Higher-power inverse functional identities and Frobenius collision obstructions

Let $D$ be a division ring, let $n\geq 2$, and let $f,g:D\to D$ be additive maps satisfying $$f(x)x^{-1}+x^n g(x^{-1})=0\qquad (x\in D^\times).$$ The paper studies when this inverse functional identity forces $f=g=0$, and where the natural vanishing statement fails. The main obstruction in positive characteristic comes from Frobenius homogeneity. We first give an exact finite-field classification. If $q=p^m$, then the $\mathbb{F}_q$-dimension of the solution space is the number of ordered pairs $(i,j)$, $0\leq i,j\leq m-1$, such that $$p^i+p^j\equiv n+1\pmod {q-1}.$$ We next prove vanishing theorems for arbitrary division rings using prime-field and central-subfield scaling. In characteristic zero the identity always forces $f=g=0$. In positive characteristic, the same method gives vanishing whenever the relevant central weights are incompatible. Finally, we examine characteristic two. The characteristic-two case is not completely resolved here. We prove a reduction theorem showing that a nonzero value $f(1)=g(1)$ produces a generalized polynomial identity except for one explicit central Frobenius-degenerate possibility. In particular, for non-centrally-finite division rings, all characteristic-two solutions reduce to a one-map identity with $h(1)=0$, apart from the exceptional case where $f(1)=g(1)$ is central and $n+1$ is a power of two. For $n=2$, this exceptional case is absent, and the reduction holds over every noncommutative division ring of characteristic two.