Appendix D — Derivation 4: Lensing from Coherence Gradient

Appendix D — Derivation 4: Lensing from Coherence Gradient (PBG view)

(Built on Foundations v1.0 · 2025-06-10)

Key idea. In Phase-Biased Geometry photons remain straight only if
their internal phase stays locked to the background coherence field
$Φ (x)$ . Spatial gradients in $Φ$ act like a
refractive-index profile, bending the ray without any spacetime curvature.

1 Photon as a latent mode

A photon envelope travels at
$c = \sqrt{α / γ}$ (Foundations §0) with internal phase

ψ_{γ} (x, t) = ρ (x - c t \hat{n}) e^{i Φ (x, t)} .

Because $β = 0$ for a mass-less mode, the only cost is the temporal
anchoring term
$\frac{1}{2} γ \int (\partial_{t} Φ)^{2} d^{3} x d t$ .
Minimising that action while crossing a static field $Φ (x)$
forces the eikonal relation

\begin{matrix} (1.1) & n (x) = 1 - \frac{2}{c^{2}} Φ (x) \end{matrix}

(see derivation in Physics/GR-Tests.md). Photons follow ordinary
Fermat least-time paths in this index profile.

2 Coherence field of a point source

For a single-winding proton / star core

Φ (r) = \frac{N}{r} e^{- k r}, N = \frac{c^{2} M}{4 π α}, k = \sqrt{β / α} (Foundations §2).

With $k R_{⊙} ≪ 1$ the Yukawa factor is ≈1 throughout the solar limb.

3 Deflection angle (solar grazing)

Index profile from (1.1):

n (r) = 1 - \frac{2 N}{c^{2} r} .

Standard straight-line eikonal integral gives

\begin{matrix} (3.1) & Δ θ (b) = \frac{2}{c^{2}} \int_{- \infty}^{+ \infty} \frac{\partial_{⊥} Φ}{c} d z = \frac{4 N}{c^{2} b} = \frac{4 G M}{c^{2} b}, \end{matrix}

using $G = c^{4} / 4 π α$ (Foundations §5).
For $b = R_{⊙}$ this yields the textbook

Δ θ_{⊙} = 1.750 arcsec .

No curved spacetime was invoked—only the phase-coherence gradient.

4 Beyond spherical symmetry

Because $n (x)$ derives directly from
$Φ (x)$ , any departure from spherical mass or any frequency-scale
dependence in $k$ yields predictable shape or chromatic lensing:

thin disc galaxy → astigmatic index → non-elliptical arcs,
polarisation-dependent phase term $θ (x, y)$
(Photon Phase Modes) → tiny birefringence in strong lenses.

These are potential discriminants versus GR.

5 Comparison to GR

Feature	GR interpretation	PBG interpretation
Light bending	Geodesic in curved metric	Fermat ray in index $n = 1 - 2 Φ / c^{2}$
Chromaticity	none (vacuum)	from Yukawa factor $e^{- k r}$ if $k$ non-zero
Spin dependence	none	possible via internal phase $θ (x, y)$

Observed solar deflection, Shapiro delay, and weak-lensing shears remain
unchanged to current precision because $k R ≪ 1$ on those scales.

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

Appendix C - Red-shift | [Index](./Appendix Master) | Appendix E - Unified Action Principle