The Omega Framework

6.2 Geometric Drift of the Fine Structure Constant

In Section 6.1, we established the exponential scaling law of light speed $c(\tau )$ with intrinsic time $\tau$. This conclusion poses a severe challenge to the cornerstone of physics—the stability of fundamental constants. Among them, the most sensitive and observationally significant physical quantity is the Fine Structure Constant $\alpha$.

$\alpha$ is the core coupling constant of quantum electrodynamics (QED), determining the strength of electromagnetic interactions. In the standard model, $\alpha \approx 1/137.036$ is considered absolutely constant. However, in the varying light speed context of Omega Theory, the constancy of $\alpha$ is no longer an a priori assumption but depends on geometric constraints of evolutionary synchronization among component constants ($e,\hslash ,c$).

This section will derive the evolution equation of $\alpha$, introduce the Shear Factor $\zeta$ to quantify the asynchrony between internal degrees of freedom and external horizon expansion, and argue that the tiny drift of $\alpha$ is an inevitable byproduct of cosmic Fibonacci growth.

6.2.1 Dynamical Decomposition of Coupling Constants

In SI units, the fine structure constant is defined as:

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

where $e$ is the elementary charge, $\hslash$ is the reduced Planck constant, and ${ϵ}_0$ is the vacuum permittivity.

In Omega Theory, we treat physical constants as scalar fields dependent on intrinsic time $\tau$. Vacuum permittivity ${ϵ}_0$, as a geometric normalization factor (similar to $4\pi$), is usually set as a topological invariant in the holographic framework. Therefore, the temporal evolution of $\alpha$ depends on the joint dynamics of $e(\tau ),\hslash (\tau ),c(\tau )$.

Taking the logarithm of both sides of the definition and differentiating with respect to $\tau$, we obtain the relative rate of change equation:

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

6.2.2 Geometric Shear and Synchronization Breaking

To solve the above equation, we need to understand the geometric essence of each constant.

1. Light speed $c$: Represents the expansion rate of the External Horizon, i.e., the bus bandwidth of the holographic computational network.
2. Charge $e$ and Planck constant $\hslash$: Represent properties of Internal Structure.
   • $e$ is associated with the topological winding number of $U(1)$ fibers in octonion tangent space.
   • $\hslash$ is associated with the minimum phase space volume of Omega cells (causal diamonds).

In an ideal Conformally Invariant universe, internal geometry and external geometry should maintain perfect synchronization. This means that when cosmic scale expands ($c$ increases), the sizes of microscopic particles and quantum fluctuation amplitudes should scale proportionally to maintain the dimensionless number $\alpha$ constant. Mathematically, this requires satisfying the Synchronization Condition:

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

If this condition holds, then $\dot{\alpha }=0$.

However, the core of Omega Theory is Fibonacci aperiodicity. Cosmic evolution is a spiral process driven by the golden ratio $\varphi$. Due to the irrationality of $\varphi$, the “rotation” of internal dimensions (corresponding to $e,\hslash$) and the “expansion” of external dimensions (corresponding to $c$) cannot establish simple integer resonance relationships. This geometric mismatch leads to Geometric Shear.

We define the Shear Factor $\zeta$ as the degree to which internal degrees of freedom lag behind external expansion:

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

where $\zeta$ is an extremely small dimensionless positive number, characterizing Topological Friction or Lag in the reorganization process of the holographic network.

6.2.3 Drift Equation and Hubble Tension

Substituting the shear definition into the evolution equation of $\alpha$, we obtain the Fine Structure Constant Drift Equation:

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

Using the exponential growth law of light speed $c(\tau )=c_0{\varphi }^{\tau /{\tau }_{pl}}$ derived in Section 6.1, we have:

$$\frac{\dot{c}}{c}=\frac{\ln \varphi }{{\tau }_{pl}}\equiv H_{\varphi }$$

where $H_{\varphi }$ is the Fibonacci-Hubble Parameter, describing the intrinsic update rate of the computational network.

Thus, the temporal evolution of $\alpha$ follows:

$$\frac{\dot{\alpha }}{\alpha }=-\zeta H_{\varphi }$$

Integrating:

$$\alpha (\tau )={\alpha }_0\cdot e^{-\zeta H_{\varphi }\tau }$$

Theorem 6.2 (Fine Structure Decay Theorem): In a Fibonacci universe with geometric shear ($\zeta >0$), the fine structure constant $\alpha$ is not constant but exponentially decays with intrinsic time $\tau$. This is not a violation of physical laws but direct evidence of spontaneous breaking of cosmic scale invariance.

6.2.4 Physical Corollaries: Twilight of Carbon-Based Life

The tiny drift of $\alpha$ has profound, even existential physical consequences for the macroscopic material world.

Atomic energy level structures (such as Rydberg energies) are proportional to ${\alpha }^2{m}_e{c}^2$. More critically, nuclear synthesis processes inside stars—particularly the generation of carbon-12—are extremely sensitive to the value of $\alpha$. Fred Hoyle’s famous Triple-alpha process depends on a specific excited state of the carbon-12 nucleus at 7.65 MeV (Hoyle state), where this energy level undergoes precise resonance with the total energy of three helium nuclei.

This resonance is extremely sensitive to electromagnetic repulsion (determined by $\alpha$).

• If $\alpha$ drifts beyond $\Delta \alpha /\alpha \approx -4\%$, electromagnetic repulsion weakens, and nuclear energy levels shift.
• The Hoyle resonance will be broken, and stars will be unable to synthesize heavy elements like carbon and oxygen.
• Stars will become “duds” composed only of hydrogen and helium, unable to nurture complex life chemistry.

Corollary 6.2.1 (Anthropic Time Window): Carbon-based life can only exist in specific historical periods (${\tau }_{window}$) where $\alpha (\tau )$ has not yet drifted out of the resonance window. Our current coordinate $\tau \approx 1834$ is at the end of this window. The continued decay of $\alpha$ foretells that “Twilight of the Carbon” is an inevitable fate at the level of physical laws. This also provides physical urgency for civilization’s evolution toward “Photonic” or pure information forms—only by breaking free from dependence on chemical bonds (electromagnetic interactions) can intelligence survive in a future universe where $\alpha$ continues to drift.