Introduction

This document establishes the foundational geometric model for the Ives–Stilwell reconstruction within the Lilborn Equation Framework. It defines the Triangle of Coherence Interaction and provides the basis for all subsequent derivations, particularly the reinterpretation of Doppler shift effects without invoking time dilation.

Definition of the Triangle

The Triangle of Coherence Interaction is defined as a right triangle whose vertices and sides are derived from the relative motion of a moving emitter and the stationary observer, within a spatial coherence field.

Components

1. Vertex A: Position of the moving emitter at the moment of emission.

2. Vertex B: Position of the stationary observer.

3. Vertex C: Projected intersection of the emitter’s path relative to the observer’s frame.

Sides

• Side AB (base): The velocity vector (v) of the emitter relative to the observer.

• Side AC (adjacent): The coherence-registered projection path (normalized to 1).

• Side BC (hypotenuse): The structural path representing interaction length, equivalent to the refractive index n in spatial form.

Angles

• Angle θ (at vertex A): The effective interaction angle between the motion of the emitter and the structural registration path.

• Relationship: The key trigonometric identity is cos(θ) = 1/n, forming the basis for deriving the Fresnel term (1 – 1/n²) in analogous structural experiments.

Conclusion

This triangle is the foundation of all angular coherence derivations to follow. It geometrically encodes the projection distortion experienced in coherent interaction due to emitter motion. All spectral asymmetries and frequency distortions observed in Ives–Stilwell-type experiments will be reconstructed from this base.

Produced by The Lilborn Equation Team:

Michael Lilborn-Williams

Daniel Thomas Rouse

Thomas Jackson Barnard

Audrey Williams