Appendix P — Derivation 16: Orbital Motion from Mutual Anchoring

(Foundations v1.0 • 2025-06-10)

Planetary orbits arise in Phase-Biased Geometry (PBG) because two
coherence fields bias one another’s motion. No Newton force, no curved
metric: each body simply follows the steepest-descent path in the
anchoring-cost landscape.

1 Two anchored modes

For bodies 1 and 2 with masses $M_{1, 2}$ the locked kernel
(Foundations §2) gives

Φ_{i} (r) = \frac{N_{i}}{r} e^{- k r}, N_{i} = \frac{c^{2} M_{i}}{4 π α}, k = \sqrt{β / α} .

Define $B_{i} = c^{2} Φ_{i}$ (units J).

2 Interaction cost

Total anchoring interaction

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

with $ρ_{i} (x)$ each envelope’s density ( $ρ_{i} \to δ$
for point mass).

3 Equation of motion (imports universal rule)

From Foundations §3:

m \ddot{r} = - m c^{2} \nabla Φ_{1} \equiv - \nabla (ρ_{2} B_{1}^{2}) .

For $k r ≪ 1$ this reduces to Newton’s
$\ddot{r} = - G M_{1} r / r^{3}$ , but the exact PBG force uses
$e^{- k r} (1 + k r)$ .

4 Orbital precession

Because $B (r)^{2} \propto r^{- 2} e^{- 2 k r}$ , the potential deviates
slightly from $1 / r$ . Expanding to first order in $k r$ yields the
well-known advance

Δ ϖ = \frac{6 π G M_{1}}{a (1 - e^{2}) c^{2}} (Mercury: 43^{″} / century),

identical to GR but here caused by the Yukawa correction rather than
metric curvature.

5 Stability & multi-body generalisation

Local minima of $C_{int}$ give stable radii.
Add further kernels → $N$ -body equations via superposition.
Resonances and tidal transfer follow by including the full
$ρ_{i} (x)$ shape (see [[3-D Envelope Implementation]]).

Take-home

Orbits in PBG are paths that keep the combined anchoring cost of all
bodies minimal. Apparent “gravity” is the gradient
$- c^{2} \nabla Φ$ , already fixed by ${α, β, γ}$ — no force
law or spacetime metric required.

Appendix O - Photon Structure | [Index](./Appendix Master) | Appendix Q - Galaxy-Scale Lensing