The Fibonacci Engine in Two-Sided Closure Theory: Recursive Admissibility, Golden-Ratio Suppression, and the Non-Adjacent Quark Transition

This paper isolates the clearest hard technical result in the Two-Sided Closure Theory (TSCT) programme: a genuine Fibonacci recursion law in the admissible Temperley–Lieb sector TL₃. The admissible composition operator has Perron–Frobenius eigenvalue φ, and its n-step amplitude scales as φ⁻ⁿ, derived from the fusion rule τ⊗τ = 1⊕τ rather than assumed. The rare non-adjacent quark transition |Vᵤb| = h₂λ³φ⁻² is therefore not a fitted small number but the natural two-step admissible amplitude, giving |Vᵤb| ≈ 0. 00361 (PDG: 0. 00369, 2. 3%). The same φ⁻² appears independently in the Fibonacci F-matrix (|F^τ_ττ, τ|² = φ⁻²) and, with corrected Wick combinatorics, in the (3+1) D bridge-field 1PI quartic vertex (Γ₄ = φ²−3 = −φ⁻²). The paper is self-contained; its results can be assessed on their own merits without reference to the broader TSCT programme.