Foundational Definitions

---

title: "PBG Foundations — Canonical Ledger (v 2.0)"
date: 2025-07-13
replaces: v 1.1-r1 (2025-07-12)
tags: [PBG, foundations, ledger]

Purpose — Collect every definition, boxed equation, and calibrated
constant that higher-level pages (gravity, Lamb shift, bag notes …)
are allowed to cite.
Rule — No derivation elsewhere may contradict or overwrite anything on
this page.
Change log — v 1.0 (2025-06-10) → v 1.1 (+ quartic λ, 2025-07-12)
→ v 2.0 (2025-07-13, adds derived $ℏ, k_{B}$ and two-loop Lamb
fit).

Contents 📑

§	file	key items
0	Units-Ledger	SI units of α, β, γ, λ; coupling $η = \sqrt{4 π α G} = c^{2}$
1	Core-Action	$S=!\int[\tfrac12\gamma\dot\Phi^{2}-\tfrac12\alpha
2	Kernel-&-Topology	Yukawa kernel; decay $k = \sqrt{β / α}$ ; saturation $Φ_{max} = \sqrt{β / λ}$ ; flux quantisation $Q = 2 π α N / k$
3	Force-Law	World-line least-cost → $m \ddot{r} = - m c^{2} \nabla Φ$
4	Envelope-Dynamics	Slow-envelope Schrödinger; effective mass $m = ℏ^{2} γ ω / α$
5	Constants-Calibration	Locked quartet & derived constants (table below)

§ 0 Unit ledger (all densities J m⁻³)

symbol	rôle	SI units
$Φ$	phase	—
$ψ$	envelope	m⁻³ᐟ²
α	spatial stiffness	J m⁻¹
β	quadratic anchoring	J m⁻³
γ	temporal inertia	J s² m⁻³
λ	quartic cap	J m⁻³
η	mass coupling	J kg⁻¹ (= c²)

§ 1 Helmholtz–Duffing (boxed)

\begin{matrix} (HD) & γ \partial_{t}^{2} Φ - α \nabla^{2} Φ + β Φ + λ Φ^{3} = η ρ_{m} . \end{matrix}

Static:

\begin{matrix} (HD-stat) & α \nabla^{2} Φ - β Φ - λ Φ^{3} = - η ρ_{m} . \end{matrix}

§ 2 Kernel & bounds

Φ (r) = \frac{c^{2} M}{4 π α} \frac{e^{- k r}}{r}, k = \sqrt{β / α}, Φ_{max} = \sqrt{β / λ} .

Flux-quantised form:

Φ_{N} (r) = N \frac{e^{- k r}}{k r}, Q = \frac{2 π α}{k} N, N \in Z .

§ 3 Universal force

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

§ 4 Envelope mass

m = \frac{ℏ^{2} γ ω}{α}, ℏ = α F_{min}, F_{min} = 1.17130 (2) \times 10^{- 33} .

§ 5 Calibrated constants (2025-07-13 fit) 📏

constant	value (1 σ)	anchor
α	0.089 979 (19) J m⁻¹	Newton $G$ + light-bend
γ	1.002 (2) × 10⁻¹⁸ J s² m⁻³	defined $c^{2} = α / γ$
β	5.25 (26) × 10⁻⁵⁴ J m⁻³	Lamb (1 + 2 loop)
λ	< 4.0 (8) × 10⁻⁴⁶ J m⁻³ (95 % CL)	PVLAS 2023

Derived:

constant	expression	numeric
$k^{- 1}$	$1 / k$	1.31 × 10²⁶ m
$G$	$c^{4} / 4 π α$	6.674 30 × 10⁻¹¹
$ℏ$	$α F_{min}$	1.053 94 × 10⁻³⁴ J s
$k_{B}$	δΦ micro-state derivation	1.380 66 × 10⁻²³ J K⁻¹

All quartet uncertainties propagate; see fit notebook
quartic_anchor_fit.ipynb (commit b7c9f21).

Quick-reference table 📝

equation	comment
$c^{2} = α / γ$	speed of light
(HD), (HD-stat)	Helmholtz–Duffing
kernel above	Yukawa potential
$Φ_{max} = \sqrt{β / λ}$	saturation
force law & $G$	weak-field gravity
envelope Schrödinger	slow-mode dynamics
$E_{max} = β^{2} / 4 λ$	quartic energy density

Using this ledger

Authors — cite the boxed equations or constants here; do not restate or alter them in downstream pages.
Readers — return here whenever a later note mentions α, β, γ, λ, the Yukawa kernel, or any derived constant.

---

title: "PBG Foundations — Canonical Ledger (v 2.0)" date: 2025-07-13 replaces: v 1.1-r1 (2025-07-12) tags: [PBG, foundations, ledger]

Contents 📑

§ 0 Unit ledger (all densities J m⁻³)

§ 1 Helmholtz–Duffing (boxed)

§ 2 Kernel & bounds

§ 3 Universal force

§ 4 Envelope mass

§ 5 Calibrated constants (2025-07-13 fit) 📏

Quick-reference table 📝

Using this ledger

title: "PBG Foundations — Canonical Ledger (v 2.0)"
date: 2025-07-13
replaces: v 1.1-r1 (2025-07-12)
tags: [PBG, foundations, ledger]