Square-to-Cube Lindelöfness and Power Separations in Hattori Spaces

Let $H(A)$ be the Hattori space associated with $A\subseteq \mathbb{R}$, and put $B=\mathbb{R}\setminus A$. Kulesza announced at the 2018 Joint Mathematics Meetings that Lindelöfness of $H(A{)}^{2n}$ forces Lindelöfness of $H(A{)}^{2n+1}$ for every positive integer $n$. The meeting abstract states this implication without a proof. This paper proves the case $n=1$ in ZFC. The same one-sided geometry gives a pointwise half-space criterion for affine discreteness and, when $H(A{)}^2$ is Lindelöf, a four-coordinate obstruction for uncountable closed discrete monotone traces. Known results on mixed Sorgenfrey–Euclidean products lead to hereditary-Lindelöf criteria for finite powers, a countable-power characterization, and a finite-power Lindelöf reduction. A transfinite construction gives $A\subseteq \mathbb{R}$ for which $H(A{)}^2$ and $H(A{)}^3$ are Lindelöf, whereas $H(A{)}^4$ contains a closed discrete affine line of cardinality $\mathfrak{c}$.