Unification from 3 Constants

Unification from α β γ

(Foundations v 1.0 · 2025-06-10 • narrative summary)

Readers’ map. This page is an executive overview of how all PBG
predictions reduce to three substrate constants
α (J m⁻¹) spatial stiffness · β (J m⁻³) bias / “mass” ·
γ (J s² m⁻³) temporal inertia.

Derivations themselves live in the locked files:

Field equation, Yukawa kernel → [[Foundations/Full-Derivations#§1-§2]]

Calibrations → [[Foundations/Full-Derivations#§5]]

Speed of light → [[Foundations/Full-Derivations#§0]]

Lamb-shift β → [[Appendix-AJ]]

1 What the three constants do

Constant	Calibrated from	Key formula (see citation)
α (J m⁻¹)	Solar-grazing light bend Δθ⊙	$Φ (r) = \frac{Q}{4 π α r} e^{- k r}$
γ (J s² m⁻³)	Photon speed $c$	$c^{2} = α / γ$
β (J m⁻³)	Hydrogen Lamb shift	$k = \sqrt{β / α}$ , β fixes $k$

After those three anchors are set nothing else is tuned.
$G, ℏ, g_{s}, H_{0}$ and dozens of observables follow.

2 Cross-domain identities (concept sketches)

The expressions below reference the audited formulas; they are not
new derivations.

2.1 Cross-domain check (solar bend vs 21 cm)

The ratio combines the Yukawa factor $k R_{⊙}$ and the hydrogen
overlap term $| ψ_{20} (0) |^{2}$ :

\frac{Δ θ_{⊙}}{Δ ν_{21} / ν_{21}} = \frac{3 M_{⊙} E_{21}}{8 π α c^{2} R_{⊙} β | ψ_{20} (0) |^{2}} = (\frac{3}{8}) k R_{⊙} \frac{E_{21}}{β | ψ_{20} (0) |^{2}} .

See Full GR Derivation and Emergence of G for $Δ θ_{⊙}$ and Appendix AJ - Lamb Shift for
the 21 cm derivation; all factors trace back to the locked constants
$α, β, γ$ .

2.2 Action scale

$S_{min} = 2 π ℏ$ (Appendix AS - Planck ℏA) with
$ℏ = α F_{min}$ links atomic and anchoring domains.

2.3 Cosmic length

$L = k^{- 1} = \sqrt{α / β}$ appears simultaneously in

Yukawa tail of nuclear kernel
Hubble radius $c / H_{0}$ (Sample Physical Results)
Large-void correlation length.

(Full proofs in their respective appendices.)

3 Collapse tables (heuristic, unit-checked work in progress)

These tables show dimension-less ratios that empirically hover near 1
once α, β, γ are locked. They are indicators, not yet rigorous proofs.

Appendix AU - Collapse Ratios

Family A – Electromagnetic fingerprints

Observable $O$	$O / α_{P B G}$
$v_{0} / c$ (Bohr electron)	≈ 1
$\sqrt{16 Δ E_{f s} / E_{R y d}}$	≈ 1
$(g_{e} - 2) / (α_{f s} / π)$	≈ 1

Family D – Yukawa / cosmic length

Observable	$O / L$
$R_{H} = c / H_{0}$	≈ 1
$\sqrt{3 / Λ}$	≈ 1

4 Why three anchors matter

Independence – each set by a different class of measurement.
No circular fits – after α, γ, β are locked, all “collapse-tests”
are predictions.
Breadth – same numbers hit atomic spectra, nucleon structure,
relativistic lensing, cosmological drift.

A single failed identity would falsify PBG.

Derivations → Foundational Derivations
Calibrated numbers → Foundational Definitions §5
Precision tests → Full GR Derivation and Emergence of G, Appendix AJ - Lamb Shift, Magnetic-Moment Structures

Home Phase-Biased Geometry Foundational Derivations Appendix Master