Foundational Definitions

PBG Foundations (re-built 2025-06-10)

Purpose – Collect in one place every definition, constant, and boxed equation
that higher-level pages (Newtonian gravity, Lamb shift, proton structure, …)
are allowed to use.
Rule – No derivation on the site may redefine or contradict anything in the Foundations.

§	File	What you’ll find
0	Units-Ledger	Phase & envelope units; SI dimensions of α, β, γ; universal coupling factor η = c²
1	Core-Action	Total action with matter term: $S = \int [γ (\partial_{t} Φ)^{2} - α \nabla Φ^{2} - β Φ^{2} - η ρ_{m} Φ] d^{4} x$
2	Kernel-&-Topology	Sourced Helmholtz equation, Yukawa kernel $Φ = N e^{- k r} / r$ ; flux quantisation $Q = 4 π α N$
3	Force-Law	Least-cost trajectory: $m \ddot{r} = - m c^{2} \nabla Φ$ ; low-range Newton limit
4	Envelope-Dynamics	Two-scale reduction to Schrödinger form; effective mass $m = ℏ^{2} γ ω / α$
5	Constants-Calibration	Final calibrated numbers: α = 0.090034 J m⁻¹, β = 5.30 × 10⁻⁵⁴ J m⁻³, γ = 1.0018 × 10⁻¹⁸ J s² m⁻³, $k^{- 1} = 1.30 \times 10^{26}$ m, $G = c^{4} / 4 π α$

All boxed equations inside those files are canonical.

Quick-reference equations

Name	Equation	Section
Speed of light	$c^{2} = \frac{α}{γ}$	§0
Sourced Helmholtz	$α \nabla^{2} Φ - β Φ = - η ρ_{m}$	§1
Kernel solution	$Φ (r) = \frac{Q}{4 π α} \frac{e^{- k r}}{r}, k = \sqrt{β / α}$	§2
Quantised source	$Q = 4 π α N, N \in ℤ$	§2
Universal force	$m \ddot{r} = - m c^{2} \nabla Φ$	§3
Effective mass	$m = \frac{ℏ^{2} γ ω}{α}$	§4
Derived Newton constant	$G = \frac{c^{4}}{4 π α}$	§5

Using this page

Authors – when you draft a new derivation, import the needed equations (e.g. “We use the universal force from Foundations §3”). Do not copy the algebra.
Readers – start here if a later note mentions α, β, γ, winding number, or the Yukawa kernel and you need the original definition.

Foundational Derivations

Last updated 2025-06-10 • Contact: Nicco Deamus

PBG Foundations (re-built 2025-06-10)

Contents

Quick-reference equations

Using this page