The Folding Number of a Tensioned Substrate: α−1 as the Maximum Compression Depth of the Cosmic Membrane

We demonstrate that the maximum number of stable circular folds nmax of a continuous

elastic substrate under its own gravitational load is given by

nmax = log2πR2Uℓ2eg≈α−1 + 1 ≈138, (1)

where ℓeg is the elasto-gravity length of the cosmic membrane, RU is the comoving radius of

the observable universe, and α ≈1/137.036 is the fine-structure constant. The calculation

uses exclusively the Universal Tension Law F(λ) = ℏc/λ2 and the dark-energy compression

scale λDE = 0.135 mm, with zero free parameters. The result is robust under variation of

H0 across the entire Planck–local range (67–73 km s−1 Mpc−1).

We further show that Maxwell’s four equations emerge identically from the transverse

sector of the linearized substrate wave equation □λ = 0, with the electric charge identified

as a topological vortex carrying a fraction α of the local substrate tension.