In this paper we disprove a conjecture by Budaghyan and Pal (DCC, 2024) on the existence of an infinite family of APN permutations of the form \(F(x)=x^{(q+3)/2}+ux^2\) over \(\mathbb {F}_q\) . Using function field theory and Hasse-Weil bounds, we prove that for \(q \ge 2719\) and \(u \notin \{0,\pm 1\}\) , the function F is not APN. Combined with computational verification for \(125< q < 2719\) , this establishes that no such infinite family exists beyond the finitely many known small-field examples. Moreover, we extend this negative result to related families including \(F(x)=x^{(p^n-1)/2+3}+ux^3\) (for \(q \equiv 1 \pmod {3}\) ) and \(F(x)=x^{(p^n-1)/2+p^k+1}+ux^{p^k+1}\) , showing these “patched monomial” constructions \((\eta (x)+u)x^d\) systematically fail to produce infinite APN families as field size grows. Our methods combine algebraic geometry (Kummer extensions, genus calculations), character sum analysis, and computational techniques. We also investigate the permutation properties and value distributions of these functions, proving general non-permutation criteria and establishing differential uniformity bounds. This work demonstrates that while these constructions yield APN functions for small finite fields, they are fundamentally unsuitable for generating infinite APN families.
