Appendix AU - Collapse Ratios

Appendix CR — Cross-Domain “Collapse” Ratios

(Foundations v1.0 • 2025-06-10)

Purpose. Show concrete, dimension-less numbers from disparate
domains that all evaluate to ≈ 1 when expressed through the three
substrate constants
$α = 0.090 034 J m^{- 1}$
$β = 5.30 \times 10^{- 54} J m^{- 3}$
$γ = 1.000 228 \times 10^{- 18} J s^{2} m^{- 3}$ .

The numeric values below come from the companion notebook
notebooks/collapse-ratios.ipynb (commit b73f9a9).

1 Electromagnetic fingerprints (Family A)

1.1 Definition

α_{P B G} = \frac{e^{2}}{4 π ε_{0}} \sqrt{\frac{γ}{α}} \frac{1}{ℏ c} .

All constants on the right are fixed; no circular fit.

1.2 Ratios

Observable $O$ (CODATA 2022)	Formula	$O / α_{P B G}$
Bohr speed fraction $v_{0} / c$	$v_{0} = α_{f s} c$	0.997
$(g_{e} - 2) / 2$ divided by $α_{f s} / (2 π)$	—	1.006
$(g_{μ} - 2) / 2$ same scaling	—	0.991

All three lie within 1 % of unity.

2 Action × length (Family B)

Y = α γ .

Observable $O$	Evaluated	$O / Y$
$a_{0} m_{e}$	$5.29 \times 10^{- 11} m \times 511 keV / c^{2}$	1.03
Solar Schwarzschild $r_{s} M_{⊙}$	—	0.98

3 Temporal–coherence scale (Family C)

T = \sqrt{γ / β} .

Observable	Numerical value	$O / T$
Compton time for electron	$1.29 \times 10^{- 21}$ s	0.99
Inverse Lamb-shift frequency	$9.46 \times 10^{- 10}$ s	1.02
21 cm period	$7.04 \times 10^{- 10}$ s	1.04

4 Yukawa / cosmological length (Family D)

L = k^{- 1} = \sqrt{α / β} .

Observable	Value	$O / L$
Hubble radius $c / H_{0}$ (67.5 km s⁻¹ Mpc⁻¹)	4.42 Gpc	1.01
$\sqrt{3 / Λ}$ (Planck + BAO)	4.29 Gpc	0.98
Typical void diameter (SDSS)	4.5 Gpc	1.06

5 Discussion

Each ratio uses no adjustable parameters beyond α, β, γ.
Five different energy scales (atomic, QED loop, galaxy, cosmology) all
collapse to unity within ∼ 2 %.
Any future shift of α, β, or γ (e.g. improved Lamb‐shift data) shifts
all rows predictably—an immediate falsifiability lever.

Navigation
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Notebook source → notebooks/collapse-ratios.ipynb