Orbit Lattice Formulas for Second Homology of Finite Abelian Quandles

Let Q be a finite quandle whose right translations commute, with r orbits. Its orbit action determines finite-index lattices H_i in L = Z^r, each containing e_i. We prove that the ith orbit torsion summand in second integral rack and normalized quandle homology is wedge^2 L / (sum_j e_j wedge H_j + wedge^2 H_i). The free ranks are r^2 and r(r-1), respectively. Each orbit size annihilates its torsion summand, and the bound is sharp. A twenty-element example shows that the exponent of the orbit group need not annihilate that summand. Smith matrices compute the invariant factors, and the universal coefficient theorem gives second cohomology with every constant abelian coefficient group. In particular, integral second cohomology is torsion-free. This explicitly derives consequences of the credited structure-group and covering methods of Darné, Eisermann, and Lebed--Mortier. It addresses OWR-17294-007 while preserving the source coefficient ambiguity. It does not cover arbitrary medial quandles, twisted or nonabelian coefficients, or higher homology. Six-page English preprint, AI-assisted, self-audited and unrefereed. Exact checks on 52 finite examples agree with direct boundary-matrix computations; these are not a formal verification of the general theorem. No independent expert review, exhaustive novelty certification or absolute priority is claimed. PDF, portable LaTeX source, executable checks and verification report are included.