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

Abstract: 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).