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

(Foundations v1.0 • 2025-06-10)

Gravity in Phase-Biased Geometry is not a force—nor does it curve
spacetime. It is the bias that arises when coherence fields overlap
and modes rearrange to minimise total anchoring cost.

1 Locked kernel recap

From Foundations §2

Φ (r) = \frac{N}{r} e^{- k r}, k = \sqrt{β / α}, N = \frac{c^{2} M}{4 π α} .

Define $B (r) = c^{2} Φ (r)$ (units: joule). Gradients in $B$ set the
anchoring landscape.

2 Cluster self-cost

For many envelopes $ψ_{i}$ inside a star/planet

B_{tot} (x) = \sum_{i} B_{i} (x), C_{self} = \int ρ_{c} (x) B_{tot}^{2} (x) d^{3} x .

The stable internal structure minimises $C_{self}$ :
self-interaction = “mass distribution.”

3 Motion of a test mode (force analogue)

Foundations §3 gives the universal rule

m \ddot{r} = - m c^{2} \nabla Φ = - \nabla [ρ_{c} B^{2}] .

Thus the test path is the steepest-descent route in anchoring cost;
looks like Newtonian gravity for $k r ≪ 1$ .

4 Two-body interaction

Interaction cost

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

→ minimising over each cluster’s path reproduces orbits, precession,
tidal effects—all with zero extra parameters.

5 Non-linear saturation

At high $ρ_{c}$ the $β ρ_{c}$ term dominates:
fields saturate instead of diverging ⇒ no singularities; “black hole”
becomes a coherence-collapse zone with finite anchoring density.

6 Effective mass

For any mode

m \propto γ \int (\partial_{t} Φ)^{2} d^{3} x,

so inertial/gravitational mass is simply temporal anchoring inertia.

Take-away

PBG reproduces gravitational phenomena because modes follow the bias of
their own coherence fields, not because a geometric force acts on them.
Gravity is internal bookkeeping of anchoring tension.

← Appendix K - Speed of Light | [Index](./Appendix Master) | Appendix M - Modal Topology

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