Packing topological entropy for amenable group actions

The paper proves the variational principle for amenable packing topological entropy by (i) a 5r-lemma-based argument showing h_Ptop(Z,{F_n}) ≥ sup_μ h^upper_loc_μ(Z,{F_n}) and (ii) a Joyce–Preiss/Feng–Huang style measure construction showing that for any s < h_Ptop(Z,{F_n}) there is a measure μ with h^upper_loc_μ(Z,{F_n}) ≥ s, hence sup_μ h^upper_loc_μ(Z,{F_n}) ≥ h_Ptop(Z,{F_n}). The candidate’s “lower bound” mirrors (ii), but their “upper bound” is flawed: it attempts to cover a μ-null leftover set N by countably many sets with zero packing pre-measure, a claim that is unjustified and generally false. Thus the model’s proof has a critical gap, while the paper’s proof is correct.