Exact thresholds for Schur positivity of the lattices m×2 and m×3

We determine the exact thresholds for Schur positivity in the two families of chain products m×2 and m×3: the former is Schur positive exactly for m≤7, and the latter exactly for m≤6. For m≥8, we prove non-Schur-positivity in both families by exhibiting explicit negative Schur coefficients obtained from Pieri’s rules and stable-composition counts. The remaining finite cases are settled by exact SageMath computations; in particular, 7×3 has a negative Schur coefficient. These results settle the n=2 and n=3 cases in the conjectural picture of Li, Qiu, Yang, and Zhang and sharpen the n=3 boundary by one. We also show that m×3 is not strongly nice for m≥44.