The Möbius Structure of Geometric Coupling Theory: Threshold, Coil, Measure

Formal verification of the Möbius transformation underlying Geometric Coupling Theory (GCT). The coupling equation α(d,β) = (2d+β)/(d+β) with β = 6/23 is shown to possess 20 exact algebraic identities, verified by symbolic arithmetic (91 sub-claims, 0 failures). The paper establishes three non-collapsing reference points — the asymptotic ceiling (2), the rational gateway (α₀ = 48/25, the image of integer 3), and the dynamical attractor (z₊ = (20+√538)/23) — and proves that the small eigenvalue λ₂ = 2 − z₊ is the total width of the coupled zone. The gateway partitions this eigenvalue gap into an irrational piece (~1/3) and a rational piece (2/25, ~2/3). The topological gap between attractor and gateway spirals in the complex plane with phase ≈ 3/11. A feedback loop analysis shows all residuals from approximate results lock to Silver Geometry fractions below 0.1% error. The rational-transcendental fork between β = 6/23 and β = φ/(2π) is characterized. Three-layer framework: Möbius = threshold flip into measurability, Fibonacci = the coil (how structure grows), Silver Geometry = the measure (how to describe what exists).