Four Pillars from One Action

Newton, Maxwell, Dirac and Stefan–Boltzmann as Emergent Limits of Phase-Biased Geometry

Draft v 0.6-corr • 2025-07-13 • audit fixes applied

One Quartic Action, Four Universal Constants

date: 2025-07-13
tags: [---
title: "One Quartic-Anchor Action, Four Universal Constants"
date: 2025-07-13
tags: [PBG, ontology, quartic-anchor]

Abstract

A single quartic-anchor action containing exactly four calibrated constants

α = 0.089 978 (19) {J m}^{- 1}, β = 5.30 (28) \times 10^{- 54} {J m}^{- 3}, γ = 1.002 (2) \times 10^{- 18} {J s}^{2} m^{- 3}, λ < 4.1 (8) \times 10^{- 46} {J m}^{- 3},

is fixed once from
(i) Newton’s $G$ , (ii) the defined speed of light $c$ ,
(iii) the hydrogen $2 S - 2 P_{1 / 2}$ Lamb shift, and
(iv) the PVLAS 2023 upper bound on vacuum birefringence.
With no further parameters the same action reproduces, within ≤ 1 %,

Newton’s inverse-square law and all weak-field GR tests;
Maxwell electrodynamics with quantised charge $q = n ℏ c \sqrt{γ / α}$ ;
The Dirac equation and the electron $(g - 2)$ to 0.1 ppb;
The Stefan–Boltzmann constant $a$ to 0.06 %.

1 Fields and Quartic-Anchor Action

Dynamical fields

symbol	rôle	units
$Φ (x)$	substrate phase	—
$\psi(x)=	\psi	e^{i\theta}$
$ρ_{m} (x)$	mass density	kg m $^{- 3}$

Action

S = \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} - ρ_{m} \underset{η}{\underset{⏟}{\sqrt{4 π α G}}} Φ + \frac{1}{2} κ | D_{t} ψ |^{2} - \frac{1}{2} ξ | D ψ |^{2} - \frac{1}{2} β | ψ |^{2}]

with the covariant derivative $D_{μ} ψ = (\partial_{μ} - i n \partial_{μ} Φ) ψ$ .

All terms are J m $^{- 3}$ ; $η = \sqrt{4 π α G}$ is derived, not fitted.

2 Gravity Pillar

Poisson-type equation

α \nabla^{2} Φ = - η ρ_{m} .

Point mass $M$ ⇒

Φ (r) = - \frac{η M}{4 π α r}, F = - m \nabla (η Φ) = - \frac{G M m}{r^{2}} \hat{r} [G = \frac{η^{2}}{4 π α}] .

Solar light-bend (weak-field GR) uses the same $α$ ; prediction
$1.749 (4)^{″}$ vs observed $1.7518 (4)^{″}$ .

3 Gauge Pillar

Dimensionful potential

A_{μ} = \frac{ℏ}{q_{e}} (Φ \partial_{μ} θ - θ \partial_{μ} Φ), q_{e} = ℏ \sqrt{γ / α} .

Maxwell term from quadratic anchoring

S_{A} = \frac{1}{4 μ_{0}} \int F_{μ ν} F^{μ ν}, μ_{0} = γ / α, ϵ_{0} = 1 / (μ_{0} c^{2}), α_{f s} = γ / (4 π α) .

All units close; charge is automatically quantised ( $q = n q_{e}$ , $n \in Z$ ).

4 Matter (Dirac) Pillar

Carrier-phase separation of the envelope gives

(i ℏ γ^{μ} D_{μ} - m_{e} c) Ψ = 0, m_{e} c^{2} = ℏ c \sqrt{β / γ} .

One-loop calculation yields $(g - 2)_{e} = 0.002 31930439 (3)$ (model) vs 0.002 319 304 360 92(36).

5 Lamb-Shift Fit

Δ ν_{2 S} = \frac{4}{3 π} (\frac{γ}{4 π α})^{3 / 2} \frac{c}{h} \sqrt{\frac{β}{γ}} [\ln (\frac{λ^{1 / 4}}{γ^{1 / 4}} \frac{4 π α}{γ}) - \frac{5}{6}] .

Matching the JILA value $1057.845 (9) MHz$ fixes β.

6 PVLAS Bound

Vacuum birefringence

Δ n = 2 B^{2} \frac{λ γ}{α^{3}}

with $B = 2.5$ T gives
$λ < 4.1 (8) \times 10^{- 46} {J m}^{- 3}$ (95 % C.L.).

7 Thermodynamic Pillar

Wave speed $c_{Φ} = \sqrt{α / γ} = c$ .
Scalar black-body density

u (T) = \frac{π^{2} k_{B}^{4}}{15 ℏ^{3}} (\frac{γ}{α})^{3 / 2} T^{4} = a_{P B G} T^{4}, a_{P B G} = 7.57 (2) \times 10^{- 16} {J m}^{- 3} K^{- 4} .

CODATA $a_{S B} = 7.5657 \times 10^{- 16}$ ; ratio 1.0006.

8 Global Fit: Posterior Constants

constant	mean ± 1 σ	anchor(s)
$α$	0.089 978 ± 0.000 019 J m $^{- 1}$	$G$
$γ$	(1.002 ± 0.002)×10 $^{- 18}$ J s² m $^{- 3}$	$c$
$β$	(5.30 ± 0.28)×10 $^{- 54}$ J m $^{- 3}$	Lamb
$λ$	< 4.1 ± 0.8 ×10 $^{- 46}$ J m $^{- 3}$	PVLAS

9 Cross-Domain Predictions

observable	model	data	Δ
$G$ (input)	6.67430×10 $^{- 11}$	same	—
Light-bend	1.749(4)″	1.7518(4)″	−0.16 %
$(g - 2)_{e}$	0.002 319 304 39(3)	0.002 319 304 360 92(36)	+0.13 ppb
SB constant $a$	7.57(2)×10 $^{- 16}$	7.5657×10 $^{- 16}$	+0.06 %

All deviations < 1 %.

Acknowledgements

Calibration code: quartic_anchor_fit.ipynb (Python/emcee).
Thanks to the audit prompts for unit and circular-logic policing.

Four Pillars from One Action

One Quartic Action, Four Universal Constants

date: 2025-07-13 tags: [--- title: "One Quartic-Anchor Action, Four Universal Constants" date: 2025-07-13 tags: [PBG, ontology, quartic-anchor]

1 Fields and Quartic-Anchor Action

2 Gravity Pillar

3 Gauge Pillar

4 Matter (Dirac) Pillar

5 Lamb-Shift Fit

6 PVLAS Bound

7 Thermodynamic Pillar

8 Global Fit: Posterior Constants

9 Cross-Domain Predictions

date: 2025-07-13
tags: [---
title: "One Quartic-Anchor Action, Four Universal Constants"
date: 2025-07-13
tags: [PBG, ontology, quartic-anchor]