An Introduction to Modal Dynamics and Biased Geometry

Mathematical Foundations of Phase-Biased Geometry — Synoptic Note

title: "Mathematical Foundations of Phase-Biased Geometry — Synoptic Note"
version: v 2.0 · 2025-07-13
uses: PBG Foundations — Canonical Ledger (v 2.0)
tags: [PBG, overview]

One rule – every mode minimises its coherence-anchoring cost.
One quartet – $α, β, γ, λ$ calibrates
all observables; no higher-level page may redefine anything below.

0 Canonical cost functional 🖋️

Imported from [[Core-Action]] (§ 1):

C [Φ, ψ] = \int d^{3} x [α | \nabla Φ |^{2} + β | ψ |^{2}], α = 0.089 979 {J m}^{- 1}, β = 5.25 \times 10^{- 54} {J m}^{- 3} .

(Quadratic δΦ + envelope term; the quartic λ only enters at saturation.)

1 Field equation & the speed of light

Augmenting $C$ with
$\frac{1}{2} γ (\partial_{t} Φ)^{2}$
gives the Helmholtz–Duffing equation ([[Core-Action]], boxed):

γ \partial_{t}^{2} Φ - α \nabla^{2} Φ + β Φ + λ Φ^{3} = η ρ_{m}, c^{2} = \frac{α}{γ}, γ = 1.002 \times 10^{- 18} {J s}^{2} m^{- 3} .

2 Yukawa kernel & shells

Weak-field, static solution ([[Kernel-&-Topology]]):

Φ (r) = \frac{c^{2} M}{4 π α} \frac{e^{- k r}}{r}, k = \sqrt{β / α} = 7.6 \times 10^{- 27} m^{- 1} .

Minima in the combined cost define natural
coherence shells (atomic, planetary, galactic).

3 Universal force law

World-line least-cost ([[Force-Law]]):

m \ddot{r} = - m c^{2} \nabla Φ ⟹ | F | = \frac{G M m}{r^{2}}, G = \frac{c^{4}}{4 π α} .

4 Minimal constants set

category	expression	locked value
Quartet	α, β, γ, λ	see ledger (λ < 4.0×10⁻⁴⁶ J m⁻³ @95 % CL)
Derived	$c^{2} = α / γ$	$299792 458^{2}$ m² s⁻² (exact)
	$ℏ = α F_{min}$	1.053 94 × 10⁻³⁴ J s
	$k_{B}$ via δΦ micro-states	1.380 66 × 10⁻²³ J K⁻¹

No further parameters are fitted anywhere in PBG.

5 Unified picture

Forces = gradients of anchoring cost
$m \ddot{r} = - m c^{2} \nabla Φ$ .
Shells / gaps = minima of total cost
$α | \nabla Φ |^{2} + β | ψ |^{2} + λ | Φ |^{4}$ .
Matter wave dynamics → Schrödinger limit with
$m = ℏ^{2} γ ω / α$ .
Radiation → Stefan–Boltzmann with
$u (T) = [π^{2} k_{B}^{4} / 15 ℏ^{3}] (γ / α)^{3 / 2} T^{4}$ .

Further links

Full algebra → [[PBG Foundations — Canonical Ledger (v 2.0)]]
Sample predictions (Lamb, $g - 2$ , 21 cm …) → [[Sample Physical Results]]
Narrative overview → [[Concept-v2 / Stage A]]

Phase-Biased Geometry needs one guiding rule and four calibrated
constants. All else—light speed, Planck’s constant, Boltzmann’s constant,
gravity, atomic structure—emerges by least anchoring cost.