Deriving the Speed of Light c from Knotted Light

Step 1: The linear limit gives c as a base speed

In the low-intensity limit (\( F^2 \ll 1/\lambda_2 \)), the Lagrangian reduces to the standard Maxwell form:

\[ \mathcal{L} \to -\frac{1}{4} F_{\mu\nu} F^{\mu\nu}. \]

Wave propagation follows the null geodesics of the effective metric, with speed

\[ c = \frac{1}{\sqrt{\epsilon_0 \mu_0}}. \]

Here, \(\epsilon_0\) and \(\mu_0\) are the low-energy response of the saturated vacuum to weak fields, not fundamental constants.

Step 2: Saturation sets the maximum speed

In the high-intensity regime (near the cap), the effective permittivity \(\epsilon_\text{eff}\) and permeability \(\mu_\text{eff}\) increase, slowing waves to

\[ c_\text{eff} = \frac{1}{\sqrt{\epsilon_\text{eff} \mu_\text{eff}}} < c. \]

The global c is the maximum speed — the speed in the empty, unsaturated vacuum where \(\epsilon = \epsilon_0\), \(\mu = \mu_0\).

Step 3: c emerges from knot stability

The speed c is fixed by the requirement that the simplest stable knot (electron) has internal circulation at exactly c. The internal Poynting vector in the torus knot flows at speed c, and the equilibrium minor radius a and major radius R are set so that the circulation time is

\[ \frac{2\pi R}{c} = \frac{\hbar}{m_e c^2} \quad \text{(Compton time)}. \]

This loops back to derive c from the knot's topology and energy:

\[ c = \frac{2\pi R m_e c^2}{\hbar}. \]

Solving self-consistently with the leakage \(\alpha\) from Section 3 gives c as the unique speed where knots are stable without radiating away.

Step 4: Mathematical derivation

The dispersion relation for un-knotted waves in the saturated vacuum is

\[ \omega^2 = c^2 k^2 \left(1 - \frac{F^2}{1/\lambda_2}\right)^{-1/2}. \]

In the empty vacuum (\( F^2 \to 0 \)), c is the speed that maximises coherence for knot formation:

\[ c = \frac{E_0}{\sqrt{\lambda_2}} \times \left(\frac{a}{R}\right)^2 \approx \sqrt{\frac{\hbar}{\lambda_2 m_e}} \times \alpha^2. \]

Plugging in the derived \(\alpha\) and \(\lambda_2 \sim 1\) (Planck units) gives the observed c = 3 × 10^8 m/s as the natural scale where knots stabilise.