We study Hamel character topologies on the real line in two parts. Part I treats the full real-orbit (TVG) and the rational-parameter (TVGQ) dual pairs. A divisibility-chain criterion decides compatibility on the rationals and on Q/Z. Compatible subsets of one chain have a compatible join, but two different chains need not: an explicit Chinese remainder construction gives two compatible topologies with an incompatible join. This answers the compact-union question negatively, disproves the Greatest Compatible TVG Conjecture for every nonzero rational-valued Hamel functional, and shows that the corresponding rational and real dual pairs admit no Mackey topology. We classify additive homomorphisms and isomorphisms of the real-valued observation spaces and their automorphism groups, prove that both prime square-denominator topologies are non-nuclear, classify sequences converging for every coefficient functional, and show that the real parameters of continuous characters of the integers with the mixed-prime metric form a set of Hausdorff dimension zero. Part II studies real vector spaces whose Hausdorff additive group topology makes every fixed rational scalar multiplication continuous (TVGQ). A rational-valued Hamel functional and a rational parameter space S generate precompact, modular and real-valued observation topologies whose exact scalar ring is the multiplier ring of S. We compute their completions and character groups, classify the precompact TVGQ topologies, characterize universal separating sequences, show that local compactness and sigma-compactness force the scalar ring to be a field, classify isomorphisms within each modular family (exactly 2^c types), identify greatest-compatible existence with the unrestricted Mackey problem, and prove that all compatible topologies for the rational Hamel pair are non-Baire.