Geometric and Arithmetic Independence of Affine Chebyshev Towers

We determine joint Galois images for finite families of normalized Chebyshev maps with nonconstant affine targets over function fields. Over \(\mathbb{C}(t)\), endpoint graphs and degree parity determine the geometric image; least common multiples merge repeated endpoint pairs, while greatest common divisors and quadratic square classes determine subfamily intersections. These formulas give exact strict-independence and finite-index criteria for the full preimage towers. Over a number-field function field \(k(t)\), globally pairwise coprime finite degrees give an arithmetic image combining endpoint cut-space signs with the cyclotomic Galois image. We compute its constant field and show that the joint index divides \([k:\mathbb{Q}]\). For full towers, finite arithmetic index is equivalent to global pairwise coprimality of the degrees; over \(\mathbb{Q}(t)\), this is equivalent to strict independence even when the geometric fields are entangled. The results are complete for this stated special family, not for arbitrary rational maps or constant number-field targets; the general AIM independence question remains open. Portable exact-check programs accompany the written proofs. The manuscript is AI-assisted, self-audited and unrefereed; no independent peer review, formal verification or absolute priority is claimed.