Appendix L — Derivation 12: Modal Self-Interaction and Gravitational Analogue

Overview

In general relativity, gravity is described as spacetime curvature sourced by mass-energy.

In modal dynamics, there is no spacetime fabric.
Mass, motion, and interaction emerge from coherence anchoring and field gradients.

This appendix derives how gravitational effects arise from the self-interaction of coherence fields emitted by modal structures—leading to orbital stability, field curvature, and deflection without invoking force or geometry.

1. Recap: The Coherence Field

From Appendix B, the coherence field emitted by a mode (or cluster) is:

B (r) = \frac{A}{r} e^{- k r}

This field represents the anchoring landscape: a structure that biases other modes' motion.

Latent modes follow gradients of $B^{2}$
Anchored modes contribute to the local field
High-density regions become coherence-saturated

2. Self-Interaction of Anchored Clusters

Let a large, extended modal structure (e.g. a planet or star) consist of many modes ${ψ_{i}}$ , each emitting coherence into space.

The total coherence field is:

B_{tot} (x) = \sum_{i} B_{i} (x)

Each $B_{i} (x)$ influences the anchoring of others. Thus, the cluster's internal coherence structure is shaped by self-field interaction.

To remain stable, the entire structure must minimise the total anchoring cost:

C_{self} = \int (ρ_{c} (x) \cdot B_{tot} (x)^{2}) d^{3} x

Where:

$ρ_{c} (x)$ is the local modal coherence density
$B_{tot} (x)$ is the total field generated by the cluster

3. Deriving the Force Analogue

Let an external (test) mode move near this structure. Its trajectory is governed by the gradient of coherence tension:

\frac{d^{2} \vec{x}}{d t^{2}} \propto - \nabla (ρ_{c} (x) B (x)^{2})

This is structurally equivalent to a gravitational force—though it is:

Emergent from anchoring tension
Density- and field-dependent
Saturable and nonlinear at high density

The motion appears gravitational but is actually bias-following through the coherence field landscape.

4. Mutual Anchoring Between Bodies

If two modal clusters emit coherence fields, their mutual anchoring shapes both trajectories.

The interaction cost is:

C_{int} = \int ρ_{1} (x) B_{2} (x)^{2} + ρ_{2} (x) B_{1} (x)^{2} d^{3} x

Minimising $C_{int}$ over each cluster's path yields their effective motion.

This produces:

Orbital precession (Appendix P)
Tidal distortion (field overlap)
Curved trajectories of latent modes (Appendix D)

All of which look gravitational, but arise from modal anchoring symmetry.

5. Nonlinear Field Contribution

At high coherence density, the field does not superpose linearly. Instead:

$B_{tot}$ saturates
Anchoring thresholds cap overlap
The field structure deforms to avoid decoherence

This replaces:

Singularities with coherence saturation
Black holes with anchoring collapse zones
Infinite forces with modal repulsion or reorganisation

6. Apparent Mass

In modal terms, mass is the resistance to coherence deformation. Modes with high $γ$ (temporal stiffness) respond slowly to field gradients, making them behave as massive.

Effective mass $m$ arises from:

m \propto γ \int | \partial_{t} ψ |^{2} d^{3} x

This connects to:

Inertia
Gravitational response
Structural persistence

Without requiring any particles or energy–momentum tensor.

Conclusion

What we call gravity is not a force.
It is the structural pressure of coherence fields interacting through anchoring gradients, guiding modal clusters into tension-minimising configurations.

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