Anchoring Cost Profiles in PBG

Appendix AQ - Anchoring Cost Profiles in PBG

1. Photon Decoherence Penalty

Let a photon be a latent coherence mode traversing a background coherence field $B (r)$ . The spherically symmetric solution of the static anchoring equation

α \nabla^{2} B - β B = 0

is the Yukawa–type kernel

B (r) = \frac{A}{r} e^{- k r}, k = \sqrt{\frac{β}{α}},

where

$α$ has units J m⁻¹,
$β$ has units J m⁻³,
$A$ carries units J m⁻¹.

The decoherence‐penalty density for a minimally anchoring mode is

Λ_{γ} (r) = γ_{0} | \frac{d B}{d r} |^{2}, \frac{d B}{d r} = - \frac{A e^{- k r}}{r^{2}} (1 + k r),

with $γ_{0}$ in J s² m⁻³. The transverse “force” on the photon follows from the tension‐gradient:

F_{⊥} = - \nabla_{⊥} Λ_{γ} (r) .

In particular, calibrating $γ_{0}$ and $A$ to yield $Δ θ_{⊙} \approx {1.75}^{″}$ reproduces the solar‐limb deflection without any spacetime curvature.

2. Mutual Anchoring & Coulomb Scaling

Define each charged mode’s coherence charge $Q$ via its phase winding (see Appendix AG). Two charges $Q_{1}, Q_{2}$ interact through the same kernel $B (r)$ . The interaction cost is

C_{i n t} (r) = α Q_{1} Q_{2} B (r) [J],

where $α$ is the spatial‐anchoring stiffness (J m⁻¹). The resulting radial force is

F (r) = - \frac{d}{d r} C_{i n t} = α Q_{1} Q_{2} | \frac{d B}{d r} | = α Q_{1} Q_{2} \frac{A e^{- k r}}{r^{2}} (1 + k r) .

In the near‐field limit $k r ≪ 1$ , this reduces to the familiar Coulomb form:

F (r) \approx \frac{α A Q_{1} Q_{2}}{r^{2}} \equiv \frac{1}{4 π ϵ_{0}} \frac{q_{1} q_{2}}{r^{2}},

identifying $q_{i} = Q_{i}$ and
$\frac{1}{4 π ϵ_{0}} = α A$ .
Thus Coulomb’s law and the constant $k_{e} = 1 / (4 π ϵ_{0})$ emerge directly from anchoring costs—no vacuum field, no extra postulates.