Appendix B — Derivation 2: Anchoring Cost Functional and the Coherence Field

Overview

This derivation constructs the coherence field $B(x)$, which mediates modal anchoring, from first principles. The field is not assumed or fitted—it is derived by minimising the anchoring cost of sustaining coherence in space.

We show that the resulting field takes a Yukawa-like form:

This coherence kernel replaces the Newtonian potential and all classical force fields in modal dynamics.

1. Physical Setup

Let a mode (or emitter) be situated at a point or region in space. It projects a coherence structure outward—a field $B(x)$—which determines the anchoring landscape for all other modes.

We assume:

• Coherence must spread out from the source
• The structure must minimise anchoring cost
• The field must decay away from the source and be well-behaved at infinity

2. Anchoring Cost Functional for a Field

We define the coherence anchoring cost of a static field $B(x)$ as:

This expression penalises:

• High gradients (phase tension, anchoring strain)
• Large field amplitudes (excessive coherence without support)

This is the structural cost of maintaining the coherence field in space.

3. Variational Principle

To find the natural field shape, we minimise the cost:

$\delta \mathcal{C}[B]=0$

We take the functional derivative of $\mathcal{C}$ with respect to $B(x)$, using the Euler–Lagrange equation for scalar fields:

$\frac{\delta \mathcal{C}}{\delta B}=-2\alpha {\mathrm{\nabla }}^{2}B+2\beta B=0$

Simplifying:

${\mathrm{\nabla }}^{2}B-\frac{\beta }{\alpha }B=0$

This is the Helmholtz equation:

${\mathrm{\nabla }}^{2}B-k^{2}B=0$

with decay constant:

$k=\sqrt{\frac{\beta }{\alpha }}$

4. Solution: Spherically Symmetric Case

In spherical coordinates (for a point emitter), the Helmholtz equation becomes:

$\frac{1}{r^2}\frac{d}{dr}\left(r^2\frac{dB}{dr}\right)-k^{2}B=0$

The finite, vanishing-at-infinity solution is:

Where $A$ is a normalisation constant set by emitter strength.

5. Interpretation

The field $B(x)$ is not a force or potential. It is a bias landscape for anchoring:

• Modes drift along gradients of $B(x{)}^2$ to reduce coherence tension
• Anchored structures emerge at local minima or stable orbits
• Coherence saturation and exclusion occur in high-$B$ regions

This field governs:

• Gravitational motion
• Photon lensing
• Orbital stability
• Anchoring interaction of all modal structures

Conclusion

We have derived the coherence field $B(x)$ from a principled cost functional with no free parameters except the physical modal stiffnesses $\alpha$, $\beta$. This field:

• Replaces gravitational and electrostatic potentials
• Provides a universal structure for bias-following motion
• Predicts saturation, decay, and structural anchoring from first principles

Appendix A | [Index](./Appendix Master) | Appendix C