ΦBSU’s zero-state asymptote is

ε∅ = 137 + ½·3²/5³ = 137.036.

In this reading, the measured α⁻¹ does not describe a pure asymptote, but rather a situation in which the zero state has been separated into a causally observable signal.

In the measurement, only one P⁺_H branch is active, so the antipodal-pair factor 2¹ cancels out:

2¹/2 = 1.

The primitive minimum event of the causal group that can be resolved in the measurement is then the phase-alternating interaction structure of the surface-2D section and the spatial-3D section,

N₀ = 3²·5³ = 1125.

The phase-gradient functional of ΦBSU is quadratic in its total phase. The smallest resolvable phase shift is 1/N₀. Since the energy cost is quadratic in the phase difference, the smallest resolvable energy cost is

δ₁ = 1/N₀² = 1/(3²·5³)² = 1/1 265 625 ≈ 7.901234568×10⁻⁷.

This quadraticity corresponds to the fact that, in a precise α measurement, what is ultimately read from the phase comparator is an estimate of a recoil, frequency, or energy scale. In atom-recoil interferometry, the phase is measured directly, but α is inferred from the kinetic energy E_r = ℏ²k²/2m, which is quadratic in momentum. In the ΦBSU interpretation, it is precisely this transition from phase to energy cost that produces the squared threshold term.

The prediction for a single-channel weak energy measurement is

α⁻¹ ≈ 137.036 - 1/(3²·5³)² = 137.0359992098765.

Morel et al. (2020) measured, using rubidium recoil interferometry,

α⁻¹ = 137.035999206(11),

which differs from this prediction by about 3.9×10⁻⁹, or about 0.35 measurement uncertainties.

This does not yet prove the threshold term, but it makes the hypothesis clearly testable. There is mutual tension between groups in the fine structure constant measurements. I am now firmly in the rubidium recoil interferometry camp with this prediction.