Appendix K — Derivation 11: Speed of Light from Anchoring Cost

Overview

In classical physics, the speed of light $c$ is a postulated constant. In field theory, it emerges from Maxwell’s equations. In modal dynamics, there are no fields—yet $c$ still emerges.

Here, we derive $c$ as the natural velocity of a coherence-preserving mode propagating through space while avoiding anchoring strain. It is the minimum-cost propagation speed for a latent mode.

A photon is represented as a coherence function $ψ (x, t)$ with:

No anchoring into the coherence field $B (x)$
Constant internal phase profile along its path
Propagation defined by coherence preservation—not force or curvature

The photon travels through the medium as a self-sustaining, latent mode, avoiding decoherence by matching its phase evolution to the modal medium.

2. Cost Functional for a Free Mode

We write the local anchoring cost density as:

L = γ | \partial_{t} ψ |^{2} + α | \nabla ψ |^{2}

This represents the total coherence tension—temporal and spatial.

Assume a trial mode moving at speed $v$ :

ψ (x, t) = e^{i (k x - ω t)}

Then:

$\partial_{t} ψ = - i ω ψ \Rightarrow | \partial_{t} ψ |^{2} = ω^{2}$
$\nabla ψ = i k ψ \Rightarrow | \nabla ψ |^{2} = k^{2}$

Substituting into $L$ :

L (v) = γ ω^{2} + α k^{2}

Using the dispersion relation $ω = v k$ :

L (v) = γ v^{2} k^{2} + α k^{2} = k^{2} (γ v^{2} + α)

To find the minimum cost path, minimise $L (v)$ with respect to $v$ :

3. Cost Minimisation and Optimal Speed

Let:

L (v) = k^{2} (γ v^{2} + α)

Minimising with respect to $v$ :

\frac{d L}{d v} = k^{2} (2 γ v) = 0 \Rightarrow v = 0

But $v = 0$ corresponds to a stationary mode—not a propagating photon. Instead, we fix $k$ and seek the velocity that keeps $L$ bounded under the constraint that the mode stays latent (no anchoring).

We require the cost density to be balanced between time and space:

γ ω^{2} = α k^{2} \Rightarrow γ v^{2} k^{2} = α k^{2} \Rightarrow v^{2} = \frac{α}{γ}

Thus, the natural coherence-preserving speed is:

c = \sqrt{\frac{α}{γ}}

4. Interpretation

$α$ : spatial stiffness—resistance to phase curvature
$γ$ : temporal stiffness—resistance to phase shift over time

The speed of light is the propagation rate that balances these tensions, allowing a phase-preserving mode to glide through the coherence medium without anchoring.

This is not imposed. It emerges from the variational structure of coherence mechanics.

5. Universality

$c$ is not local to the emitter
It depends only on the medium’s stiffness constants
It is the same for all latent modes of the same structural class

This explains the universality of light speed, Lorentz invariance, and photon coherence preservation across cosmological distances.

Conclusion

The speed of light is not a constant inserted into the theory.
It is the natural drift rate of a latent phase structure through a coherence medium, derived from the cost of preserving internal order without anchoring.

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