Maximal Operator-Schmidt Rank in a Two-Fold Abelian Wreath-Product Permutation Module

Let A be a finite abelian group, B=A×A, and W=B⋊⟨τ⟩=A wr S₂, where τ swaps the two factors. For an arbitrary subgroup S≤B, consider the induced permutation module C[W/S] with its natural sheet/base tensor factorization C²⊗C[B/S]. We derive a closed formula for the maximum operator-Schmidt rank attained by the image of the complex group algebra C[W]. The formula depends only on three orbit statistics (d,p,u) describing how the annihilator Q=S⊥ intersects the fixed points and two-cycles of the swap involution. The non-swap-invariant case introduces half-orbits, encoded by u. We also prove that the same maximum is attained by unitary elements of the image algebra. This yields, in Nielsen's exact one-use coherent communication model, a symmetry-determined Hartley lower bound on the communication cost of unitaries in the class. The magnitude is necessarily at most two qubits for the present C² sheet factor; the point is the exact symmetry-forced value, not a large asymptotic bound. Supplementary files: bilingual (English/French) README and two Python verification scripts covering 131 exhaustive subgroup cases for the theorem and 390 for the unitary corollary.