Transversal Difference Numbers in Finite Abelian Quotients

Given $H\leq G$ finite abelian groups, a transversal $T\subseteq G$ for $G/H$ has fixed size $|G/H|$, but its ambient difference support $D(T)=T-T$ can vary with the embedding of $H$ in $G$. We call $\delta(G,H)=\min_T |D(T)|$ the transversal difference number of the pair $(G,H)$. This invariant is related to finite abelian factorisation, tiling complements, and small-sumset questions, and is motivated by recent work regarding ambient Galois labels in CRT transforms for cyclotomic-subfield homomorphic encryption. We prove various results regarding this invariant, including a general lower bound $\delta(G,H)\geq 2|G/H|-m(G,H),$ where $m(G,H)$ is the largest order of a subgroup of $G$ disjoint from $H$. The bound is sharp for cyclic quotients, and Kneser's theorem gives a cross-transversal estimate leading to exact product families with one nonsplit cyclic coordinate and arbitrary split factors. These results isolate the first genuinely new residual obstruction, namely the same-prime square plane $$G=(\mathbb Z/p^2\mathbb Z)^2,\qquad H=pG.$$ For odd $p$, this case is the technical core of the paper. Here transversals are graphs of functions $\mathbb F_p^2\to \mathbb F_p^2$, and $D(T)$ decomposes into carry-corrected finite-field derivative images. We conjecture that $$\delta(G,H)=(2p-1)^2$$ for all odd primes $p$, prove the unconditional lower bound $3p^2-p-1$, and give small-prime, probabilistic, and fixed-polynomial evidence for the conjecture.