\label{universe_intersection} The surface \(S_j\) (see (\ref{comptonsurface})) can have a physical meaning only in a universe-lattice \(U_{i, j}\) that contains the two distinct lattices \(U_i\) and \(U_j\). This can be achieved only in a universe-lattice intersection \(U^*\) of the two lattice-universes \(U_i\) and \(U_j\): \[U_i \cap U_j = U_{ij}^*\] In a lattice-universe the spatial step is given by \(\lambda_c\), therefore to build the lattice-universe \(U^*\) as the intersection between \(U_i\) and \(U_j\), the respective spatial steps should be in superposition and intersected.
We distinguish two cases concerning:
if \(\lambda_i\) and \(\lambda_j\) are geometrically commensurable, then the intersection will be represented as superposition of their direct lattices;
if \(\lambda_i\) and \(\lambda_j\) are incommensurable then the intersection can be built using an intermediary field that constitutes a background field where the two respective lattices can be in superposition.
Whatever the case into consideration, however, we could assume that \(R^*\) is the radius of \(U^*\). The same will be for a lattice-universe in the critical phase. We will then: \[\label{radius}
R^* = \sqrt{n_i} \sqrt{n_j} \lambda_i\] Let us consider the intersection of the two universe-lattice \(U_p\) and \(U_e\) (see also the figure \ref{fig_intersection}) \[U^*_{p, e} = U_p \cap U_e\] Note that the actual universe \(U\) consists mainly of protons and electrons or hydrogen, then we could assume that U-universe is built on an universe \(U^*_{p,e}\) , intersection of the two lattices \(U_p\), \(U_e\): \(U \equiv U^*_{p,e}\).
We ask ourselves, therefore, if the hydrogen atom builds (through the intermediation of the photons, the quanta of the electromagnetic field) the intersection between the two lattices, or \(U^*_{p,e} = U^*_H\).
If we consider the diameter of the hydrogen atom, we can assert that along the diameter a bond between the proton and the orbiting electron is established. The diameter is given by an IQuO-chain where the oscillations of three particles (proton, electron and photon) can be in superposition.
In the hydrogen atom, the electron wavelength is \(\lambda_{\text{bond}} = \lambda_e / \alpha_{\textbf{em}}\), where \(\alpha_{\textbf{em}}\) is the fine structure constant. We observe that \(\lambda_{\text{bond}}\) is the hydrogen Bohr radius.
We will say then that the action of the electromagnetic field (photon) adapts the Compton wavelength of the electron to the proton in order to allow the intersection of the two lattices.
If this is true, then \(\lambda_{\text{bond}} / \lambda_p = k_s\), where \(k_s\) is a rational number (less to a scale factor about \(10^4\)).
However the electron is a particle “orbiting” around the nucleus (proton), so the oscillation that describes it takes place both along the diameter both along the circumference. Because the superposition of the two lattices \(\lambda_H = \lambda_{\text{bond}}\) must be projected along the diameter since the \(\lambda_p\) is projected along the diameter: in other words we should be divided \(\lambda_H\) with \(2 \pi\) in order to relate it with \(\lambda_p\); so it follows \[\frac{\lambda_H}{2 \pi} \frac{1}{\lambda_p} = k_s \cdot 10^4\] And we define a universal adaptation factor \(k_U\): \[k_U = \frac{k_s}{10^4} = 4.00464080\] Let us return in \(U^*_{p,e}\), and following (\ref{radius}), we have: \[\label{radiuspe}
R_{p, e} = \sqrt{n_p} \sqrt{n_e} \lambda_p\] This relation is correct in a critical universe.
Then we could develop the following system of relations: \[\begin{aligned}
R^*_{p_e} & = n_p \lambda_e \nonumber\\
R^*_{p_e} & = c \tau^*_c\end{aligned}\] These relations could allow us to derive the critical age of the universe \(U^*_{p,e}\). Since \(U^*_{p,e}\) is the intersection between two universe-lattices with a variable metric, we can also assert that it evolves from an expansive phase with an open metric to a phase of contraction with a closed metric passing through a phase a flat metric.
Now, the multiplicative constant in relation (\ref{unverse_surfaces}) can be expressed in \(U^*_{p,e}\): \[n_e S_p = k n_p S_e\] where the numerical constant \(k\) will be connected with the ratio between the particles’ density of the two lattices \(U_p\), \(U_e\). Following (\ref{radiuspe}), we have: \[R^*_{p,e} = k_U n_p \lambda_e\]