Appendix W — Derivation 23: Medium Constants $\alpha$, $\beta$, and $\gamma$ from Modal Anchoring

Overview

The PBG coherence medium is characterised by three structural constants:

• $\alpha$: spatial phase stiffness
• $\beta$: anchoring cost per unit modal density
• $\gamma$: temporal phase inertia

This appendix derives all three from first principles—based on phase geometry, coherence turnover, and structural propagation.

1. Anchoring Cost Functional

The full cost functional is:

Each constant governs a distinct tension in the coherence field:

• $\gamma$ penalises fast phase change in time
• $\alpha$ penalises spatial phase curvature
• $\beta$ penalises anchoring in saturated regions

2. Deriving $\alpha$ — Spatial Phase Rigidity

Assume $\psi (x)=\rho (x)e^{i\varphi (x)}$.
For slowly varying $\rho$, we approximate:

We define $\alpha$ such that a full $2\pi$ winding across a distance $\lambda$ stores unit coherence energy:

Solving gives:

In standard units with $\lambda =1$, $\rho =1$:

3. Deriving $\gamma$ — Temporal Phase Inertia

From Appendix K, the latent propagation speed is $c=\sqrt{\alpha /\gamma }$.
Solving gives:

Substituting the previous result:

4. Deriving $\beta$ — Anchoring Cost from Saturation

Anchoring becomes unstable as ${\rho }_c\to {\rho }_{\text{crit}}$.
To model this, $\beta$ must diverge at saturation.

From Appendix F:

To produce this divergence, define:

At low density:

This sets our energy scale.

5. Summary of Derived Constants

All medium constants are now fully derived from modal structure and coherence saturation.

Conclusion

The constants $\alpha$, $\beta$, and $\gamma$ are not free parameters.
They emerge from phase geometry, anchoring constraints, and coherence propagation. This completes the structural grounding of the PBG modal medium.

Appendix V | [Index](./Appendix Master) | Appendix X