Appendix Q — Derivation 17: Galaxy-Scale Lensing from Coherence Structure

Overview

Observations of galaxy clusters and spiral galaxies reveal gravitational lensing effects too strong to be accounted for by visible mass. This leads to the dark matter hypothesis in standard cosmology.

In modal dynamics, there is no mass-based gravity.
Instead, light deflection emerges from gradients in the coherence field $B (x)$ produced by modal emitters (stars, cores, halos).

This appendix derives:

The coherence structure of galaxies
The photon deflection mechanism across that structure
How observed lensing arises without invisible mass

1. Coherence Field of a Galaxy

Each galaxy is composed of billions of coherence-emitting structures (stars, cores, modal shells).

The total coherence field is:

B_{gal} (x) = \sum_{i} \frac{A_{i}}{| x - x_{i} |} e^{- k | x - x_{i} |}

Where each $x_{i}$ is a modal source location. In the continuum limit, this becomes:

B_{gal} (x) = \int ρ_{emit} (x^{'}) \frac{A}{| x - x^{'} |} e^{- k | x - x^{'} |} d^{3} x^{'}

This field:

Extends well beyond visible matter
Remains coherent over tens of kiloparsecs
Produces long-range phase gradients

2. Lensing Mechanism: Coherence Gradient Penalty

Photons are latent modes, and they follow paths that minimise decoherence:

A_{γ} = \int Λ_{γ} (x (t)) d t

Where:

Λ_{γ} (x) = γ_{0} {| \nabla B_{gal} (x) |}^{2}

The path of least accumulated decoherence curves in response to $\nabla B^{2}$ .

The deflection angle is:

δ θ = \int \frac{d^{2} x_{⊥}}{d t^{2}} d t \propto \int \nabla_{⊥} ({| \nabla B |}^{2}) d t

This reproduces observed light bending without invoking any gravitational field.

3. Coherence Shells and Gradient Amplification

Galaxies naturally form modal shells: coherence zones surrounding dense cores.

These shells:

Reinforce field strength in outer regions
Create steep coherence gradients at large radii
Extend lensing zones well beyond visible matter

This produces strong deflection where Newtonian gravity predicts weak pull.

4. Directional Asymmetry

Unlike spherical mass models, galaxy coherence structures are:

Disc-like
Anisotropic
Internally phase-aligned

As a result:

Lensing varies with photon path direction
Vertical (polar) deflection differs from edge-on passes
This is testable and unique to modal dynamics

5. Observational Match

Simulated galaxy fields with:

$10^{6}$ – $10^{9}$ structured coherence sources
Modal envelopes extending 50–100 kpc
Nonlinear saturation

produce photon deflection curves consistent with:

Strong lensing arcs
Einstein rings
Weak lensing shear fields

All without dark matter or geometric curvature.

Conclusion

Galaxy-scale lensing is not mysterious.
It arises from coherence field geometry and the structural necessity for photons to preserve internal phase in steep anchoring landscapes.

No invisible matter. No mass fitting.
Just coherence, anchoring, and structural drift.

Appendix P | [Index](./Appendix Master) | Appendix R