We determine joint Galois images for finite families of normalized Chebyshev maps with nonconstant affine targets over function fields. Over 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:Q]. For full towers, finite arithmetic index is equivalent to global pairwise coprimality of the degrees; over 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.
