Square-to-Cube Lindelöfness and One-Sided Obstructions in Hattori Spaces

Let H(A) be the Hattori space associated with A ⊆ ℝ, and put B = ℝ \ 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. H(A)² is Lindelöf ⇒ H(A)³ is Lindelöf. The proof starts from an uncountable closed discrete subset of the cube, passes to a perfect set with injective coordinate projections, and uses the one-sided local bases to obtain three cocountable Pareto-maximality sets with countable triple intersection. The same one-sided geometry gives a pointwise half-space criterion for affine discreteness. If A ≠ ∅, no positive-dimensional affine subspace is discrete in H(A)^m for m ≤ 3. Under the square hypothesis, every uncountable closed discrete monotone trace has at least two increasing and two decreasing coordinates. Known hereditary-Lindelöf results control the mixed Sorgenfrey–Euclidean strata. Combining these results with closed-discrete arguments gives hereditary-Lindelöf criteria for finite powers, a countable-power characterization, and a finite-power Lindelöf reduction. For n ∈ ℕ, the power H(A)ⁿ is hereditarily Lindelöf if and only if the Sorgenfrey power Bⁿ is Lindelöf. Also, H(A)^ω is hereditarily Lindelöf if and only if every finite Sorgenfrey power of B is Lindelöf. For the ZFC example constructed in an earlier preprint, the square-to-cube theorem gives Lindelöfness of the third Hattori power, whereas the earlier construction shows that the fourth Hattori power is not Lindelöf. For the same example, H(A)² and H(A)³ are Lindelöf but not hereditarily Lindelöf.