Joint Ergodicity of Hardy Field Sequences

The paper proves the joint ergodicity theorem for Hardy-field sequences under the stated ‘logarithmic separation from CZ[t]’ hypothesis and identifies the L2-limit as the product of integrals. The model’s write-up cites Tsinas (2021) itself as a black-box theorem to conclude exactly the statement that the paper is proving, without providing an independent argument or reduction; this is circular. The paper’s proof is complete and coherent, relying on Host–Kra factors, reduction to seminorm estimates plus equidistribution, and careful polynomial-approximation arguments, whereas the model’s solution does not engage with these steps.