromagnetic field;
• \(\chi\) describes the putative internal scalar mode;
• \(\Phi_g\) is the resulting gravitational potential created by baryonic matter and the energy \(\chi\).

Separating them eliminates ambiguity and allows each level of the model to be independently tested.

A flat rotation curve is necessary, but far from the only check. For the geometric scalar mode to be a candidate for dark matter, the model must explain:

1. the different rotation curves of galaxies and their relationship to baryonic mass;
2. gravitational lensing by the same distribution that governs the motion of stars;
3. the behavior of the scalar component in galaxy cluster collisions;
4. the growth of cosmological inhomogeneities and the formation of large-scale structure;
5. the transition from the galactic profile to the intergalactic background;
6. the absence of inadmissible effects in the Solar System.

Cluster collisions, in which the distribution of the lensing mass can be separated from the bulk of the hot gas, are particularly significant. If \(\chi\) is an independent field with its own dynamics, it can, in principle, preserve such a separated distribution. If it is determined solely by the instantaneous baryon density, explaining such observations will be significantly more difficult.

The Hubble parameter cannot be directly derived from the characteristic galactic velocity by dividing it by some "typical" galaxy radius. Galactic radii vary, and the Hubble parameter characterizes the global expansion of space. To obtain it, one must know the cosmological field density \(\chi\), its pressure, and its evolution in the expanding metric.

For this reason, this paper does not combine dark matter and dark energy and does not propose a formula for \(H_0\). Such a connection can only be discussed after deriving the potential \(V(\chi)\) and solving the cosmological equations. The galactic result (6)–(13) is independent of this.

Within the framework of idempotent wave geometry, one can assume the existence of a coherent scalar mode \(\chi\), which does not have the usual transverse electromagnetic projection, but has energy and therefore creates a gravitational effect.

From the flat rotation curve, the logarithmic potential unambiguously follows \[ \Phi_g(r)=v_{\rm flat}^2 \ln\left(\frac r{r_s}\right), \] and the corresponding effective density has the profile \[ \rho_{\rm eff}(r)= \frac{v_{\rm flat}^2}{4\pi G r^2}. \] It is this profile that the future scalar mode equation must reproduce.

The fine structure constant defines a remarkable characteristic scale \[ v_*=\afs^{3/2}c\approx187\;\text{km/s}, \] but does not yet determine the individual velocity of each galaxy. To complete the model, it is necessary to derive the coefficient \(\eta_g\), the dependence on the baryon mass, the dynamics of the field \(\chi\), and its behavior during gravitational lensing and cluster collisions.

Thus, dark matter is considered here not as an already proven consequence of algebra or as an electromagnetic scalar potential, but as a testable hypothesis: part of the internal wave geometry can be preserved in a non-transverse coherent channel and manifest in outer space primarily through gravity.