Orbital Modelling in PBG

Orbital Modelling in PBG: Mercury and Earth–Moon Systems

Core Principle

In Phase-Biased Geometry (PBG), planetary motion arises from minimising an anchoring cost functional defined over overlapping coherence fields. There is no gravity, spacetime, or curvature—only modal coherence and bias-following trajectories.

Coherence Fields

Sun/Earth modal shell:

B_{⊙} (r) = \frac{A}{r (1 + ℓ r + ϵ r^{2})}

$A$ : Anchoring amplitude
$ℓ$ : Decay rate (field steepness)
$ϵ$ : Shell curvature parameter

Planet/Moon modal patch:

B_{M} (r) = B_{M} \cdot \exp (- \frac{r^{2}}{2 σ^{2}})

$B_{M}$ : Coherence amplitude
$σ$ : Modal coherence width

Anchoring Cost Functional

Total coherence field:

B (\vec{r}) = B_{⊙} (\vec{r}) + B_{M} (\vec{r})

Anchoring cost:

C [B] = \int [α | \nabla B |^{2} + μ (\vec{r}) B^{2}] d^{3} x

$α = 0.090034$ J·s $^{2}$ /m $^{3}$ (anchoring stiffness)
$β = 5.30012 \times 10^{- 54}$ J/m $^{3}$ (coherence penalty, appears in kernel)
$γ = 1.000228 \times 10^{- 18}$ J·s $^{2}$ /m (for photon/EM-related effects)
$μ (\vec{r})$ : Coherence density (localised for each modal patch)

Equation of Motion (Bias-Following)

The trajectory follows the gradient of cost:

\vec{a} = - \frac{1}{m} \nabla C

Motion is always along the steepest descent in total anchoring cost—no force or field is invoked.

Model Parameters

Mercury–Sun System

Parameter	Value	Description
$A$	$8.0 \times 10^{25}$	Solar field amplitude
$ℓ$	$1.5 \times 10^{- 7}$ km $^{- 1}$	Solar decay rate
$ϵ$	$5.0 \times 10^{- 15}$ km $^{- 2}$	Shell curvature
$B_{M}$	$2.0 \times 10^{23}$	Mercury coherence amplitude
$σ$	$5.0 \times 10^{4}$ km	Mercury patch width

Earth–Moon System

Parameter	Value	Description
$A_{E}$	$7.9 \times 10^{20}$	Earth field amplitude
$k_{E}$	$4.7 \times 10^{- 7}$ km $^{- 1}$	Earth decay constant
$ρ_{M}$	$1.9 \times 10^{- 6}$	Moon coherence density
$σ$	$6.0 \times 10^{4}$ km	Anchoring width

Results

Mercury

After approximately 415 orbits (one century), PBG predicts:

Precession = {41.85}^{″} per orbit ⟹ \sim 43^{″} per century

This matches the observed general relativistic value, with no gravity, curvature, or tuning.

Earth–Moon

A stable orbit is reproduced, with PBG binding energy:

E_{bind}^{PBG} = + 3.67 \times 10^{28} J

This matches the Newtonian binding energy in magnitude (sign reflects coherence enhancement).

Method Summary

Define coherence fields for each body.
Compute the total cost functional for all positions.
Obtain the cost gradient (bias-following acceleration).
Integrate motion stepwise—no curvature or gravitational potential at any stage.
Parameters are set by modal structure, not tuned to data.

Physical Significance

Precession and orbital stability emerge from mutual modal anchoring alone.
No gravity, spacetime, or force fields are invoked.
The approach generalises naturally to any planetary system:
- Define all modal fields
- Sum anchoring costs
- Integrate motion via cost gradients

Universal Anchoring Constants

Constant	Value	Units
$α$	$0.090034$	J·s $^{2}$ /m $^{3}$
$β$	$5.30012 \times 10^{- 54}$	J/m $^{3}$
$γ$	$1.000228 \times 10^{- 18}$	J·s $^{2}$ /m

All planetary motion, including precession and binding energy, is recovered from coherence structure alone within PBG. No gravity, curvature, or external fields are required.