This paper establishes structural limits on the determinability of dimensionless gauge couplings in renormalizable four-dimensional quantum field theories. Within the standard framework of Lorentz invariance, local gauge symmetry, renormalizability, unitarity, anomaly cancellation, and canonical normalization, it is shown that dimensionless couplings such as the fine-structure constant cannot be internally fixed by symmetry or algebraic consistency alone. The analysis formalizes coupling space as a space of moduli — continuous parameters labeling structurally admissible but physically inequivalent theories — and interprets renormalization group flow as motion on this theory space. The paper derives explicit perturbative stability basin constraints, presents worked examples from QED and QCD (including dimensional transmutation and the emergence of the QCD scale), and proves that Gaussian fixed points in four dimensions cannot uniquely determine gauge couplings because they possess marginal directions. A strengthened uniqueness criterion is given: unique determination of a dimensionless coupling requires a strongly isolated ultraviolet fixed point with no relevant or marginal directions. A determinability taxonomy is also provided, identifying the only structural mechanisms by which a dimensionless coupling could become calculable (fixed-point determinacy, dimensional transmutation, dynamical stabilization, ultraviolet matching, or environmental selection). Absent such mechanisms, dimensionless couplings remain continuous moduli rather than derived quantities. The work is not a proposal for deriving the value of the fine-structure constant. Instead, it establishes boundary theorems clarifying when such a derivation would be structurally possible and when it is not. The analysis is confined to renormalizable four-dimensional quantum field theory; extensions to infinite-dimensional Wilsonian effective field theories or quantum gravity would require additional structural analysis.