Three Confined Channels from a Logarithmic Scattering Density

A scattering density of 1/ln n on the positive integers, interpreted as a radial coordinate, produces an effective potential V =  -π²/u² via WKB inversion. The potential admits exactly three confined angular momentum channels (ℓ = 0, 1, 2) separated from a free continuum (ℓ ≥ 3) by a mass gap ΔE = √((π²-6)(12-π²)) = 2.8712. The gap is strictly positive because π is transcendental. The predicted value falls within the range 2.73–2.90 from current lattice and phenomenological determinations of the QCD confinement scale.