Foundational Derivations

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title: "Phase-Biased Geometry — Foundational Compendium v 2.0"
date: 2025-07-13
replaces: v 1.1-r1 (2025-07-12)
tags: [PBG, foundations, quartic-anchor]

Purpose. Provide one audit-clean scroll that

sets unit conventions,

states the complete quartic-anchor bulk action,

derives the sourced Helmholtz–Duffing equation, kernel, force law,

shows the slow-envelope (Schrödinger) limit, and

lists the latest quartet calibration
$α, β, γ, λ$
obtained from the global fit (G, c, Lamb (1 + 2 loop), PVLAS).

All later derivations must cite—not overwrite—these results.

§ 0 Unit Ledger (all densities J m⁻³)

symbol	rôle	SI units	note
$Φ$	dimension-less phase	1	—
$ψ$	complex envelope	m⁻³ᐟ²	$
$α$	spatial stiffness	J m⁻¹	fixes $G$
$β$	quadratic anchoring	J m⁻³	sets Yukawa mass
$γ$	temporal inertia	J s² m⁻³	fixes $c$
$λ$	quartic cap	J m⁻³	saturation bound
$η$	mass-substrate coupling	J kg⁻¹	$η = \sqrt{4 π α G} = c^{2}$

(Every term in the action density below evaluates to J m⁻³.)

§ 1 Quartic-Anchor Action with Source

S [Φ; ρ_{m}] = \int d^{4} x [\frac{1}{2} γ (\partial_{t} Φ)^{2} - \frac{1}{2} α | \nabla Φ |^{2} - \frac{1}{2} β Φ^{2} - \frac{λ}{4} Φ^{4} - η ρ_{m} Φ]

Helmholtz–Duffing equation

\begin{matrix} (1.1) & γ \partial_{t}^{2} Φ - α \nabla^{2} Φ + β Φ + λ Φ^{3} = η ρ_{m} . \end{matrix}

Static limit ( $\partial_{t} Φ = 0$ ):

\begin{matrix} (1.2) & α \nabla^{2} Φ - β Φ - λ Φ^{3} = - η ρ_{m} . \end{matrix}

§ 2 Kernel & Topological Charge

2.1 Yukawa solution (weak-field, $| Φ | ≪ Φ_{max}$ )

For a point mass $ρ_{m} = M δ^{3} (r)$ and neglecting the
cubic term:

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

2.2 Saturation amplitude

Φ_{max} = \sqrt{β / λ} .

Yukawa form valid only while $| Φ | ≪ Φ_{max}$ .

2.3 Flux quantisation

\oint_{Σ} \nabla Φ \cdot d S = 2 π N, N \in Z ⟹ Q = \frac{2 π α}{k} N .

Integer-charge kernel:

Φ_{N} (r) = N \frac{e^{- k r}}{k r}, | Φ_{N} | ≪ Φ_{max} .

§ 3 Universal Force Law

World-line action for a point mode of inertial mass $m$ :

S_{mode} = \int d t [\frac{1}{2} m {\dot{r}}^{2} + m c^{2} Φ (r)] .

Euler–Lagrange:

m \ddot{r} = - m c^{2} \nabla Φ .

For $k r ≪ 1$ and $N = 1$ :

| F | = \frac{G M m}{r^{2}}, G = \frac{c^{4}}{4 π α} .

§ 4 Slow-Envelope (Schrödinger) Limit

Write $ψ = φ e^{- i ω t}$ with $ω ≫ \dot{φ}$ .

i \partial_{t} φ = - \frac{α}{2 γ ω} \nabla^{2} φ + \frac{β - γ ω^{2}}{2 γ ω} φ .

Identify

m = \frac{ℏ^{2} γ ω}{α}, V_{bias} = m c^{2} Φ_{ext} (x),

to obtain the ordinary Schrödinger equation.

§ 5 Quartet Calibration (2025-07-13 global fit)

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

Derived:

Yukawa length $k^{- 1} = 1.31 \times 10^{26} m$ .
$G_{PBG} = 6.674 30 \times 10^{- 11} m^{3} k g^{- 1} s^{- 2}$
(locked by construction).
Planck constant $ℏ = α F_{min}$
with $F_{min} = 1.17130 \times 10^{- 33}$ from the photon action appendix.
Boltzmann constant $k_{B} = 1.38066 (11) \times 10^{- 23} J K^{- 1}$
from the δΦ micro-state count.

Quick Reference (copy-box)

Expression	Role
$Φ (r) = \frac{c^{2} M}{4 π α} \frac{e^{- k r}}{r}$	Yukawa kernel
$Φ_{max} = \sqrt{β / λ}$	saturation amplitude
$F = - m c^{2} \nabla Φ$	universal force
$m = ℏ^{2} γ ω / α$	envelope mass
$E_{max} = β^{2} / 4 λ$	quartic energy density

Version log

v 2.0 (2025-07-13) — adds quartic cap units fix, η = √(4παG),
Planck & Boltzmann derivations, two-loop Lamb fit, updated quartet.
v 1.1-r1 (2025-07-12) — Yukawa integer form $+ k$ patch.
v 1.0 (2025-06-10) — initial quadratic draft.

(Any future update must cite this ledger and record its diff.)

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title: "Phase-Biased Geometry — Foundational Compendium v 2.0" date: 2025-07-13 replaces: v 1.1-r1 (2025-07-12) tags: [PBG, foundations, quartic-anchor]