This paper develops the mathematical framework of φ-damped spiral convergence — the mechanism by which self-referential iterations converge to the fixed point σ = 1/φ. The Golden Ladder residuals are the corrections at each rung of convergence. The spiral structure explains why the α series converges rapidly (each term is suppressed by w = 1/(360φ⁴) ≈ 6.6×10⁻⁴ per rung) and why the corrections involve prime-numbered rungs. The theory provides the mathematical foundation for the α derivation.