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

(Foundations v1.0 • 2025-06-10)

Result. In Phase-Biased Geometry the universal light speed
$c^{2} = \frac{α}{γ}$
is the rate at which a mass-less (“latent”) coherence mode can propagate
while keeping its temporal and spatial anchoring tensions perfectly
balanced. It is not an imposed postulate.

1 Anchoring Lagrangian (import)

From Foundations §1

L [Φ] = \frac{1}{2} γ (\partial_{t} Φ)^{2} - \frac{1}{2} α | \nabla Φ |^{2}, (β = 0 for a photon) .

2 Wave equation → dispersion

Euler–Lagrange gives

γ \partial_{t}^{2} Φ = α \nabla^{2} Φ .

For a plane wave $Φ \propto e^{i (k \cdot x - ω t)}$ :

γ (- ω^{2}) = α (- k^{2}) ⟹ ω^{2} = c^{2} k^{2}, c^{2} = \frac{α}{γ} .

No additional assumptions are required.

3 Energy-balance interpretation

Spatial term $α | \nabla Φ |^{2}$ — cost of phase curvature.
Temporal term $γ (\partial_{t} Φ)^{2}$ — cost of phase advance.

Equalising the two costs ( $α k^{2} = γ ω^{2}$ ) yields the
minimal-tension propagation speed $v = \sqrt{α / γ} = c$ .

4 Universality

α and γ are substrate constants, fixed once (solar deflection and
$c$ calibration).
Every latent (β = 0) mode satisfies the same dispersion → same $c$ .
Lorentz invariance in PBG is a direct consequence of this balanced
anchoring cost, not a separate symmetry postulate.

Units check

$α$ [J m⁻¹], $γ$ [J s² m⁻³] → ratio = m² s⁻² ✔︎

Last edited 2025-06-10 • cites only locked equations from Foundations

Appendix J - Casimir Pressure | [Index](./Appendix Master) | Appendix L - Modal Self-Interaction