An Introduction to Modal Dynamics and Biased Geometry

# Mathematical Foundations of Phase-Biased Geometry

(synoptic page • references Foundations v1.0, 2025-06-10)

Phase-Biased Geometry (PBG) rests on one guiding rule

Every mode evolves so as to minimise its coherence-anchoring cost.

A mode = envelope $ψ (x)$ (spatial density) + phase field $Φ (x, t)$ (its influence on others).

1 Cost functional (import)

From Foundations/Full-Derivations#§1.-Core-Action-with-Matter-Coupling:

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

Constants (locked in Foundations §5)

$α$	$β$	Units
0.090 034	$5.300 \times 10^{- 54}$	J m $^{- 1}$ / J m $^{- 3}$

2 Field equation & speed of light

Adding the temporal term $\frac{1}{2} γ (\partial_{t} Φ)^{2}$ and minimising the action gives (Foundations §1):

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

The ratio $α / γ$ fixes the emergent light speed

c^{2} = \frac{α}{γ} (Foundations §0) .

3 Kernel solution & coherence shells

Static sourced Helmholtz solution (Foundations §2):

Φ (r) = \frac{N e^{- k r}}{r}, k = \sqrt{\frac{β}{α}} .

Minima in the combined envelope + kernel cost produce natural shells: atomic, planetary, galactic.

4 Three universal constants — no more

Constant	Calibrated from	Locked value
$α$	solar light-bending	0.090 034 J m $^{- 1}$
$β$	hydrogen Lamb shift	$5.30 \times 10^{- 54}$ J m $^{- 3}$
$γ$	speed of light ( $c$ )	$1.000228 \times 10^{- 18}$ J s $^{2}$ m $^{- 3}$

Once fixed, ${α, β, γ}$ power all PBG predictions—no further fitting.

5 Unified picture

“Forces” = gradients of anchoring cost
$m \ddot{r} = - m c^{2} \nabla Φ$ (Foundations §3).
Shells & gaps = minima / nodes of the Gaussian envelope width
$σ^{2} = α / (β + λ)$ (Foundations §1).
Mass, charge, magnetic moment, red-shift… each trace back to the same kernel and cost functional.

Further links

Full algebra → [[Foundations/Full-Derivations]]
Cross-scale examples → [[Sample Physical Results]]
Concept narrative → [[Concept-v2/Stage-A]]

PBG unifies atoms, planets, and galaxies by one rule: minimise coherence-anchoring cost. New data refine ${α, β, γ}$ —the equations remain the same.