From Wikipedia entry on Hensel's Lemma (paraphrased):
if ${\displaystyle f(r)\equiv 0{\bmod {p^{k}}}\quad {\text{and}}\quad f'(r)\not \equiv 0{\bmod {p}}}$,
then there exists an integer s such that${\displaystyle f(s)\equiv 0{\bmod {p^{k+m}}}\quad {\text{and}}\quad r\equiv s{\bmod {p^{k}}}.}$
Furthermore, this s is UNIQUE ${\bmod {p^{k+m}}}$.
Ultimate question: What does UNIQUE mean here?
I keep finding seemingly contradictory examples:
${\text{Find all x satisfying: } x^2-1 \equiv 0 \bmod 4}$
We find only 1 works (mod 2), and all requirements for the lemma are met: ${f(1) \equiv 0 \bmod 2} \quad and \quad f'(1)\not \equiv 0 \bmod 2$
However, against my expectation, the next iteration doesn't give a UNIQUE result mod 4, because the next binary digit may be either 1 or 0, resulting in multiple solutions, specifically it has a mod 4 solution set of {1,3}. It seems to be NOT-UNIQUE for that reason. In what sense is the result UNIQUE?
NOTE: I'm aware {1,3} = {1,-1} because ($3 \equiv -1\bmod4$).