The first three Kaluza–Klein masses do not uniquely determine the one-loop Wilson-line curvature

We study whether three low-lying fermion masses determine a finite Wilson-line response in a five-dimensional Abelian theory on a circle. A neutral angular field generates a periodic fermion mass profile on a winding minimum of a local energy with two cosine couplings. Together with the Yukawa amplitude, these couplings are the three varied inputs. The first three antiperiodic Kaluza–Klein masses locally identify these inputs in a specified weak-coupling region. Nevertheless, we construct two separated inverse branches with exactly the same three masses and different leading fermion one-loop Wilson-line curvatures. Their difference has a negative leading term with an explicit positive coefficient involving the induced fifth and seventh harmonics; the fourth mass distinguishes the pairs. The proof retains all harmonics generated by the field equation and uses sequential weak-shape and weak-amplitude limits. Numerical matching and a transfer-difference calculation realize the construction. We also calculate angular and gauge corrections with an order-by-order adjustment of three real resonance masses in a specified local subtraction. At an illustrative coupling point their known sum is 1.56% of the leading curvature difference in magnitude. Unassigned effective-field-theory coefficients and higher loop orders remain outside this numerical sum. The result separates local spectral identification from uniqueness of the Wilson response across parameter neighborhoods.