Orbit Lattice Formulas for Second Homology of Finite Abelian Quandles

Let \(Q\) be a finite quandle whose right translations commute, with \(r\) orbits and stabilizer lattices \(H_i\) in \(L=\mathbb Z^r\). We give an explicit presentation \[T_i=\bigwedge^2 L\big/\left(\sum_j e_j\wedge H_j+\bigwedge^2 H_i\right)\] for the torsion summand of each orbit in second integral rack and quandle homology. The free ranks are \(r^2\) and \(r(r-1)\). Each orbit size annihilates its summand, and the bound is sharp; a twenty-element example disproves the stronger orbit-exponent bound. Smith normal form computes the groups. The universal coefficient theorem gives second cohomology for every constant abelian coefficient group, with integral second cohomology torsion-free. This makes explicit consequences of the credited methods of Darné, Eisermann and Lebed–Mortier. AI-assisted, self-audited and unrefereed; no independent review, formal verification or absolute priority claim.