Appendix E — Derivation 5: Unified Action Principle

Appendix E — Unified Action Principle (PBG)

(Foundations v 1.0 · 2025-06-10)

Phase-Biased Geometry replaces the standard zoo of fields and metrics with
one scalar coherence phase $Φ (t, x)$ governed by a single,
three-term action. Every later PBG result — from $c$ and $G$ to the 21 cm
line — traces back to this Lagrangian.

E.1 Motivation

Conventional physics needs separate actions for gravity, gauge fields and
quantum waves. PBG collapses them into one coherence-anchoring principle,
hence its predictive economy.

E.2 Coherence-anchoring action

S [Φ] = \int d^{4} x L [Φ],

L [Φ] = \frac{1}{2} γ (\partial_{t} Φ)^{2} - \frac{1}{2} α | \nabla Φ |^{2} - \frac{1}{2} β Φ^{2} (Foundations §1) .

Constant	Locked value	Units	Role
$α$	0.090 034	J m⁻¹	spatial stiffness
$β$	5.30 × 10⁻⁵⁴	J m⁻³	bias / mass term
$γ$	1.000 228 × 10⁻¹⁸	J s² m⁻³	temporal inertia

E.3 Why only these terms?

With global phase shift, spatial isotropy, time translation, $P$ and $T$
invariance, the only relevant or marginal scalar operators (mass
dimension ≤ 4) are

(\partial_{t} Φ)^{2}, | \nabla Φ |^{2}, Φ^{2} .

Higher powers or higher derivatives are either forbidden or
infra-red–irrelevant.

E.4 Euler–Lagrange equation

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

E.5 Immediate corollaries

Wave dispersion
$ω^{2} = c^{2} k^{2}$ with $c^{2} = α / γ$ (relativistic).
Static Yukawa kernel
$Φ (r) = \frac{Q}{4 π α r} e^{- r / ℓ}, ℓ = \sqrt{α / β}$ (Foundations §2).
Red-shift law
$D_{L} (z) = \frac{c}{k} (1 + z) \ln (1 + z)$ with $k = \sqrt{β / α}$ (Appendix Dₓ).

E.6 Dimensional summary

Quantity	Mass dimension
$Φ$	0
$(\partial_{t} Φ)^{2}$	4
$	\nabla\Phi
$Φ^{2}$	2
$α$	+1
$β$	+3
$γ$	+3

(Mass dimension counted in ℏ = $c$ = 1 units.)

E.7 Links out

Lamb shift with $Φ$ → [[Appendix-AJ]]
Gravitational lensing via index $n = 1 - 2 Φ / c^{2}$ → [[Lensing-from-Coherence]]

Closing line

With just $α, β, γ$ and the action above, PBG reproduces a
remarkable fraction of observed physics — no separate metric, gauge
potentials or Higgs field required.

Appendix D - Lensing | [Index](./Appendix Master) | Appendix F - Thermodynamic Structure