Inverse Falconer Distance Theorems over the Integer Residue Rings ${\mathbb{Z}}_n$

We establish an ideal-theoretic rigidity principle for quadratic distance images over integer residue rings. Specifically, we prove that near-extremal collapse of the distance set in ${\mathbb{Z}}_n^d$ forces strong algebraic structure supported on annihilator submodules arising from the arithmetic of $n$. As a consequence, we obtain the first inverse theorem for the Falconer distance problem over ${\mathbb{Z}}_n$ for composite moduli. We show that if a set $E\subset {\mathbb{Z}}_n^d$ of size $\mathrm{\mid }E\mathrm{\mid }\asymp n^{(d+1)\mathrm{/}2}$ determines only $O(n)$ distinct squared distances, then $E$ must be supported on a coset of an annihilator submodule on which the distance form is algebraically degenerate. The proof introduces a divisor-depth decomposition intrinsic to ${\mathbb{Z}}_n$, together with a lifting mechanism that transfers local degeneracies at prime moduli into global ideal-theoretic constraints. This yields a complete classification of near-extremizers for the Falconer distance problem in the ring setting, revealing a rigidity phenomenon with no analogue over fields.