Local Zero Branches and Degenerate Splitting in a Multiplicative Residual Operator

We study the local zero structure of the multiplicative residual operator Δ_q F(x) = qF(qx) − F(q²x) in the neighborhood of q = 1. A second-order expansion identifies the regularity condition governing local zero branches and yields an explicit first-order displacement law for nondegenerate zeros. The analysis shows that, when F''(x*) ≠ 0, a zero satisfying F(x*) = x*F'(x*) generates a unique local branch whose leading displacement is −(3/2)x*(q−1). Degenerate zeros, for which F''(x*) = 0, require higher-order analysis. An explicit example exhibits complex conjugate leading branches, illustrating that degeneracy does not guarantee a real local branch. A numerical verification for a regular example and an exact homogeneous-family calculation are included as complementary evidence. The results concern the local mathematical structure of the operator and do not constitute a general classification of all of its zeros or a derivation of external physical theories.