Foundational Principles from TUS

The TUS model provides the necessary framework for this analysis by defining the electron not as a point particle, but as a self-coherent toric structure governed by geometric laws.

• Toric Structure: The electron is a torus whose major radius is the reduced Compton radius (\(R_u\)) and whose minor radius is the classical electron radius (\(r_e\)). The fine-structure constant \(\alpha\) is the intrinsic geometric ratio \(\alpha = r_e / R_u\).

• Spin as Internal Dynamics: The electron’s spin is not a classical rotation but a manifestation of the internal, helical flow of confined energy within the torus. Its spin-1/2 nature requires a \(4\pi\) rotation (a double turn) to return the structure to its initial state.

• Interaction as Geometric Perturbation: An external field or the presence of a nucleus imposes an external dynamic constraint on the torus, perturbing its fundamental state.

Geometric Origin of the Fundamental g-factors

Before deriving the composite factor \(g_J\), TUS must first provide a geometric origin for the fundamental factors \(g_L\) and \(g_S\).

\(g_L = 1\): The Signature of a Simple Dynamic

The orbital angular momentum (\(L\)) corresponds to the large-scale circulation of the entire electron torus around a nucleus. This is a simple, singular dynamic. In classical and quantum physics alike, a simple current loop produces a magnetic moment directly proportional to its angular momentum. Therefore, the proportionality factor is unity. Within TUS, \(g_L = 1\) is the geometric signature of a simple, non-composite dynamic circulation.

\(g_S = 2\): The Signature of a Dual Dynamic

The spin magnetic moment arises from the complex internal dynamics of the torus. Its g-factor, \(g_S=2\) (ignoring the anomalous correction for this derivation), is not arbitrary. It is the direct consequence of the dual-circulation topology of energy within the torus:

1. Toroidal Circulation: The flow of energy along the major radius (\(R_u\)) of the torus. This constitutes a primary current loop, contributing a factor of 1 to \(g_S\).

2. Poloidal Circulation: The simultaneous helical winding of this energy flow around the cross-section of the torus (along the minor radius \(r_e\)). This constitutes a second, intrinsic current loop, also contributing a factor of 1 to \(g_S\).

Thus, the total spin g-factor, \(g_S = 2\), emerges as the sum of these two inseparable geometric contributions. The first ‘+1‘ stems from the primary toroidal circulation of energy, while the second ‘+1‘ is a direct consequence of the poloidal circulation—the helical winding of this energy flow within the torus tube.

It is crucial to emphasize that this winding is not arbitrary; its precise geometry (its "torsion") is strictly determined by the fine-structure constant, \(\alpha\), in its role as the fundamental ratio of the torus’s radii (\(\alpha = r_e/R_u\)). Therefore, \(\alpha\) is at the very origin of the second component of the spin g-factor.

The experimentally observed value, \(g_S^{real} = 2(1+a_e)\), then incorporates the further corrections (\(a_e\)) which, according to TUS, arise precisely from the dynamic response and topological stabilization of this same \(\alpha\)-defined structure.

The Landé \(g_J\) as a Geometric Interference Factor

The Landé g-factor emerges when these two dynamics—the simple external (orbital) and the complex internal (spin)—are superimposed. The total angular momentum \(\vec{J} = \vec{L} + \vec{S}\) becomes the conserved quantity, and both the orbital and spin axes precess around it.

The total magnetic moment is the vector sum of the moments from each dynamic: \[\vec{\mu}_J = \vec{\mu}_L + \vec{\mu}_S\] Within the TUS framework, these are proportional to their respective angular momenta, scaled by their geometric g-factors: \[\vec{\mu}_J \propto -(g_L\vec{L} + g_S\vec{S}) = -(\vec{L} + 2\vec{S})\] The interaction energy with an external magnetic field depends only on the component of \(\vec{\mu}_J\) that is projected onto the stable axis of the system, \(\vec{J}\). The Landé factor \(g_J\) is defined as the scaling factor for this effective moment, \(\mu_{J, \text{eff}} = \vec{\mu}_J \cdot \frac{\vec{J}}{|\vec{J}|}\). \[g_J |\vec{J}| \propto (\vec{L} + 2\vec{S}) \cdot \frac{\vec{J}}{|\vec{J}|}\] By substituting \(\vec{L} = \vec{J} - \vec{S}\), we get: \[g_J |\vec{J}| \propto (\vec{J} - \vec{S} + 2\vec{S}) \cdot \frac{\vec{J}}{|\vec{J}|} = (\vec{J} + \vec{S}) \cdot \frac{\vec{J}}{|\vec{J}|} = \left( \frac{\vec{J} \cdot \vec{J}}{|\vec{J}|} + \frac{\vec{S} \cdot \vec{J}}{|\vec{J}|} \right)\] Using the vector identity \(2\vec{S}\cdot\vec{J} = J^2 + S^2 - L^2\), we can solve for \(g_J\): \[g_J = 1 + \frac{J^2 + S^2 - L^2}{2J^2}\] Replacing the squared magnitudes with their quantum mechanical eigenvalues (\(J(J+1)\), etc.) yields the familiar Landé g-factor formula.

Conclusion

The Theory of Sealed Universes provides a compelling physical origin for the Landé g-factor. It is not an abstract consequence of quantum rules, but a quantifiable coefficient of geometric interference. It measures how the magnetic response from the electron’s complex internal dual-circulation dynamic (spin, \(g_S=2\)) is modulated by the superposition of a simple external circulation dynamic (orbit, \(g_L=1\)). The Landé formula thus emerges as a direct calculation of the effective response of this composite geometric system, reinforcing the TUS postulate that fundamental physical properties are projections of a self-coherent internal geometry.