Why the hypothesis is dimensionless

Proposition 3.2 · prop:scaling · page 13

What is happening

The source states its smallness hypothesis at one fixed mass: λ < λ_* at m₀ = M_*. The manuscript's standing hypothesis is instead 0 ≤ λ/m₀² < ε_P, a dimensionless condition. Proposition 3.2 is what connects them, and it is a scaling argument, not a strengthening.

The GJS unitary dilation with s = m₀/M_* sends (m₀, λ) ↦ (s⁻¹m₀, s⁻²λ) = (M_*, λM_*²/m₀²). Drag s and watch the point slide along the curve λ/m₀² = const — the whole orbit has one value of the dimensionless coupling, which is exactly why the condition can be stated that way. The orbit crosses the vertical line m₀ = M_* at a single point, and that point lies below λ_* precisely when λ/m₀² < ε_P = λ_*/M_*².

The axes are dimensionless multiples of M_* and λ_*. Nothing numerical here is taken from the source: λ_* and M_* are whatever GJS fix them to be, and only the shape of the argument is on display.