Appendix Y — Derivation 25: Continuum Mechanics of Coherence Media

Overview

To model large-scale coherence structures (e.g. galaxies, plasma-like fields, decohering regions), modal dynamics must support a continuum formulation.

This appendix derives:

• The fluid-like evolution of modal coherence
• The elastic-like response of anchoring tension
• The continuum analogues of stress, flow, and decoherence
• A unified set of modal field equations for coherent media

1. Modal Coherence Fields

Let:

• $${\rho }_c(x,t)\$\$bethe\ast \ast coherencedensity\ast \ast$$
• $$B(x,t)\$\$the\ast \ast anchoringfield\ast \ast$$

Define a coherence current:

which encodes the transport of coherence structure through the medium.

2. Continuity Equation for Modal Density

From conservation of coherence content:

Where:

• $${\mathrm{\Gamma }}_{\text{dec}}\$\$isthe\ast \ast localdecoherencerate\ast \ast ,increasingas\$\${\rho }_c\to {\rho }_{\text{crit}}$$

This defines the modal fluid continuity equation, with loss term from structural collapse.

3. Anchoring Tension as Stress

The anchoring cost from Appendix A is:

In continuum terms, this defines an effective modal stress tensor:

Where:

• The first term encodes directional coherence tension
• $$P_{\text{anchor}}=\beta {\rho }_c\$\$isscalaranchoringpressure$$

4. Momentum-Like Evolution of Coherence

Define a “modal momentum” field:

Its evolution obeys:

Where:

• $$\mathrm{\nabla }({\rho }_c{B}^2)\$\$represents\ast \ast externalanchoringbias\ast \ast$$

This is the analogue of the Navier–Stokes momentum equation, with structural (not viscous) tension and modal alignment driving flow.

5. Decoherence and Saturation Terms

The decoherence rate term $$\Gamma_\text{dec}$$ depends on:

• Local overlap of phase surfaces
• Proximity to $$\rho_{\text{crit}}$$
• Internal structural tension

We define:

This diverges as $$\rho_c \to \rho_{\text{crit}}$$, enforcing modal turnover and self-limiting coherence density.

6. Full System of Continuum Equations

Together, the modal continuum equations are:

1. Coherence continuity:

2. Phase momentum flow:

3. Anchoring field evolution:

This set defines the modal fluid-elastic field of the PBG framework.

Conclusion

The coherence medium behaves as a structurally reactive fluid with:

• Stress from phase gradients
• Flow from anchoring tension
• Decoherence-driven dissipation

Modal dynamics thus possess a full continuum mechanics formulation, enabling simulation and prediction of large-scale systems.

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