Fine Structure Constant as Geometric Projection Parameter

Key Discovery

The fine structure constant α is not an independent fundamental constant.

Within WILL Geometry framework, it is shown that:

$\boxed{\alpha = \beta}$

where $\beta$ is the kinetic projection parameter for the ground state of hydrogen.

By analogy with the Schwarzschild radius $R_s = \frac{2GM}{c^2}$, we define the electromagnetic critical radius:

$R_e = \frac{2e^2}{4\pi\varepsilon_0 m_e c^2} = 2r_e$

where the potential energy equals half the electron rest mass energy.

Fundamental condition: $n\lambda_n = 2\pi r_n$

For the ground state ($n=1$): $\lambda_1 = 2\pi r_1$

In the WILL Geometry system:

$\kappa_1^2 = \frac{R_e}{r_B} = \frac{2e^4}{(4\pi\varepsilon_0)^2 \hbar^2 c^2} = 2\alpha^2$

From the fundamental relation $\kappa^2 = 2\beta^2$:

$2\alpha^2 = 2\beta_1^2 \quad \Rightarrow \quad \boxed{\beta_1 = \alpha}$

$E_1 = \frac{\alpha^2}{2} m_e c^2 = \frac{(7.297 \times 10^{-3})^2}{2} \times 0.511 \text{ MeV} = 13.606 \text{ eV}$

Exact agreement with experimental hydrogen ionization energy.

$v_1 = \beta_1 c = \alpha c = 7.297 \times 10^{-3} \times 2.998 \times 10^8 = 2.188 \times 10^6 \text{ m/s}$

Independent verification through force balance: $v_1 = \sqrt{\frac{e^2}{4\pi\varepsilon_0 m_e r_B}} = \alpha c$

Complete consistency.

α is not a fundamental constant! The fine structure constant is:

Unification of interactions: Electromagnetism and gravity have one geometric basis
Explanation of “fine-tuning”: $\alpha \approx 1/137$ is not coincidence, but geometric necessity
New physics: Possible local variation of $\alpha$ depending on gravitational environment

Remarkable discovery: the critical value $\kappa^2 = \frac{2}{3}$ corresponds to:

The same geometric angle $\theta \approx 54.74°$ governs:

The derivation $\alpha = \beta$ from pure geometry represents:

Methodological breakthrough: Fundamental constants as geometric inevitabilities
Conceptual revolution: Rethinking the nature of physical laws
Practical significance: New approaches to physics unification

Central result of WILL Geometry:
Electromagnetic interaction is geometrically predetermined by the structure of energy projections in spacetime.