The involution kernel and the dual potential for functions in the Walters’ family

The paper (within the Walters family R(Ω)) proves that the series-defined involution kernel exists if and only if the potential has Walters regularity, via explicit case-by-case calculations and known characterizations of Walters regularity in that class. It also shows that convergence of the series implies Walters regularity. By contrast, the model’s (⇒) direction hinges on a uniform Cauchy bound that misuses variation estimates var_{n+k}(S_n A) and incorrectly concludes that the limit kernel W is independent of x. It also assumes continuity from pointwise convergence without proof. These errors conflict with the paper’s formulas where W depends on both coordinates and with the correct logic linking series convergence and regularity.