An Introduction to Modal Dynamics and Biased Geometry

Mathematical Foundations of Phase-Biased Geometry

From Concepts to Equations

Phase-Biased Geometry (PBG) translates its core ideas into mathematics with one guiding rule:

Nature minimizes the cost of coherence anchoring for every mode.

Every “particle” or system is a mode—defined by an envelope (its spatial structure) and a coherence field (its influence on others).

The Cost Functional

The core mathematical object is the cost functional:

C [Φ, ψ] = \int d^{3} x [α | \nabla Φ |^{2} + β | ψ |^{2}]

$Φ$ : The coherence field (phase structure) at every point.
$ψ$ : The envelope of a mode—its spatial “shape.”
$α, β$ : Universal constants, calibrated once from observation.

Equations of Motion

By minimizing the total cost (and including a time term), we get:

γ \partial_{t}^{2} Φ = α \nabla^{2} Φ - β Φ

with the “speed of light” emerging naturally as:

c^{2} = \frac{α}{γ}

Shell Structure and Kernel Solutions

Bound modes (planets, electrons, belts) settle at radii that are minima of the cost—this creates shells (like atomic orbitals, planetary belts, galactic rings).
The kernel solution for the coherence field is:

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

with

k = \sqrt{\frac{β}{α}}

Universal Constants

PBG requires only three constants:

$α$ : Sets phase stiffness (calibrated from light bending, lensing)
$β$ : Sets envelope anchoring (from Lamb shift, quantum structure)
$γ$ : Sets temporal cost (from $c$ , the speed of light)

Once fixed, these constants let PBG predict phenomena at every scale—without further fitting.

Full Variational Derivations
Unification from 3 Constants

The Unified Law

All motion, structure, and interaction arise as modes follow least-cost paths in the field of coherence.

“Forces” are gradients in cost.
Shells and gaps are natural in the landscape of minima and nodes.

Next Steps

PBG’s mathematics unifies the physics of atoms, planets, and galaxies—not by inventing new forces, but by revealing the universal rule of coherence cost minimization.