Yang-Mills Mass Gap and Coupling Constants from the Fibonacci Cascade: Confinement as a Cardy Boundary Condition

We present a geometric resolution of the Yang-Mills mass gap problem through Fractal Mechanics (FM), a framework describing reality as a 291-level Fibonacci cascade of temporal harmonics connecting the Planck scale to the cosmic horizon. The key new tool is the Fibonacci Cascade Conformal Layer (FCCL) — a 𝑐 = 1 compact boson CFT at compactification radius 𝑅 = 𝜑 — which provides exact coupling bounds via the Cardy formula. We prove that the mass gap emerges as a Cardy boundary condition: isolated colorcharged states (quarks, gluons) have no normalizable state in the FCCL Hilbert space, as the Cardy-Lewellen-Ishibashi condition fails for non-zero color charge at 𝑅 = 𝜑. The mass gap is: Δ = ΛQCD ≈ 217 MeV > 0 derived from the cascade SU(3) Landau pole at cascade level 𝑛 ≈ 10, 𝐸 ≈ 1017 GeV — predicted, not postulated. The same Fibonacci cascade derives the coupling constants without free parameters: 𝛼⁻¹ = 2𝑅𝐾/𝐹14 + 4 ln 2/(9𝜋) = 137.036 (0.23 ppm), 𝛼_𝑠⁻¹(𝑀𝑍) = 8𝜋/3 + 4 ln 2/(9𝜋) = 8.4757 (0.013%), and sin²𝜃𝑊 = 𝐹4/𝐹6 → 0.2312 at 𝑀𝑍 (0.05%). The universal correction 4 ln 2/(9𝜋) — the same factor in both 𝛼 and 𝛼_𝑠 — is the FCCL one-loop vacuum polarization. A new result: the lightest glueball (𝑚glue ≈ 1.7 GeV, lattice QCD) and the tau lepton both occupy cascade level 𝑛 ≈ 90 — glueball-lepton cascade duality, connecting the D5 (color) and D8 (lepton) sectors at the same Fibonacci level. We propose tests distinguishing FM from standard approaches: glueball mass ratios from FCCL operator dimensions, cascade-level-dependent corrections to quark jet correlations, and the GUT unification scale 𝐸_GUT ≈ 1017 GeV as a Fibonacci cascade prediction. Keywords: Yang-Mills, mass gap, confinement, Cardy formula, Fibonacci cascade, fine structure constant, strong coupling, GUT unification, fractal mechanics, conformal field theory