Inverse Multiwise Davenport Problems in Rank Three

Let \(_k(G)\) be the least length forcing \(k\) pairwise disjoint nonempty zero-sum subsequences in a finite abelian group \(G\). We study the inverse problem for zero-sum sequences of length \(_k(G)\) whose maximum factorization length is at most \(k\). A group-independent reduction shows that the equality \(_k(G)-ₖ₋₁(G)=(G)\) forces every extremal sequence to have zero-sum girth at least \((G)\); if \(_k(G)<(k+1)(G)\), this condition is also sufficient. For every fixed group, the stabilized inverse family is Presburger-definable, hence semilinear, and its labeled counting function is eventually quasi-polynomial. Combining the reduction with Luo's exact short-zero-sum theorem yields an exact membership criterion for every rank-three \(p\)-group \(Cp^a⊕ Cp^b⊕ Cp^c\) with \(a≤ b<c\). We then give complete classifications for \(C_2^3\) and \(C_2^2⊕ C_4\). In the first case all higher extremals are obtained from the seven-point Fano sequence by adjoining square pairs. In the second, extremality is equivalent to a signed-cap condition and admits a unique finite-core decomposition; an exact exhaustive census gives 928 labeled cores, ten automorphism orbits, and the formula \(e_k=112/3k(k+1)(2k+1)\). Explicit examples in \(C_3^3\) delimit the stable-increment converse and show that binary block stripping has no naive odd-prime analogue. Deterministic verification code accompanies all finite enumerations. The factorization number is one-Lipschitz in edit distance, metrically separating all strata and making every inverse classification robust.