Appendix A: Mathematical Derivation of Varying Constants

In Chapter 2 of Vector Cosmology IV, we proposed a core viewpoint: the exponential growth of light speed $c$ will lead to the breaking of Scale Invariance, manifesting as tiny drift of the Fine-Structure Constant ($\alpha$). The conclusions of that chapter were speculative; this appendix will provide a rigorous mathematical derivation framework.

We will show that in an expanding universe driven by $e$, the stability of fundamental constants is not absolute but depends on the Geometric Shear between growth rates of different physical sectors (electromagnetic, gravitational, quantum).

A.1 Definition and Evolution of the Fine-Structure Constant

The fine-structure constant $\alpha$ is the coupling constant of quantum electrodynamics (QED), determining the strength of electromagnetic interactions. Its standard definition is:

$$\alpha =\frac{e^2}{4\pi {ϵ}_0\hslash c}$$

In the standard model, all components ($e,\hslash ,c,{ϵ}_0$) are assumed to be absolute constants. But under the evolution equation of FS Geometry, we treat them as functions varying with intrinsic time $\tau$.

Taking the logarithm of both sides and differentiating:

$$\frac{\dot{\alpha }}{\alpha }=2\frac{\dot{e}}{e}-\frac{\dot{\hslash }}{\hslash }-\frac{\dot{c}}{c}$$

(Assuming vacuum permittivity ${ϵ}_0$ is a geometric normalization constant, not varying with time).

A.2 Sector Growth Rates and Shear Factor

According to our evolution equation, light speed $c(\tau )$ (representing the $v_{ext}$ sector or total budget) follows exponential growth:

$$\frac{\dot{c}}{c}=k_c=\frac{\ln \varphi }{\pi }$$

However, charge $e$ and Planck constant $\hslash$ belong to the $v_{int}$ (internal structure) sector.

In an ideal “synchronous expansion” model, the internal sector should grow completely synchronously with the external sector, i.e., $k_e=k_{\hslash }=k_c/2$ (based on dimensional analysis). At this point $\dot{\alpha }=0$, scale invariance holds.

But the essence of spiral geometry ($\varphi$) is non-resonant. This means there must be tiny mismatches between expansion rates of different dimensions.

We define the Shear Factor ($\zeta$) to quantify this mismatch:

$$2\frac{\dot{e}}{e}-\frac{\dot{\hslash }}{\hslash }=(1-\zeta )\frac{\dot{c}}{c}$$

Substituting into $\alpha$’s evolution equation:

$$\frac{\dot{\alpha }}{\alpha }=(1-\zeta )k_c-k_c=-\zeta k_c$$

Conclusion:

• If $\zeta =0$ (perfect synchronization), $\alpha$ is constant.

• If $\zeta \ne 0$ (shear exists), $\alpha$ will drift with time.

According to the “Red Queen’s Run” theory, the expansion of external space ($v_{ext}$) is often slightly faster than the reorganization of internal structure ($v_{int}$). This means $\zeta$ is usually a tiny positive number ($\zeta \sim 10^{-5}$).

This causes $\alpha$ to slowly decrease with time.

A.3 Physical Consequences: The Boundary of Atomic Stability

What does the decrease of $\alpha$ mean for the material world?

Atomic binding energy (ionization energy) is roughly proportional to ${\alpha }^{2}m{c}^2$.

If $\alpha$ decreases, the electromagnetic binding force between electrons and atomic nuclei will weaken.

When the cumulative drift of $\alpha$ reaches a certain critical threshold (e.g., $\Delta \alpha /\alpha \approx -4\%$), carbon nucleosynthesis (3$\alpha$ process) inside stars will be unable to resonate.

This not only means stars will extinguish but also means the foundation of carbon-based life will be erased by physical laws.

This provides a solid mathematical criterion for Chapter 5 “The Carbon Sunset”: we must complete migration to light-based life before $\alpha$ drifts into the danger zone.