Collatz and Fibonacci as Dual Operations of Icosahedral Geometry

The Fibonacci recurrence and the Collatz iteration are shown to be dual operations on the same geometric architecture: the icosahedron in D = 3 dimensions with Euler characteristic χ = 2. Fibonacci generates the growth constant φ = (1+√5)/2 from the vertex coordination number √5, with contraction ratio |−1/φ| < 1. Collatz generates the stability constant from D = 3 and χ = 2, through the lattice operator T(n) = (Dn+1)/χ, with average contraction D/χ² = 3/4 < 1. The runway numbers R_k = (χ^{2k}−1)/D that guarantee convergence are built entirely from lattice parameters. A Divisibility Theorem is proven: integer runways exist for all k if and only if D|(χ²−1), which for χ = 2 uniquely selects D = 3. Combined with the stability window χ < D < χ² (also uniquely selecting D = 3) and the no-cycle guarantee from GCD(D,χ) = 1, three independent conditions all point to the same unique dimension. Generalised Collatz maps T(n) = (Dn+1)/χ are tested across dimensions: D = 2 is ill-defined, D = 4 admits non-trivial cycles (no integer runways), and D ≥ 5 diverges. The continued fractions of the two constants reveal their complementary roles: φ is maximally irrational (CF all 1s, featureless) while α = log₂(D/χ) is lattice-structured (CF encoding χ, D, 5). The ground state {1, 2, 4} = {χ⁰, χ¹, χ²} is the first runway and the topology cycle. Second of four papers connecting the Collatz conjecture to icosahedral geometry in the Bootstrap Universe framework. Version 1.2: Addresses all adversarial review issues. Title updated from "Dual Eigenvalues" to "Dual Operations" with clarification of terminology. Divisibility condition promoted to formal Theorem with proof. No-cycle guarantee via GCD(D,χ) = 1 added. CF comparison integrated. Cross-references Paper 89 (doi:10.5281/zenodo.18674329) for rotation angle, runway singularities, and three-part proof.