Appendix AE — Derivation 30: Continuum Mechanics from Modal Anchoring

Overview

Classical continuum mechanics describes the behaviour of fluids and solids via:

Density $ρ (x)$
Momentum $\vec{p} (x)$
Stress tensor $T_{i j}$

In PBG, matter and motion are replaced by coherent modal structure.
This appendix constructs a full modal continuum formulation, with:

Coherence density $ρ_{c} (x)$
Phase velocity ${\vec{v}}_{ϕ} = \nabla ϕ$
Anchoring tension as structural stress

The result is a fluid–elastic hybrid framework governed by phase geometry and coherence evolution.

1. Coherence Density and Phase Flow

Let:

$ρ_{c} (x, t)$ be the total modal coherence density
$ϕ (x, t)$ be the local phase field
${\vec{v}}_{ϕ} = \nabla ϕ$ the phase velocity field

The coherence current is:

{\vec{J}}_{c} = ρ_{c} {\vec{v}}_{ϕ}

This governs the flow of modal structure through space.

2. Continuity Equation for Coherence

Total coherence is conserved except during decoherence. The evolution is:

\frac{\partial ρ_{c}}{\partial t} + \nabla \cdot {\vec{J}}_{c} = - Γ_{dec}

where $Γ_{dec}$ is the local decoherence rate, which diverges near saturation.

3. Anchoring Stress Tensor

Anchoring cost from Appendix A:

C = \int (γ | \partial_{t} ψ |^{2} + α | \nabla ψ |^{2} + β (ρ_{c}) | ψ |^{2}) d^{3} x

leads to an effective anchoring stress tensor:

T_{i j} = α ρ_{c} \partial_{i} ϕ \partial_{j} ϕ + P_{anchor} δ_{i j}

Where:

First term is directional coherence tension
$P_{anchor} = β (ρ_{c}) ρ_{c}$ is scalar anchoring pressure

Define “phase momentum”:

\vec{p} (x, t) = ρ_{c} \nabla ϕ

Its evolution obeys:

\frac{\partial \vec{p}}{\partial t} + \nabla \cdot T = - \nabla (ρ_{c} B^{2}) - {\vec{f}}_{dec}

$\nabla (ρ_{c} B^{2})$ is the coherence gradient force from the bias field
${\vec{f}}_{dec}$ models structural drag from decoherence onset

This is the analogue of Newton’s second law for modal continua.

5. Elastic Analogy and Coherence Deformation

Let the phase displacement field be:

u_{i} = ϕ (x_{i})

Then the coherence strain tensor is:

ε_{i j} = \frac{1}{2} (\partial_{i} u_{j} + \partial_{j} u_{i})

The stress–strain relationship is encoded in anchoring cost gradients.
For nearly linear phase structures, we define modal stiffness:

T_{i j} = λ δ_{i j} \nabla \cdot \vec{u} + 2 μ ε_{i j}

with effective Lamé parameters $(λ, μ)$ derived from $α$ and $β$ .

Thus, modal coherence media behave like elastic continua under structural distortion.

Dissipative terms from phase disruption or decoherence can be modelled as:

T_{i j}^{viscous} = η (\partial_{i} v_{j} + \partial_{j} v_{i} - \frac{2}{3} δ_{i j} \nabla \cdot \vec{v})

This allows fluid-like coherence media with:

Vortex damping
Shear resistance
Structural delay in phase rearrangement

7. Full Field Equations

The modal continuum is governed by:

Coherence continuity:

\frac{\partial ρ_{c}}{\partial t} + \nabla \cdot (ρ_{c} \nabla ϕ) = - Γ_{dec}

Momentum evolution:

\frac{\partial (ρ_{c} \nabla ϕ)}{\partial t} + \nabla \cdot T = - \nabla (ρ_{c} B^{2}) - {\vec{f}}_{dec}

Anchoring field:

α \nabla^{2} B - β B + ρ_{c} = 0

This describes a coherence-based fluid–elastic hybrid, with structure, tension, and decoherence naturally coupled.

Conclusion

PBG supports a full continuum mechanics formalism where:

Coherence density replaces mass
Phase gradients define velocity and strain
Anchoring cost generates stress and tension
Decoherence governs dissipation and instability

The modal medium flows, resists, and reorganises just as classical matter does—yet its behaviour is entirely derived from coherence principles.

Appendix AD | [Index](./Appendix Master) | Appendix AF